Looking For Another Relative Engel's Law François Gardes University of Paris I Panthéon Sorbonne, Paris School of Economics, Crest1 April 2007

Abstract 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. Lastly, the article demonstrates the existence of a new Engel's law relating to the social positioning of the agents within their reference groups. 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. An application to US and Polish panel data confirms this relative version of the Engel law and the existence of Duesenberry’s demonstration effect. This relative Engel’s law is finally discussed in a domestic production scheme which helps to explain the differences in food and services expenditures patterns between the U.S. and Europe.

Introduction Engel's law (discussed in detail and with subtlety by C. Berthomieu, 1965) 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, as Stigler has remarked (1954), 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. Such imprecision indicates clearly the lack of theoretical foundations for the law, as it has been stated simply over the last two centuries, in terms of a hierarchy of needs2. It is a hierarchy which nothing explains.

1

Centre d’Economie de la Sorbonne, 106-112 Boulevard de l’Hôpital, 75647, Paris cedex 13, France. Special thanks are due to Greg Duncan, Patrice Gaubert, B. Górescki, and Christophe Starzec for making available the data for the U.S. and Poland. I fully acknowledge comments and suggestions from Franck Arnaud, Andrew Clark, Marc Diaye, David Margolis, Philippe Merrigan, Claude Montmarquette and participants to seminars in Cirano (Montréal), Crest-Ensai, Charles Gide Colloquium, Journées de Microéconomie Appliquée, Universities of Caen, Cergy, Lausanne and Paris. Section 2 was written while I was invited at the University of Montréal. 2

“A reader who dislikes the term `law' because of the belief that there are no laws in the social sciences can use the term `monotonicity' instead”, W. Hildenbrand, 1994, p.3. To be sure, the translation of a hypothesis in terms of parameters with no dimension, following on from the relationship between two economic variables, makes such a hypothesis similar to the classical concept of a law in natural science. Engel puts forward his law in his original research (Berthomieu, p. 81): “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”. Stigler explains

This permanent relationship is generally considered as a necessary and self-evident relationship between the scarcity of goods (following on from the budget constraint and the price mechanism), and needs as felt by consumers. From this point of view, the law is an expression of rational consumer behaviour in the face of scarcity. The law is nowhere presented as stemming from agents' endogenous needs and hence their preferences, whereby needs perceived by households change with income. Engel's law is usually defined as a purely empirical proposition, linked to a time and place. It is seen as merely translating ranked values of income elasticity for various consumption functions. As this automatically implies that there is a dependency between these elasticities and the population considered, it would be more appropriate to talk of Engel's hypothesis. The latter would consist of a qualitatively stable hierarchy of income elasticities between the dominant consumption functions. This bears out the dependence of the law on a twofold choice in the classification of goods and the definition of social classes. (Social classes were defined a priori in the first publication in 1857, as working class families in need, low income families receiving no state support, comfortably-off families. Engel subsequently drew on Carrol Wright's classification of income categories (1875), which he used in his definitive paper, in 1895, see Stigler, 1965, pp. 203-206). In my view, it is not this second interpretation of the law which explains its widespread acceptance (were this the case, it would be possible to speak with as much conviction about the laws relating to the dependency of different types of investments on interest rates, of the dependency of output on the price of inputs etc.). The law's acceptance seems to stem from the simplicity of a bivariate relationship between income and consumer spending; its easy interpretation in terms of ranked needs; the ease with which the proposed ranking of income elasticities may be tested (comparing average budget shares between income classes, different periods or different nations; the fact that the law allows consumer spending to be predicted on the basis of income growth. Indeed, the translation of an empirical hypothesis into a law no doubt explains its permanency in textbooks and common economic literature. This may be criticised, even if the law's mathematical formalisation as an elasticity relating income and consumption allows consumer choices to be sorted out from the point of view of economic analysis. Several questions follow from these observations, beginning with how Engel's observation was structured as a law. Section 1 discusses the origins of Engels’s law and presents some problems in its estimation. Section 2 describes a model of social interactions giving rise to a relative version of Engel’s law which is tested on an two panels of households expenditures. Section 3 presents a relative version of Engel’s law as the consequence of the characteristics of the households domestic production functions. Section 1. The origins of Engel's law The law may generally be translated by the following hypotheses: i.

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

this difference by the fact that household incomes were not considered, in the 19th century, as a variable whose fluctuations were important in the short term. This justified ignoring the consequences of such minor variations.

first place, in the link between individual and household spending, using scales of equivalence). ii.

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 made on the basis of individual budget data. Engel was surprised by the fact that Ducpetiaux and Le Play “restricted themselves to a simple presentation of numerical data”. Engel himself appears just to have established the data with the aim of estimating the structure of consumption in the Kingdom of Saxony, only on the basis of Ducpetiaux and Le Play's consumption data, and not with respect to their theoretical interest. Nevertheless, he felt the data to be sufficiently stable to carry out a statistical reconstruction exercise, which no theory really supported. The hypotheses were quickly accepted as natural by economists at the end of the 19th century. Charles Gide, for example, in his Course on Political Economy (1919, 4th edition, p. 476-8), discusses them in detail, adding expenditures on social solidarity (union contributions, taxes etc.). But he also notes that “Engel's laws”, which were in fact Wright's, were partly contradicted in fact (the greater-than-unity elasticity of clothing), and that trends in spending according to living standards were not necessarily the same for all social classes (Gide's reasoning is a priori, and he justifies his comments by observing the life of couples). Gide also notes there is an even more complex issue, namely the link between present and future needs, which raises the question of the endogeneity of needs. Nevertheless, this does not seem to be interesting from a theoretical point of view (Pareto, for example, does not mention the issue in his textbook). It is odd that the relationship between consumption and income, which today seems so natural, was of such interest to economists at the end of the 19th century, and that it was promoted as a law, comparable if not the same, as laws in natural science. This is all the more surprising, as Stigler has pointed out, given that the relationship was not linked to economic theory at the time, as were, for example, the relationships between consumption and prices. Income-effects theory was indeed only developed 70 years later by Slutsky and Hicks, whereas the price-effects theory had been put forward 40 years earlier, prior to the first, systematic, empirical analyses. Discussing the law It may be noted that Engel extends his relationship between food consumption and social position (viewed as corresponding to the standard of living) from the individual to the whole of the population. This makes its status uncertain, raising the issues about which process may be used to aggregate individual behaviour permanently for groups of individuals, and whether or not cross-section and time variations may be held to be the same. Three questions may be asked regarding the law: 1.

Is the statistical relationship between spending and income an explanation of the nature of a population's consumer needs or does it also depend on supply conditions and procedures for aggregating the two variables to be compared? In the first case, 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 (which would make the demand equations endogenous)?

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 (as suggested by P.C. Pradier). In this case, the law could predict the evolution of consumption. In the second case, the law would correspond to the usual tests, on cross-section data, of hypotheses relating to ranked needs. 3.

Should changes in living standards which affect the whole of the 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?

The endogeneity of needs To address these vast issues, I draw on two articles (Gardes-Loisy, 1995; GardesMerrigan, 2006) 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 function variables, which rules out considering optimisation programmes based on constant preferences. 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 a bias in endogeneity in at least one of the two estimations. Such a bias has been proved to exist in American (PSID) and Polish panel suveys (Gardes et al., 2005): for example, the income elasticities of consumer spending on food are in the order of 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. An indicator for this lies in calculating the price differences households face as a function of their socio-economic status: if food costs more for rich households compared to poor ones, given that the quantity consumed is unchanged, then it may be assumed that costs for the former include higher quality. A way of calculating virtual consumer prices and income elasticities of these prices is presented in Gardes et al. (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. It thus appears that Engel's law holds both for cross-section changes as well as changes over time, though to a lesser extent in the second case. The differentiation of the law depending on whether different households or the same household over time are considered, allows the two Engel laws to be integrated: the law specific to an income elasticity of around

unity and a law which assumes a variation in the quality of food consumption according to populations' standards of living. Section 2. An absolute or a relative Engel's law? The following analysis puts forward a way for estimating the effect of a change in status within a household's reference class, without these social classes being defined in a specific manner, a priori. This is therefore a semi-parametric test for the effects of social interactions within the reference populations. The analysis proves that Engel's law covers changes in relative household income within the households' social classes. If two households have same income per unit of consumption, but one belongs to a poorer class (in which it is relatively rich), while the other belongs to a richer class (in which it is relatively poor), the latter will have a significantly higher coefficient for its food budget, even though it is identical to the former in terms of all explanatory variables, including food. This relative version of Engel's law demonstrates its universality and may allow statements of the law in terms of social class and income to be better distinguished. The original Engel law for food consumption relates it to the household income variations, its elasticity being supposed to be positive but smaller than one. Empirical test of this law are generally based on cross-section data, but references to the law also discuss the changes of aggregate time-series (the budget share for food increasing while the aggregate income of the nation increases3). In contrast, when the food income elasticity is obtained comparing social classes (as in the first paper by Engel), it is not evident that it measures only income effects, as social interaction may be present in the cross-section income elasticity (comparing households in different social classes) by means of the relative position of the household in the income distribution. The schedule presented in section 3 allows to separate absolute and relative income effects on food consumption. The relative income hypothesis proposed by Duesenberry (1948) has been fairly neglected in the empirical literature, perhaps because it was primarily used for the macroeconomic consumption analysis, and was finally replaced by the Permanent Income Hypothesis. Some recent articles take up this hypothesis, explaining its theoretical foundations (BagwellBernheim, 1996; Dybvig, 1995). In recent research (Clark-Oswald, 1996; Kapteyn et al., 1997), this type of interdependence of preferences is generally proved to be consistent with the empirical evidence (though serious interpretation problems arise, as shown by Manski in his discussion of the reflection problem, Manski, 1995). It deserves more thorough appreciation from the micro point of view, for it builds a bridge, as a Veblen effect, between the sociological and economic explanations of household behavior. Parametric tests of the relative income model can be criticized because of the assumptions necessary to define relative income and specify consumption functions. For instance, Duesenberry, in his macro applications, defines relative income as the largest past income observed by the households, while, in his micro analysis, relative income is measured as the relative position of households within a given population. There is no doubt that the first definition is quite remote from the original micro concept, while the second, sociological, definition requires defining the reference population with objective characteristics (if no 3

This pattern is also discussed by Engel in his second contribution, but he does not distinguish it from the crosssection. It seems that Engel interpreted his law as time variations, cross-section estimation being a second best in the absence of panel data.

subjective information from the household is available), and specifying the relative income effect on consumption. Section 2.1 discusses the testing of a relative version of the Engel law and relative income effects. Section 2.2 describes the tests and Section 2.3 applies it to the U.S. Panel Study of Income Dynamics and to a Polish panel. 2.1. Framework Suppose the relative income position of the household within its reference population can be measured by (yh-myh), with yh as the income of household h (per consumption unit) and myh the average income in the reference population. A relative income effect can be revealed by relating this relative income position to the similar residual of household consumption, over the average consumption of the reference population: ch-mch.

Table 1. Correlation between Food Expenditures and Relative Income

Survey

1987

1988

1989

1990

my

0.496

0.495

0.444

0.668

0.526

(0.036)

(0.039)

(0.034)

(0.039)

(0.019)

0.533

0.499

0.443

0.551

0.506

(0.009)

(0.009)

(0.009)

(0.009)

(0.005)

ys

average

Data-set: Polish Panel; 52 reference populations have been defined according to five cohorts, three education levels of the household head, and four quarters (some populations have been grouped to obtain a sufficient size). Logarithmic specification with average total expenditure for the reference population: my and the residual total expenditure for household h: ysh=yh-my. Control variables: proportion of children, log of head’s age, location, education level, relative prices, 16 quarter dummies as proxies of macroeconomic shocks (similar results for linear AI Demand System specification).

Such an estimation is presented in Table 1, for the Polish surveys. The elasticity between residual income and consumption is not systematically smaller than the elasticity as concerns average income, which contradicts the observation that (relatively) poor households may spend more money for food than the (relatively) rich. In addition, the income elasticities as concerns the average income of the reference population and the residual income are not significantly different. Moreover, this correlation does not truly indicate a relative income effect because residual consumption may be influenced by all the determinants of individual consumption, which are not used as criteria to define the reference populations or as explanatory variables, because they are absent in the data-set. These latent variables may be also correlated to the endogenous household income, thus creating an indirect relationship between consumption and income. Thus, it seems necessary to take into account, by panel estimation, the specific component of consumption which cannot be explained by the

explanatory variables present in the data-set, but which persists over time and which characterizes the household consumption. Indeed, when estimating a consumption function on panel data, the specific consumption of a household h can be identified as the permanent component of the residual. The correlation between the permanent part of the individual error and the relative income position of the household within its reference population indicates the presence of Duesenberry effects. However, it is well known that the computation of such an individual effect is difficult. Thus, this study proposes to estimate directly the endogeneity bias in panel data due to the relative income effect. For grouped data, the remaining correlation between the specific effect and income variables can no longer represent Duesenberry effects since all relative positions within reference populations have been cancelled by aggregation. Consequently, the difference between the estimation on individual data and the estimation on grouped data is that part of the correlation which disappears when the data is grouped into cells corresponding to the reference populations. This estimation procedure clearly shows the role of aggregation in reducing the endogeneity bias, as well as the statistical assumptions needed to estimate the relative income effect. Moreover, the procedure gives an indicator of the importance of the Duesenberry effect (for instance, the ratio between the bias obtained on individual or grouped data) which may be used to compare different groupings and to define the reference populations (the social groups or vignettes to which the individuals compare themselves) and the relative impact of each household’s consumption on this definition.

2.2. Theory Consider a model of consumption for individual h at time t: zht = Xht β + uht with uht = αh + εht.

( 1)

When estimating with panel data, Mundlak (1978) shows that between estimates are biased if the specific effects αh are correlated with the explanatory variables: E(αh|Xht) ≠ 0. The specific effect αh may be related to the between form of the explanatory variables, such that: αh = BXhtπ+ ζh with ζh ∼N(0,σζ2), i.i.d., E(ζh,BXht) =0.

(2)

Thus, the model can be written in between form (i.e. for the average over periods): Bzht = BXht β + BXht π +ζh+Bεht

(3)

which implies for the between estimates of Bzht over BXht: E(βb|X) = E[(Xht’BXht)-1 Xht’Bzht] = β + π while the within estimate (computed on the difference between equation (1) and equation (3)) is unbiased: E(βw|X) = β as the within operator suppresses all specific effects, and therefore the endogeneity biases caused by the correlation between these effects and the explanatory variables. Such endogeneity biases are shown to exist in the estimation of consumption functions for at least half of the commodities (Gardes et al., 1996, 2005).

Aggregating data can be operated to correct for measurement errors, or to build pseudo-panel data in order to estimate dynamic models when only separate surveys are available. Such data grouping may change the endogeneity biases, but in a way which is difficult to predict by considering directly the aggregation of (1), (2), (3). In order to analyze how aggregation affects the endogeneity bias, the individual effect αh is split here into the collective specific effect (common to all individuals in the sub-population) and the residual effect, specific to the individual. Suppose that the estimation is performed on a population H of individuals h = 1 to N, surveyed within the whole population (H ⊂ ). Sub-populations are defined by crossing characteristics kj, j=1 to J such that: i = {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 i ∩ H4. 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,σ2yi) and αh ∼ N(µi,σ2αi), i.i.d., with yi = E(yh|h∈Hi ), µi = E(αh|h∈Hi) < ∞. The average yi in Hi is computed by regressing yh on the vector of characteristics K: yHi = Kai + ξi so that yHi = 1/ni (Σh∈Hi yh ) with ni the number of individuals in Hi. So the distribution of the mean is: yHi ∼ N(yi,σ2yi/ni). The specific income (which may be considered as the relative income5 of individual h in its reference population i ) is defined as ysh = yh - yHi so that ysh ∼ N(0, σ2yi - σ2yi/ni). By the same reasoning, µHi = 1/ni (Σh∈Hi αh ) and µHi ∼ N(µi,σ2αi /ni). Consider now the decomposition of the specific effect into the specific effect of the reference population Hi and an individual effect: αh = µHi+νh. We obtain the distribution for ν as: ν h ∼ N(0,σ2µi(1-1/ni)). The covariance on individual data between αh 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).( αh-Eα)} = E{ [(yHi-yi) + (yi-y) + ysh].[(µHi-µi) + µi + νh]} 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 = π = (βb - βw)panel = p (βb -βw) grouped data +(1-p) γ V(y) 4

It is assumed that this grouping, according to a priori exogenous criteria (age and education), is exogenous to household consumption. 5 Note that ysh corresponds to the log-ratio of a household’s income and average income yHi, if incomes are defined in logarithms.

V(yHi) and γ is the coefficient resulting from the correlation between the specific V(y) effect νh of household h and its specific (relative) income.6 Thus, this coefficient γ and its standard error can be computed in terms of the difference between the estimates of β on individual and grouped data in the between and within dimensions:

where p =

γ (νh/ysh) =

1 1− p

{(βb - βw)panel - p.(βb - βw)grouped data}

(4)

2.3. Empirical application Since 1968, the Panel Study of Income Dynamics has followed and interviewed annually a national sample that began with about 5,000 U.S. families. Only four years (19841987) are used here in the estimation of the consumption equation which is comparable with the Polish data. In all cases the data are restricted to households in which the head did not change over the six-year period, and to households with major imputations on neither food expenditure nor income variables (in terms of the PSID’s “Accuracy” imputation flags, we excluded cases with codes of 2 for income measures and 1 or 2 for food “at home” and food “away from home” measures). In order to construct cohorts for the pseudo-panels, a series of variables was defined, based on the age and education levels of the household head. Specifically, this study defines : (i) 6 cohorts of age of household head (under 30 years old, 30-39, 40-49, 50-59, 60-69, and over 69 years old); and (ii) three levels of the household head education (uncompleted high school (12 grades), completed high school but with no additional academic training, at least some university-level schooling completed). The population is randomly divided into four sub-samples, each of which is used to aggregate data for the different years. This prevents the same household from being included in the same cell for more than one period. The PSID cell size varies from 9 to 183 households with a mean of 65.5 (see Gardes et al., 2002, for details). The annual Polish expenditure surveys contain about thirty thousand households, which represent approximately 0.3% of all households in Poland. For every annual subsample between 1987 and 1990, it was possible to identify 3707 households participating in the surveys during all four years and interviewed in the same quarter each year, about their expenditures, income and various socio-economic variables. The period covered by the Polish panel is unusual even in Polish economic history. It represents the shift from a centrallyplanned, rationed economy (1987), to a relatively unconstrained, fully-liberalized market economy (1990). Real GDP grew by 4.1% between 1987 and 1988, but fell by 0.2% between 1988 and 1989, and by 11.6% between 1989 and 1990. Price increases across these pairs of years were 60.2%, 251.1% and 585.7%, respectively. Thus, the transitory years 1988 and 1989 produced a period of a very high inflation and a mixture of a free-market, a shadow and an administrated economy (see Gardes et al., 2002, for a presentation of the data-set and the estimation procedures, and Lednicki, 1982, Górecki, 1992, for a full description of the master sample generating procedure). The pseudo-panel is built for the whole surveys from 1987 to 1990. It contains 224 cells with 107 households per cell on average. 6

For p=1, each cell contains only one household, so that the panel and the pseudo-panel coincide. For p=0, all the population is grouped into one cell, and ys is the difference between y and its average for the whole population, so that γ just indicates the endogeneity bias estimated on the panel.

Identification for the estimation on Between transformed data needs more cells than the number of regressors: 18 cells are used for the PSID and 224 for the Polish panel (for four years), compared to 9 and 35 cells for the explanatory variables. The precision of the estimators depends on the number of cells, but the errors of measurement (due to the fact that the cells contain different households for two periods) decrease with cell size, so that a tradeoff exists between numerous cells in the pseudo-panel, but with possible errors of measurement, and a small number of large cells without much error of measurement7. A priori, the estimators may be more efficient in the Polish case, with a great number of numerous cells. The specification uses the linear Almost Ideal Demand System for the PSID. For the Polish panel, prices for four professional groups and each quarter and year are available, so that it is possible to estimate the price elasticities and the integration coefficient8 of the quadratic system (see Banks et al., 1995, for the estimation method by convergence on the integration coefficient).

7 8

Also, the use of limits in probability in the proof (Appendix I) requires numerous cells. Which indicates the ability to integrate the demand system, see Gardes-Garrouste (2007).

Table 2. Income Elasticities and Relative Income Effects

Individual dataa

Between Within

Pseudo- Between Panel datab Within (βB - βw) : Ind. Data Pseudo-Panel V(yHi) V(y) Budget Share p=

γ

Food at Home

Food Away

PSID

Polish Panel

PSID

0.494

0.186

1.050

(0.012)

(0.027)

(0.142

0.755

0.507

0.443

(0.012)

(0.112)

(0.205)

0.452

0.311

1.387

(0.022)

(0.045)

(0.068)

0.542

0.240

0.800

(0.023)

(0.095)

(0.150)

-0.261

-0.321

0.607

(288.10-6)

(0.0132)

(0.0438)

-0.090

0.071

(0.00101)

(0.0111)

(0.0271)

0.587

0.3294

0.2731

0.2731

0.508

0.138

0.0328

-0.1753

-0.0646

0.0202

(0.0151)

(0.0097)

(0.0137)

Student for γ

11.60

2.09

1.36

Relative income Elasticity

0.655

0.532

1.616

(0.030)

(0.070)

(0.418)

a

panel data (3630 households for the Polish panel, 2430 households for the PSID). Polish Pseudo-panel: 224 cells according to six cohorts, three education levels of the head, location and four quarters (some cells are grouped to obtain a sufficient cell size). PSID Pseudo-panel: 18 cells according to six cohorts and three education levels of the head. Specifications: PSID: Almost Ideal Demand System specification with instrumented income and 8 control variables: equivalence scale and its square, log of head’s age and its square, 4 survey dummies as proxies of relative prices and macroeconomic shocks. Polish surveys: QAIDS specification with instrumented total expenditure and its square and 33 control variables: proportion of children, log of head’s age, location, education level, relative prices, 16 quarter dummies as proxies of macroeconomic shocks; estimation by convergence on the integration parameter. b

Results: (i)

For food away, γν/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.

(ii)

γν/ys are negative and significant for food at home, both in US and Poland. It indicates a negative Duesenberry effect on food 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, as a share of its income, than a relatively rich household-belonging to another reference population P1-which has the same total income Y0 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.

Y0

Figure 1. Income distribution for Population P1 and P2 (iii)

For food away, γν/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

(iv) 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. (v)

The estimation of the relative income effect is much better for the Polish panel, as was expected for this much larger data-set.

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: πi = pmi + ti.ω

(5)

with pmi the monetary price for goods used in consumption activity i, ti the time spent to consumption i and ω the opportunity cost for time. The later is supposed to be proportional to the ratio of the minimum wage rate (for market services) to the household wage rate, so that (independently of any tax correction and for constant hours of work) the income elasticity of the opportunity cost is –1. Formula (5) gives, whenever the monetary price does not depend on the household’s income: ∂πi/∂Y = ti.(∂ω/∂Y) = ti.(ω/Y) = Eπi/Y. (πi/Y) = Eπi/Y. [pmi + ti.ω)/Y] so that: ti = [(Eπi/Y/(1-Eπi/Y)].(pmi/ω)

(6)

Formula (6) indicates the minimum time for activity i such that the household would prefer too buy the market service, instead of accomplishing himself the domestic production corresponding to consumption i: ti > [(Eπi/Y/(1-Eπi/Y)].(pmi/ω) Ù market substitute > domestic production

(7)

For example, if the monetary price for food at home in France is 5 euros per person, and the opportunity cost for one hour of domestic activity 10 euros, the maximum duration of the domestic food at home activity must be 4.5 hours. For ω = 50 euros, this maximum duration is 1.9 hours. For the U.S., with (Eπi/Y/(1-Eπi/Y) set at 15, pmi = 3 $ and ω = 8 $ (respectively 50 $), the limit ti is 5.6 hours (respectively 0.9). These figures seem plausible9. The limit duration ti increases with the ratio of monetary price to the opportunity cost of time, and also with the income elasticity of the complete price for activity i. As the latter is positively related to the income inequality in the population, the decision rule (7) indicates that market services expenditures may be greater for the more inequal countries (at least when income inequality is measured by the ratio of maximum to minimum income, for instance by the inter-decile ratio).

9

Note that they are sensible to the proximity of Eπi/Y to the unity.

We first calibrate the elasticity of the budget share over an income inequality index. Then, we calculate the difference in the Theil index between France and the U.S. and compute by three different methods the resulting difference in the budget share for services.

Changes in the income distribution and Opportunity Costs Suppose that the income distribution differ between two countries by the sole higher income class, with populations Π1 (for instance France) and Π2 (U.S.), and YDk the kth income decile: (Π1) YD9 = 1.5 Y D5 = 3 Y D1 (Π2) YD9 = 2 Y D5 = 4 Y D1 The Theil index measuring the income inequality increases by 0.1 ln 1.33 ≈ 2.9% between countries 1 and 2 (supposing that the distributions of income over the ninth decile are proportional in the two countries). Therefore, because of the greater inequality in country 2, the opportunity cost for domestic production for the rich in terms of the minimum wage rate is smaller by 25% in country 2 (1/4 compared to 1/3). The own price elasticity can be calibrated, under Frisch hypothesis, as minus half of the income elasticity, around -0.75 for all services (according to the income elasticity for all services in Gardes-Starzec, 2004, Table 5, 6 and in Cardoso-Gardes, 1996). Suppose also that the budget share for all services is 0.5 for the rich. Then, the change in the budget share is equal to –0.75 . 0.5 . -0.25 = +0.094 for 10% of the population. This aggregates to + 3.1% for the whole population if the rich are supposed to consume 30% (respectively a proportion p) of the total expenditure in services. Thus, the marginal propensity of the aggregate budget share over the Theil index Γ is ∂ wserv/∂Γ = 0.031/0.029=1.07 (respectively 0.094p/0.029). Therefore, the influence of income distribution is much under-estimated in an Oaxaca decomposition of budget shares (see Gardes-Starzec, 2004, Table 6 and Kalwij et al., 2006, tables 10 and 12). Moreover, the Theil index is calculated, in these estimations, for the whole population instead of a comparison between very high and very low income classes. Income inequality in France vs the U.S. and market services expenditures We use three different method to assess the influence of the income distribution on the budget share for services. In the first calculation, we calibrate the differential of the budget share as concerns the Theil index ∂ wserv/∂Γ as in section4.4, considering the changes in the relative price for services due to income inequality, and compute the change in the budget share corresponding to the observed change in the Theil index. Between countries i and j: dwserv=(∂wserv/∂Γ).dΓij. In the second method, we calculate directly the change in then relative price corresponding to the ratios of hight and low income in the two countries, and use the price elasticity eps to compute the change in services expenditures: d wserv=eps ln(pi/pj). Finally, the third method uses the difference in the Theil indexes corresponding to the ratio of hight and low incomes in each country, then compute the change in complete prices for services using then income elasticity of complete prices calculated in section 4.2 : dπi/πi = Eπi/Y.dY/Y. Then, the change in the budget sharedepends on the price effect of complete prices on the budget share : dwserv = wserv.Ewserv/πi.(dπi/πi). Thus, the first method computes directly the effect of a change in then Theil index on the budget share, while the second estimates the price effect due to the difference in the income distributions, and the third estimates the effect of the income

distribution on the relative price using the relation between relative income and the complete price for services. Note that these three methods are not completly independent, especially the first and the second. First, we compare only the two extreme decile incomes in the U.S. and in France (Table 3 in Kalwij et al.). The Theil index Γ is greater in the U.S. by 0.019, which imparts a greater budget share for services of 0.020 (for ∂wserv/∂Γ=1.07). The income inequality between higher and lower income classes, which influences the opportunity cost of domestic production, concerns also households pertaining to the 8th or the 7th decile classes. Also, the Theil index of 0.019 does not take into account the distribution of income above the 9th decile. So, the influence of the Theil index difference due to the comparison between the extreme income classes, may be rather a figure which is very similar to the actual difference observed, around 7.5% (Table 3 in Gardes, Starzeec et al., 2006). By a more direct method based directly on the price effect, we could calculate the difference between the budget shares in the two countries by the formula: dwserv = -0.75 . ln(πFi/πU.Si) = 0.75. ln(3.87/4.76).p = +0.078 for p=0.5 and 0.047 for p=0.3. The third method consists to estimate the change in the relative income of some group of households corresponding to a change of the Theil index, then computing the change in the complete price corresponding to the relative income elasticity of this price, and using the own price elasticity of services to calculate the change in their budget shares. The change in Theil index writes, supposing xi =x + dxi: ∆Γ’ = ∑i ln{(x+dxi)/x} ≈∑i dxi/x ≈ Y9- Y1 /Y1 = ln(4.76/3.87) = 0.207 Thus, with Eπi/Y = -0.64 for all services (respectively –1.8 for domestic services) as previously estimated, we obtain: dwserv = ∂wserv/dln πi . Eπi/Y . dYr/Yr = wserv . Ewserv/πi . Eπi/Y . dYr/Yr => dwserv = 0.5 . (-0.75) . (-0.64) . 0.207 = + 0.0497 and dwdom.serv = 0.025 . (-0.8) . (-1.8) . 0.23 = 0.0075. Finally, income inequality explains more than a half of the difference (amounting to 7.5%) between the French and the U.S. budget shares for services. We conclude that the difference between the American and European budget shares for services may be correctly accounted using the different ratios of the maximum and minimum incomes in these countries, these ratios being proxies for the relative price of services for the richer households which are the more likely consumers of domestic services in industrial countries10.

10

A calculation of the effects of the income distribution in Europe and the U.S. necessitates precise estimates of the income elasticity of complete prices for all types of services expenditures, an estimate of the difference between the ratio of maximum and minimum wages in the two countries (or the difference in the Theil index due to the extreme income classes), and the proportions of services for each income class.

Conclusion The Engel’s law depends on changes of the quality of goods (change of the characteristics structure). As noted by Engel himself, this tends to diminish t(he change of the budget share as income increases, but it may also explain the different income elasticities between the social classes. Nevertheless, the Engel’s law is quite general, as it appears in its relative income version. This may explain its permanence in the literature, since empirical regularities are generally sensible to aggregation and not stable through time. The law hiddens a fundamental difference between the comparison of social classes and through time. This difference shows the importance of the social inscription of individuals and its theoretical representation in economic models. The estimations of relative income effects confirm the existence of a demonstration effect for food in both countries and for the reference populations defined by age cohorts, education and location. The effect is negative for food at home and positive for food away from home, which corresponds to the predictions. A more important effect could be obtained by defining more precisely the reference populations. Moreover, another application of these results could be to distinguish better homogenous groups for clustering. The magnitude of the correlation coefficients γν/ys between the residual of the dependent variable and those of the independent variables, may serve as an indicator for the homogeneity of the reference populations, thus providing a criterion to define them.11 The test is said to be semi-parametric because the specific household consumption is computed as a residual by panel analysis, thus taking indirectly into account the influence of all latent variables which are constant over time. In contrast, the relative income is predicted by the criteria used to define the reference populations, so the method is parametric as concerns this definition12. On the whole, the test is much more general than usual parametric tests and is based on independent definitions of relative income and consumption.

11

For instance, given that households are grouped into sub-populations according to some exogenous criteria, their specific expenditures for a set of commodities and the correlation coefficients γν/ys can be recovered by panel analysis. As these specific residuals are used in a second step used to group optimally households (so that the specific effect is homogenous within each cell and can be removed efficiently by a pseudo-panel estimation), the correlation coefficients γν/ys estimated in the second step are used to compare the homogeneity of this second grouping to the first. 12 Income would be also predicted by panel analysis, using life cycle models, but this would require much longer series than the two panels.

Appendix I : Decomposition of the endogeneity bias The limits in probability are examined when Hi↑ products of the endogeneity bias for individual data:

, for the nine cross-

A=E{(yh-Ey).(αh-Eα)}=E{((yHi-Eyi)+(Eyi-Ey)+ysh).((µHi-µi)+µi+νh))}. Supposing as usual that all means and variances are bounded, the limits in probability are equivalent to the limits in mean square, so the proof is given for limits in probability. Note that plim yHi=Eyi= yi ,plim µHi=µi, plim (Eyi-Ey)=Eyi-Ey,=yi-y, Hi ↑ plim pHi=pi=proportion of the sub-population in the whole population and plim (ph/pHi)=1/Ni with Ni=Card( ). (i) plim (Σh∈H ph.1Hi.(yHi-Eyi).(µHi-µi))=plim (ΣiΣh∈Hi ph.(yHi-Eyi).(µHi-µi)) =Σiplim pHi.plim (yHi-Eyi).plim (µHi-µi) =0. (ii) plim (Σh∈H ph.(yHi-Eyi).µi)=Σipi.plim ((yHi-Eyi).µi))≤(Supiµi)Σipi.plim (yHi-Eyi)=0. (iii) plim (Σh∈H ph.(yHi-Eyi).νh)=Σi pi plim (yHi-Eyi).Σh∈Hi plim (ph/pHi)νh =Σipi.plim (yHi-E yi).( 1/Ni)Σh∈Hi νh=0 as ph/pHi=1/Ni and Σh∈Hiνh=0. (iv) plim (Σh∈H ph.(Eyi-Ey).(µHi - µi))=Σi pHi.plim (yi-Ey).plim (µHi-µi)=0. (v) plim Σh∈H ph.(Eyi-Ey)µi=Σi pHi.plim (Eyi-Ey).plim µi ΣipHi.(Eyi-Ey).µi =ΣipHi.(Eyi-Ey).(µi-µ) = covariance on grouped data as ΣipHi.Eyi → Ey. (vi) plim Σh∈H ph.(Eyi-Ey).νh=ΣipHi.plim (Eyi-Ey).Σh∈Hi plim (ph/pHi).νh =(ΣipHi.(Eyi-Ey)).(1/Ni).Σh∈Hi νh=0 as Σh∈Hiν h =0.

Hi ↑

(vii) plim Σh∈H ph.ysh.(µHi-µi)=Σi plim (µHi-µi).Σh∈Hi plim ((ph/pHi).ysh) =Σi plim (µHi-µi).(1/Ni).Σh∈Hi ysh=0. (viii) plim Σh∈H ph.ysh.µi=Σi(pi.µi.Σh∈Hi plim ((ph/pHi).ysh))=Σi(pi.µi.Σh∈Hi (1/Ni).ysh) =Σi(pi.µi.(1/Ni).Σh∈Hi ysh)=0. (ix) plim Σh∈H ph.ysh.νh=covariance due to relative specific effects on the individual data. Thus we obtain: A = (v)+(ix) and ( v ) V(yHi) V(y) - V(yHi) (ix ) A = . + . V(y) V ( yHi ) V(y) V(y) V ( ys)

so that, according to (2): A = π= (βb -βw)panel = p (βb -βw) grouped data +(1-p) γ V(y)

with p = V(yHi)/ V(y) and γ = (ix)/ V(ys) = the correlation coefficient of the specific effect νh in equation (1) over the specific (relative) income.

Appendix II : Description of the data sets. Table 2.1: Means and standard deviations of the variables used in the PSID analyses

Budget share for food at home

1983

1984

1985

Level

Level

Dif.

1986

Level

Dif.

1987

Level

Dif.

Level

Dif.

.147

.144

-.003

.129

-.015

.137

.008

.134

-.003

(.103)

(.098)

(.084)

(.095)

(.086)

(.100)

(.082)

(.096)

(.081)

% with athome share = 0

0.0

0.0

53.2

0.0

74.0

0.0

41.5

0.0

51.3

Budget share for food away from home

.033

.034

.001

.031

-.003

.033

.002

.033

.001

(.040)

(.038)

(.034)

(.038)

(.033)

(.041)

(.032)

(.034)

(.033)

9.5

8.9

5.7

9.6

5.5

10.3

5.5

8.9

5.7

ln household income

9.9254

9.9985

(.648)

Ln age Head

% with away-fromhome share >0

ln family size (Oxford scale)

.0731 10.1714

.1729 10.1238

-.0475 10.1671

.0432

(.657)

(.280)

(.716)

(.320)

(.686)

(.308)

(.694)

(.299)

3.7044

3.7306

.0262

3.7573

.0267

3.7801

.0228

3.8044

.0242

(.377)

(.368)

(.013)

(.359)

(.013)

(.351)

(.012)

(.343)

(.012)

.6741

.6837

.0096

.6896

.0060

.6894

-.0002

.6912

.0018

(.404)

(.401)

(.162)

(.405)

(.168)

(.409)

(.159)

(.410)

(.171)

Table 2.2: Means and standard deviations in the Polish panel

Budget share for food at home Budget share for food away from home % with away-from home share > 0 Ln household Expenditure Ln head’s age

1987

1988

1989

1990

0.508

0.484

0.486

0.554

(.14)

0.0006

(.15)

(.18)

0.0006

0.0005

(.02)

(.03)

(.02)

(.03)

28.4

29.7

26.7

20.5

10.65

11.17

12.25

14.14

(.45)

(.49)

(.79)

(.50)

3.789

3.809 (.33)

(.32) Log of family equivalence scale

(.15)

0.0005

1.140 (.59)

1.121 (.60)

3.824 (.32) 1.095 (.61)

3.842 (.32) 1.081 (.61)

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