A. Chai

MONETA, ALESSIO

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           Volume 29, Issue 2   Comparing shapes of engel curves     Andreas Chai  Productivity Commission, Australian Government Alessio Moneta  Max Planck Institute of Economics, JenaAbstract We measure how different the shapes of Engel curves are across 59 commodity groups. The same analysis is carried out for their derivatives and variances. While Engel curves possess a relatively homogeneous shape, significantly more heterogeneity is present in derivatives and when particular sub-classes of income are considered.We thank Nancy Heckman, Nadine Chlaß, and the participants of the Evolutionary Economics group seminar at MPI for useful suggestions. We also thank Sebastian Müller and Michael Enukashvili for research assistance. We retain responsibility for any error.  Citation: Andreas Chai and Alessio Moneta, (2009) ''Comparing shapes of engel curves'', Economics Bulletin, Vol. 29 no.2 pp. 1156-1162.Submitted: Feb 09 2009.   Published: May 28, 2009.       1 IntroductionEngel curves (ECs) capture a fundamental aspect of consumer behavior: the sensitivity of con-sumption to income. However, there is significant variation in how consumption expendituresreact to changes in income across different commodity groups (see among others Prais 1953,Aitchison and Brown 1954). Nonparametric methods have been used to tackle this hetero-geneity since they do not require researchers to impose any functional form in the estimation(see e.g. Ha¨rdle and Jerison 1991, Engel and Kneip 1996). In parallel, economic theory hasdeveloped new demand systems allowing for the observed diversity in expenditure-income re-lationships (Banks et al. 1997, Blundell et al. 2007). Yet, no study so far has tried to measurethe magnitude of this diversity across commodity expenditures. We assess to what extent ECsdiffer in shapes for a large variety of expenditure categories. To achieve this, we exploit theflexibility both in estimation and in curve comparison that nonparametric techniques provide.Our findings indicate that ECs do not differ much in shape. They do, however, substantiallydiffer in terms of their derivative, which is the empirical analogue of the marginal propensityto consume one particular commodity. Intriguing differences are also found for sub-classesof income. We finally estimate the variance of expenditure at each point of income, shortlyreferred to as “ECs variance.” Each EC variance describes how the variability in consumptionbehavior depends on income for a given commodity group. Our findings reveal here noteworthydifferences across commodity groups.2 Data and Estimation MethodWe use data from the U.K. Family Expenditure Survey 2001. The data set consists of familybudgets related to 59 types of expenditures. These refer to housing costs (1 type of expendi-ture), fuel (2 types), food (14 types), alcohol (2 types), tobacco (2 types), clothing and footwear(5 types), household goods (7 types), domestic services (5 types), personal goods and services(6 types), motor expenditures (4 types), travel expenditures (3 types), TV, Books, Newspa-pers, etc. (4 types), entertainment and education (4 types). Expenditures are expressed inBritish pounds.1An EC is estimate for each type of expenditure g by regressing the level of expenditure Yifor g on total expenditure Xi:Yi = m(Xi) + ²i, (1)for households i = 1, . . . , n.2 For this purpose we use the Gasser-Mu¨ller kernel type of smooth-ing (Gasser and Mu¨ller 1984, Gasser et al. 1991). This choice has two reasons. First, thisestimate has an asymptotic bias that is preferable to the Nadaraya-Watson method. Second, itis the most suited to estimating the derivatives of regression functions. We use a fourth-orderkernel (necessary for derivative estimation) as weighting function. The bandwidth parameteris selected via the Herrmann’s (1997) plug-in approach, in order to deal with heteroscedastic-ity, which is a typical feature of expenditure data. The variance of Y at each point of X (ECvariance) is estimated using the Engel and Kneip’s (1996: 204) binned convolution method.1Sample sizes range from 173 (non-motor vehicles) to 6382 observations (cereals). The average sample sizeis 3287.2We normalize X to take values in [0, 1]. This is done dividing, for each category of commodity, both Xiand Yi by max (Xi).13 Measure of Shape SimilarityTo measure the similarity in shape between the estimated curves we use the rank correlationmethod proposed by Heckman and Zamar (2000). A comparative advantage of this methodis the ability to capture qualitative features of the curves such as kinks and spikes. Methodsbased on L2 distance do not perform well in this respect (Marron and Tsybakov 1995). Therank correlation used here is a generalization of the rank correlation between two finite vectorsof numbers (cf. Gibbons 1991). If two curves have the same shape our coefficient is equal toone. Two curves y = m1(x) and y = m2(x) are said to have the same shape if there exists astrictly increasing function g such that m1(x) = g{m2(x)}, that is the plot of y = m1(x) is thesame of y = m2(x) after a deformation of the y axis (Heckman and Zamar 2000: 136). TheHeckman-Zamar method presupposes the definition of a probability measure µ on the intervalin which m1(x) and m2(x) are defined (which is the unit interval after standardizing the data).Our proposed measure is µ(A) = (#x ∈ A)/(#x ∈ [0, 1]) (i.e the proportion of x points thatare in A), for any subinterval A of the unit interval. The rationale for using this measure isto give more weight to the portion of the curve for which there are more observations. Therank correlation between m1(x) and m2(x) is defined as:ρµ(m1,m2) =∫ {rm1(w)−Rm1}{rm2(w)−Rm2}dµ(w)√∫ {rm1(w)−Rm1}2dµ(w) ∫ {rm2(w)−Rm2}2dµ(w) , (2)where rm1(x) = µ{t : m1(t) < m1(x)} + 12µ{t : m1(t) = m1(x)} and Rm1 =∫rm1(w)dµ(w).rm2(x) and Rm2 are defined analogously. A consistent estimator of ρµ is given by Heckmanand Zamar (2000: 139).4 Empirical ResultsWe measure to what extent ECs are similar in shape by calculating the distance d = (1− ρµ)for each pair of the 59 estimated ECs. Since −1 ≤ ρµ ≤ 1, 0 ≤ d ≤ 2. The same procedureis applied to derivative and variance curves.3 The first important result is that there is morevariety in the shapes of derivatives than in the shapes of simple ECs and variances. This isevident when comparing the calculated distances between derivatives with both the calculateddistances between ECs and those between variances. The second column of table 1 displayshow the distances between ECs are distributed across different intervals. Specifically, 642out of 1711 (= 59 · 58/2) pairs of ECs (37.5% of the total) consist of two curves distant0 < d ≤ 0.01 from each other. In the same way, 519 (30.3%) pairs have a within distance0.01 < d ≤ 0.1, 114 pairs have 0.1 < d ≤ 0.25, and so on. In contrast (see third column intable 1), the majority of pairs of ECs derivatives possess a within distance situated in higherintervals. Indeed, in the 18.8% of cases 0.1 < d ≤ 0.25, in the 24.9% 0.25 < d ≤ 0.5, in the 21%0.5 < d ≤ 0.75. EC variances (fourth column) display a pattern which is intermediate betweensimple ECs and derivatives: more pairs than simple ECs but less pairs than derivatives fallin high intervals. A Kolmogorov Smirnov test confirms that derivatives display a distributionof within-pairs distances different from that displayed by simple ECs and variances. Wecompare the proportions of pairs of ECs having a distance falling in 200 sub-intervals (of thesame length) between 0 and 2 with analogous proportions of pairs calculated for derivatives3Six types of expenditure are excluded in the analysis of variance because containing less than 100 obser-vations.20.0 0.2 0.4 0.6 0.8 1.00.0000.0010.0020.0030.0040.0050.006Total ExpenditureExpenditures on g207412622433749Shapes Engel Curves 20010.0 0.2 0.4 0.6 0.8 1.0051015Total ExpenditureExpenditures on g46712164348Shapes Variances Engel Curves 20010.0 0.2 0.4 0.6 0.8 1.00.0000.0050.0100.0150.0200.025Total ExpenditureExpenditures on g46121649748Shapes Derivative Engel Curves 2001Figure 1: Engel curves (first diagram), ECs derivatives (second diagram), and ECs variances (third diagrams)for (some of) the following expenditures categories: cereals(4), eggs(6), fats(7), sugar(12), food at work & school(16), cigarettes (20), outwear(22), legal costs(37), spectacles(43), driving insurance & lessons (48), non-motorvehicles(49). Solid-line curves belong to the same cluster, dashed-line curves are clustered outside of it.and variances. The hypothesis that the distances among ECs have the same distribution ofdistances among derivatives is strongly rejected. The same happen when we compare distancesamong derivatives with distances among variances.Table 1: Synopsis of the results for different types of curvesNumber (proportion) of pairs of curvesDistanceintervalEngel EC EC ECs low ECs ECs highcurves Derivatives Variances income medium incomeincome(0-0.01] 642 (.375) 17 (.009) 285 (.199) 253 (.147) 37 (.021) 5 (.002)(0.01 - 0.1] 519 (.303) 110 (.064) 568 (.396) 325 (.189) 159 (.092) 18 (.010)(0.1 - 0.25] 114 (.066) 323 (.188) 328 (.229) 303 (.177) 219 (.127) 83 (.048)(0.25 - 0.5] 298 (.174) 427 (.249) 125 (.087) 316 (.184) 230 (.134) 156 (.091)(0.5 -0.75] 75 (.043) 360 (.210) 56 (.039) 213 (.124) 248 (.144) 245 (.143)(0.75 - 1] 3 (.001) 327 (.191) 49 (.034) 138 (.080) 260 (.151) 336 (.196)(1 - 2] 60 (.035) 257 (.150) 20 (.013) 163 (.095) 558 (.326) 868 (.507)Note: each entry in columns 2-7 denotes the number of pairs of curves (in brackets the proportion) whosedistance falls within the interval specified in the first column. The distance is measured as 1− ρµ.We also attempt to group the 59 estimated ECs on the basis of the shape. We performa hierarchical cluster analysis using as distance measure d = (1 − ρµ). We find that ECs3Table 2: Synopsis of the results across yearsNumber (proportion) of pairs of curvesDistanceintervalEngel Engel Engel EC EC ECcurves curves curves Derivatives Derivatives Derivatives1974 1988 2001 1974 1998 2001(0-0.01] 737 (.626) 782 (.457) 642 (.375) 26 (.022) 25 (.014) 17 (.009)(0.01 - 0.1] 391 (.332) 356 (.208) 519 (.303) 69 (.058) 110 (.064) 110 (.064)(0.1 - 0.25] 0 (0) 185 (.108) 114 (.066) 153. (.130) 284 (.165) 323 (.188)(0.25 - 0.5] 48 (.040) 62 (.036) 298 (.174) 237 (.201) 416 (.243) 427 (.249)(0.5 -0.75] 0 (0) 149 (.087) 75 (.043) 143 (.121) 250 (.146) 360 (.210)(0.75 - 1] 0 (0) 15 (.008) 3 (.001) 211 (.179) 331 (.193) 327 (.191)(1 - 2] 0 (0) 162 (.094) 60 (.035) 337 (.286) 295 (.172) 257 (.150)Note: see note in Table 2.do not group in few equinumerous clusters or in clusters corresponding to the macro typesof expenditures listed in section 2. For any agglomeration criterion used (average, single,complete linkage, Ward) we find that most curves fall into one very large cluster. The otherclusters are composed of one or maximum two elements. Specifically, with the average linkagecriterion, it emerges a large cluster of ECs consisting of 54 elements when we impose 5 splits.This same cluster contains 53 elements when the splits are six, up to 50 when the splits areten. The first diagram in Figure 1 displays result from the imposition of 10 splits. Five ECsdrawn in a solid line all belong to the same large cluster. The curves in dashed lines are ECsthat belong to other, smaller clusters. Curves are readjusted in the y axis so that they canbe easily compared by shape. The same analysis is performed for variances and derivatives.Variances group in a manner very similar to ECs. 40 out of the 53 commodity groups fallinto one cluster when the sample 8 split are imposed. The second diagram of figure 1 displaysboth variance curves belonging to the large cluster (solid lines) and variance curves belongingto smaller clusters (dashed lines). Remarkably different results emerge from the analysis ofthe derivatives. Derivatives exhibit much more heterogeneity in shape, as the third diagramof figure 1 shows. Although a large cluster still emerges, it is smaller relative to the onefound in the analysis of simple ECs. More specifically, when 10 splits are imposed the largecluster of derivatives includes 42 commodity groups possessing an average within distance ofd = 0.3311. In contrast, the large cluster of simple ECs contained (for the same number ofsplits) 50 commodity groups with average within distance d = 0.0288. Moreover, in the resultspertaining to the derivatives, the size of the smaller clusters has also grown, with one clustercontaining 6 commodity groups and another with 4 commodity groups.We also investigate how the distances are distributed in sub-classes of income, dividing theX axis in three equidistant intervals. In general there is a higher variety of shapes in sub-classes of income (see table 1, columns 5-7). The most relevant result in this respect is thatin the high class of income this heterogeneity dramatically increases. Indeed half of the pairs(50.7%) show a within distance larger than 1. Finally, we repeat our analysis considering twodifferent years: 1974 and 1998. The results are displayed in table 2. The patterns displayed in2001 emerge to some extent in 1974 and 1998 as well. Kolmogorov-Smirnov tests do not rejectsimilarities in the patterns between 1998 and 2001 (both for ECs and derivatives). Howeversimilarities in patterns between 1974 and 1998 are confirmed only as regards EC derivativesand rejected between 1974 and 2001 (both at 0.05 and 0.01 significance level).45 Discussion and ConclusionsECs display a very similar shape across different commodities. Noteworthy differences inshapes are observed for only very few expenditure categories. However, a wider variety ofshapes emerges when considering derivatives and higher classes of income. When groupingcurves (ECs, derivatives or variances) on the basis of their shape, we do not obtain classifi-cations typically made in consumption research. We do, in particular, find no distinction interms of goods versus services or durable versus non-durable.The overall dominant EC (see again figure 1) is strictly increasing up to a certain point ofX. Thereafter, we find a class of shapes for which it goes up again, while in others it remainsflat or goes down. This confirms Prais’ (1952) early hypothesis that the “typical shape” of anEC for a commodity g reflects that g has an income elasticity greater than unity at low incomelevels for which g is a luxury. For higher income levels where g has become a necessity theEC displays an income elasticity between 0 and 1. At a certain level of income expenditure itdoes not react to increases of income anymore. This is in line with the common assumptionthat a point of satiety occurs at which the expenditure on g becomes insensitive to furtherincreases in income (Aitchison and Brown 1954, Pasinetti 1981, Witt 2001).The observed tendency of ECs to change direction at very high levels of X should be cau-tiously interpreted. For this range, there is typically a dramatic decrease in the number ofobservations. This fact, however, does not bias our results on comparison. This is becausethe measure incorporated in the rank correlation method weights the sub-intervals of totalconsumption according to the frequency of observations. Intervals containing few observa-tions thus contribute much less to the overall rank correlation than those which have manyobservations. How may one now explain the observed heterogeneity of EC derivatives relativeto the homogeneity of simple ECs? First, small kinks in the ECs become magnified in theshape of their derivatives. Thus, if an EC has a kink at some level of X and another does not,the respective derivatives of the two curves will differ. Second, the rank correlation method ishighly influenced by whether two curves are monotonic over the same intervals. Derivativesof monotonic curves, however, are not necessarily monotonic themselves. Hence, while therank correlation between two particular ECs may be high, the rank correlation between thederivatives of the same ECs can turn out to be much lower.In sum, our empirical findings provide new insights into the heterogeneity patterns ofconsumption behavior. We suggest the following:1. Some fundamental characteristics of consumption behavior are invariant across expen-diture categories, independently of the sort of commodities involved. These consist oftwo features as evidenced by the “typical shape” of ECs: (i) families with low incomeare very sensitive to income increases, and (ii) at a certain level of income a point ofsatiation is reached.2. The richer are families, the more various is their consumption behavior. This equallyholds across types of expenditures as evidenced by a certain homogeneity in the shapeof EC variances. It reflects that households with higher income know less constraintswhen allocating their budget than poorer households. However, this feature shows lesshomogeneity across commodities than the characteristics of consumption behaviors.3. The fact that heterogeneity in consumption behavior becomes higher as income increasesis confirmed by the diversity of ECs for rich families.54. Marginal propensities to consume differ in their sensitivity to income across groups ofexpenditures. This evidenced by the fact that shapes of EC derivatives vary widelyacross expenditure categories.ReferencesAitchison, J. and J.A.C. Brown (1954), A Synthesis of Engel Curve Theory, The Review ofEconomic Studies, 22(1), 35-46.Banks, J., R. Blundell, and A. Lewbel (1997), Quadratic Engel Curves and Consumer Demand,The Review of Economics and Statistics, 79(4), 527-539.Blundell, R., X. Chen, and D. Kristensen (2007), Semi-Nonparametric IV Estimation of Shape-Invariant Engel Curves, Econometrica, 75(6), 1613-1699.Engel, J. and A. Kneip (1996), Recent Approaches to Estimating Engel Curves, Journal ofEconomics, 63(2), 187-212.Gasser, T., A. Kneip, and W. Ko¨hler (1991), A Flexible and Fast Method for AutomaticSmoothing, Journal of the American Statistical Association, 86(14), 643-652.Gasser, T. and H.G. Mu¨ller (1984), Estimating Regression Functions and Their Derivativesby the Kernel Method, Scandinavian Journal of Statistics, 11, 171-185.Gibbons, J.D. (1991), Nonparametric Measures of Association, Newbury Park CA: Sage Pub-lications.Marron, J.S. and A.B. Tsybakov (1995), Visual Error Criteria for Qualitative Smoothing,Journal of the American Statistical Association, 90(430), 499-507.Heckman, N. E. and R. H. Zamar (2000), Comparing the shapes of regression functions,Biometrika, 87(1), 135-144.Herrmann, E. (1997), Local Bandwidth Choice in Kernel Regression Estimation, Journal ofComputational and Graphical Statistics, 6(1), 35-54.Pasinetti, L. (1981), Structural Change and Economic Growth, Cambridge: Cambridge Uni-versity Press.Prais, S.J. (1952), Non-Linear Estimates of the Engel Curves, The Review of Economic Studies,20(2), 87-104.Witt, U. (ed) (2001), Escaping Satiation: The Demand side of economic growth, SpringerVerlag, Berlin.6

Comparing shapes of Engel curves

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