Author aliation Werner Hildenbrand Rheinische Friedrich-Wilhelms-Universitat Bonn Lennestrae 37 D{533 Bonn Germany Telefon (0228) Telefa (022

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1 Projektbereich A Discussion Paper No. A{560 Demand Aggregation under Structural Stability by W. Hildenbrand A. Kneip October 997 Acknowledgement We would like to thank K. Utikal for his collaboration in the statistical analysis of the U.K. Family Ependiture Survey. The Figures in this paper have been prepared by him. We would also like to thank the participants of the Bonn- Workshop on Aggregation for their critical comments and helpful suggestions. We are particularly grateful to R. Blundell, J.M. Grandmont, M. Jerison, A. Lewbel and T. Stoker. Financial support by Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 303 at the University of Bonn is gratefully acknowledged.

2 Author aliation Werner Hildenbrand Rheinische Friedrich-Wilhelms-Universitat Bonn Lennestrae 37 D{533 Bonn Germany Telefon (0228) Telefa (0228) E{Mail: Alois Kneip Rheinische Friedrich-Wilhelms-Universitat Bonn C.O.R.E. Lennestrae Voie du Roman Pays D{533 Bonn B-348 Louvain-La-Neuve Germany Belgien Telefon (0228) Telefon (0032) Telefa (0228) Telefa (0032) E{Mail: E{Mail: Kneip@stat.ucl.ac.be

3 Abstract The goal of this paper is to model the mean (aggregate) consumption ependiture of a large and heterogeneous population of households. The aggregation process is based on assumptions of how the income distribution and the composition of the population evolves over time (structural stability). It is shown that the change in the aggregate consumption ependiture ratio can be decomposed into an eect of changing income dispersion, an eect of income growth, an eect of price-ination and an eect of changing composition of the population. JEL Classication System: D, D 3, C 43, E 0 Keywords: aggregate consumption ependiture, aggregation, structural stability, income distribution, distribution of household attributes

4 Introduction. The goal of this paper is to model the change over time of mean consumption ependiture C t of a large and heterogeneous population H t of households: C t = #H t h2h t c h t () where c h t denotes the consumption ependiture of household h in current prices during period t on all commodities that belong to a certain consumption category, such as food, housing or non-durables. The starting pointofany analysis of aggregation across households is a model of individual household behaviour. To concentrate in this introduction on the essential we start directly from a micro-relation c(; ) c h t = c( h t ; h t ) ;h2 H t (2) where h t denotes disposable income in period t of household h and h t = ( h t;; h t;2;:::) denotes avector of household characteristics that are used as eplanatory variables in the underlying model of household behaviour (e.g. preferences). We do not eplicitly mention prices and interest rates in the introduction; this amounts to assuming that they do not change over time. The population of households in period t is described by the joint distribution t of household income h and characteristics h across the population H t. Given the micro-relation (2), one obtains for mean consumption ependiture C t = c(; )d t : (3) Thus, given the micro-relation c, C t is a function of t ; C t = C( t ). The distribution t, however, is not a useful eplanatory variable for mean consumption ependiture, because it is a far too detailed description of the population. The goal of aggregation theory is to simplify the function C( t )by reducing the entire distribution t to certain relevant characteristics of t, such as mean or dispersion. Obviously, such a simplication { even if one is satised with an approimation to C( t ) { is only possible if one restricts the way in which the distribution t changes over time and/or if one appropriately species the micro-relation. There is a large literature on aggregation starting with Antonelli in 886. For a general discussion of the various aspects of aggregation theory we recommend Malinvaud (993). 2

5 In order to illustrate this point we give a simple eample. If the Engel curve of the population 7! c(; )d t j =: c t () is time-invariant, i.e., c t = c (denitely, an unrealistic assumption), then (3) becomes C t = c() t ()d where t denotes the density of the income distribution in period t. Thus, if c is linear, then mean consumption ependiture C t = c( ), and hence depends only on mean income without any restriction on the changes in the income distributions. On the other hand, if changes in the income distributions are restricted to proportional changes in household income, hence the relative income distribution is time-invariant, say equal to, then one obtains C t = R c( ) ()d. Thus, mean consumption ependiture depends only on mean income without any restriction on the Engel curve c. In this paper we want to avoid, as far as possible, any assumption on the micro-relation (other than being smooth in the relevant variables, e.g., income). Thus, the micro-relation is merely a notation; it just species the set of eplanatory variables for consumption ependiture on the household level. To achieve the desired simplication of C( t ) we must therefore restrict the evolution over time of the distribution t. Since some of the household characteristics { that are eplanatory variables in the micro-relation { are unobservable, we consider in addition to household characteristics also observable household attributes, such as age and employment status or household size. Household characteristics that are observable may be listed among attributes as well. Household income and attributes are used to stratify the population. Then we obtain C t = c(; )d t j(; a) d t that is to say, we rst consider mean consumption ependiture of the subpopulation consisting of all households with income and attribute prole a and then we average over the subpopulations, i.e., we integrate with respect to the joint distribution t of income and attributes. 2. In section 2 we model the changes over time of the conditional distribution t j(; a) of household characteristics (Hypothesis ) and of the joint distribution t of household income and attributes (Hypotheses 2 and 3). 3

6 It is our goal to \eplain" the observed changes over time of C t bychanges in the observable income-attribute distribution t. Such an \eplanation" is only satisfactory if changes in C t are not attributed to changes in the unobservable distributions t j(; a). Therefore we must somehow link changes in t j(; a) to changes in t. This is achieved by Hypothesis, which is called \structural stability 2 of household characteristics with respect to household attributes". In the special case where the conditional distribution t j(; a) does not depend on, the hypothesis simply epresses that the distribution t ja of household characteristics across all households with attribute prole a changes very slowly over time such that the distributions s ja and t ja can be considered as identical for periods s and t that are not too far apart from each other (local timeinvariance). Hypothesis 2 describes how the income distributions are allowed to change over time. Since we want to allow for changing income dispersion (e.g., changing Gini-coecient) we can not rely on the simple assumption of time-invariance of the relative income distribution. Obviously, the actual evolution of household income is more comple than just a proportional change. No single assumption can eactly describe the comple evolution of income. We have chosen the simple hypothesis of local time-invariance of the standardized log income distribution (Hypothesis 2). Of course, this hypothesis should only be considered as an approimation to the comple actual changes of income distributions. The descriptive accuracy of Hypothesis 2 is illustrated in Figures and 2. Finally, we model how the attribute distributions are allowed to change over time. Hypothesis 3 epresses that the income-conditioned attribute distribution s j s in period s is \approimately" equal to the income-conditioned attribute distribution t j t in period t for two periods s and t that are close to each other, provided the income levels s and t are in the same percentile position (quantile) in the income distribution in period s and t, respectively. As in the case of Hypothesis 2, this Hypothesis should be interpreted as an approimation, capturing the main tendency of the actual very comple change of attributes. The empirical content of Hypothesis 3 is illustrated in Figures 3, 4 and The propositions in Section 3 are based on a strong version of Hypothesis 3. It is assumed that the dierence between the attribute distributions s j s and t j t can be neglected (Hypothesis 3 + ). This requires, of course, that the periods s and t are close to each other. 2 The idea of \structural stability" is borrowed from Malinvaud (98), chapter 2.3 and (993), section 0. 4

7 We then derive for the change of the mean consumption ependiture ratio a rst-order approimation (Proposition 2): C t =, C s = s = s log t + s log s s + O ma (log t s ) 2 ; (log s ) 2 where the coecients s and s are determined by the micro-relation c and the distribution s and s is a measure of income dispersion. Consequently, neglecting second-order terms, the change in the mean consumption ependiture ratio is the sum of two terms: the eect of the changing income dispersion, s log t s, and the eect of mean income growth, s log t s. The sign of the coecients s and s are, of course, important. For eample, a negative s implies that increasing income dispersion, (hence log t s is positive), decreases the mean consumption ependiture ratio. In Section 3 we show what kind of information on the micro-relation and the distribution s is required to infer the sign or magnitude of the coecients s and s. It turns out that no assumption on the micro-relation alone determines the sign of the coecient s. In certain circumstances, that are eplained in Section 3, one can estimate the coecients s and s from cross-section data in period s. In this case one does not need the micro-relation to determine s and s ; only the actual household consumption ependiture in period s is needed. 4. In Section 4we etend the approimation of Section 3 to the case where the dierence of the attribute distributions s j s and t j t is not negligible. 5

8 2 Notation, Denitions and the Modelling Methodology 2. The goal of this analysis is to eplain or more modestly to model the change over time of aggregate consumption ependiture C t on a certain consumption category (such as Food, Housing and Non-Durables) in current prices of a large and heterogeneous population. What are the relevant eplanatory variables for modelling the change in C t? Every analysis that accounts for aggregation over economic agents must begin with a model of individual behavior. Consequently, the starting point of our analysis is a micro-relation which relates household h's consumption ependiture c h t in period t to the price system p t, the interest rate r t, the disposable income h t and certain theoretical household characteristics h t =( h ;t; h 2;t;:::): c h t = c(p t ;r t ; h t ; h t ) () The nature of these household characteristics h depends on the specication of the micro-model of individual behavior. Typically, some of the household characteristics are unobservable. If individual behavior is modelled as an intertemporal decision problem (foreward looking households) then, the micro-relation depends also on past information, for eample, past income h t,;::: and past prices p t, ;::: since this information is needed to predict future income and future prices. In the present paper, however, we do not treat this general case, since it greatly complicates the analysis. We start from a micro-relation () dened by a function c which is the same for all households; households dier, however, in income h 2 IR + and in characteristics h 2. We assume that space of household characteristics can always be considered as a metric space. Given the micro-relation () the population of households in period t is described by the joint distribution t of income 2 IR + and household characteristics 2;thus t is a distribution on IR +. Mean consumption ependiture C t in period t is then dened by C t := IR + c(p t ;r t ;;)d t = C(p t ;r t ; t ) (2) In addition to the theoretical household characteristics we shall consider observable household attributes, such as age, employment status, household size and composition etc. Such a prole of household attributes is denoted by a =(a ;a 2 ;::: ;a n ) which takes values in a set A, asubset in IR n. 6

9 Household characteristics which are observable belong also to the set of attributes (e.g. stock of nancial assets). Typically, there are household attributes which are not household characteristics, that is to say, the micro-model is not formulated in terms of these attributes. Notation t on IR + A the joint distribution of income and household attributes a t j on A the conditional distribution of household attributes given income t a on A the marginal distribution of household attributes t on IR + he joint distribution of income and household characteristics t j(; a) onhe conditional distribution 3 of household characteristics given (; a) t j on he conditional distribution of household characteristics given income t on he marginal distribution of household characteristics c t (p t ;r t ;;a):= R c(p t;r t ;;)d t j(; a) the regression function of consumption ependiture versus income and household attribute a. ~c t (p t ;r t ;):= R c(p t ;r t ;;)d t j the regression function of consumption ependiture versus income ; ~c t (p t ;r t ; ) is called the Engel curve in period t. Prototype eample: There are two distributions and 2 on. There is a function t from IR + into [0; ] and t denotes the density of an income distribution. We consider one attribute which can take two values; A = fa ;a 2 g. 3 this conditional distribution has to be derived from the joint distribution of (; ; a) 7

10 Dene t j := 8 < : a with probability t () a 2 with probability (, t ()) a t := 8 < : a with probability R t () t ()d a 2 with probability R (, t ()) t ()d t j := t () +(, t ()) 2 t j(; a ):= ; t j(; a 2 ):= 2 c t (p t ;r t ;;a i ) = c(p t ;r t ;;)d i ; i =; 2 ~c t (p t ;r t ;) = t ()c t (p t ;r t ;;a )+(, t ())c t (p t ;r t ;;a 2 ) Note that in this eample, income and household characteristics are independently distributed if one conditions on household attributes. Consequently c t (p t ;r t ;;a i ~c t (p t ;r t ;) 6= [@ c(p t ;r t ;;)] d t j(; a i c(p t ;r t ;;)d t j: The evolution over time of t is determined by the evolution of t and t. Note that without specic assumptions on the evolution of t and t the marginal distribution t of household characteristics is not time-invariant. In order to model the change over time of mean consumption ependiture C t = IR + A c(p t ;r t ;;)d t j(; a) we have to formulate hypotheses on the change over time of d t (3) ) the conditional distribution t j(; a) of household characteristics given income and attribute prole a, or alternatively, the regression function c t (p t ;r t ;;a) 2) the joint distribution t of income and attributes. 8

11 2.2 Structural stability of household characteristics Hypotheses on the change over time of the conditional distribution t j(; a) are delicate. Any hypothesis on the change of the distribution t j(; a) is purely theoretical, that is to say, speculative, since t j(; a) describes a distribution of unobservable household characteristics. It is our goal to \eplain" the observed changes over time of C t bychanges in the observable distribution t (thus, in particular, invariance of t ;p t and r t must imply invariance of C t ). Therefore we must link possible changes in t j(; a) with changes in t. Otherwise t cannot serve as a satisfactory eplanatory variable since changes in C t can then always be attributed to unobservable changes in t j(; a). Of course, it might turn out that the observed changes in C t cannot be \eplained" by the observed changes in t since the set A of attributes is not suciently comprehensive. The distribution t of household characteristics of the whole population may change over time, yet this change should be caused by achange in the distribution t of income and attributes. The simplest way to achieve this is to postulate that the distribution t ja of characteristics of all households with attribute prole a is time-invariant or, at least, changes very slowly over time (local timeinvariance). This postulate is based on the idea that households typically keep their characteristics if their attribute prole does not change over time. In the case where the conditional characteristic distribution t j(; a) does not depend on (i.e., within the subpopulation of all households with attribute pro- le a, income and household characteristics are independently distributed), the above postulate can serve as a denition of \structural stability" of household characteristics with respect to a set of household attributes. In this case the regression function c t (; ; ;a), dened by c t (p; r;;a)= c(p; r;;)d t j(; a) is time-invariant; a property which usually is assumed in applied micro-econometrics (e.g. Stoker (993), Jorgenson et.al. (982) and Blundell et. al. (993)). We would like to dene \structural stability" also in the case where t j(; a) might depend on. Then, with household income and prices changing over time, it does not seem to us meaningful to postulate time-invariance of t j(; a), since denotes nominal income. Rather one should condition on \real" income or on quantiles in the income distribution. Denition: The income levels s and t in period s and period t, respectively, are in the same percentile position in the income distribution in period s and t, 9

12 respectively, if s 0 s ()d = t 0 t ()d : The following heuristic argument motivates the denition of structural stability that is basic for our analysis. Consider two periods s and t and the income densities s and t. For periods that are close to each other one epects a high positive association between household's income in period s and the later period t. If this association were perfect then households would remain in the same percentile position in the income distributions s and t. Furthermore, if households whose attribute prole does not change in going from period s to period t keep their characteristics then the distributions of household characteristics s j( s ;a) and t j( t ;a) will approimately be the same if s and t are in the same percentile position in s and t, respectively. Hypothesis : structural stability of household characteristics Structural stability of household characteristics with respect to the set A of household attributes is dened by the following property: if s and t are in the same percentile position in the income distribution of the whole population in period s and t, respectively, then, for every a 2A, s j( s ;a)= t j( t ;a): Structural stability implies the above postulate that the distribution of household characteristics of all households with attribute prole a is time-invariant. This justies the label \structural stability" (in the sense of time-invariance). This property, in turn, implies structural stability if t j(; a) does not depend on, that is to say, conditioned on household attribute prole a, income and household characteristics are independent. We remark that the prototype eample is structurally stable. 2.3 The change over time of the distribution t In formulating hypotheses on the change over time of the observable distribution t we have to face the following well-known empirical facts: ) t describes a multi-variate distribution whose components are not independently distributed, that is to say, the joint distribution t of income and attributes 0

13 is not the product of its marginals (e.g. the distributions of income, age, household size, etc.). Typically there is a high correlation between income and the various household attributes. 2) There is no satisfactory a priori given functional form (determined up to some few parameters!) for the distribution t and even for its marginals. For eample, the shape and the change over time of income distributions are the outcome of many dierent forces some of which are operating in dierent directions. Furthermore, the shape and the change over time of income distributions depend on the notion of income (e.g. \disposable income"), on the units over which the distribution is dened (e.g. \household") and on the population (e.g. \full-time employed household head"). By \modelling the change over time" of the distribution t we do not mean \to predict the evolution of t " but to suitably restrict the possible changes, that is, we want a \parametrization of the transition" from s in period s to t in period t. To eplain this \parametrization of the transition" we consider rst the case of income distributions. Eample: Time-invariance of the relative income distribution The relative income distribution in period t is obtained by dividing the income h t of every household in the population by mean income. If t and t denote the density of the income and relative income distribution, respectively, then one obtains t () = t ( ) : (4) Time-invariance of t then implies that for two periods s and t t () = s s s : (5) Thus, the transition from s to t is parametrized by mean income, that is to say, if one knows s and then t is determined. Time-invariance of the relative income distribution cannot serve as a suitable hypothesis for our analysis. Indeed, it implies that the income dispersion, measured, for eample, by the Gini-coecient or the standard deviation of log income, remains constant over time. It is however a well-established empirical fact (e.g. Gottschalk and Smeeding (997)) that for most countries the income dispersion changes over time, even though, for some countries, the change is very slow. To take into account a changing income dispersion we consider the following

14 Hypothesis 2: Time-invariance of the standardized log income distribution Instead of income consider the logarithm of income, log. Let m t and t denote the mean and standard deviation, respectively, of the distribution of log in period t. of The standardized log income distribution is then dened as the distribution t (log(), m t ): The density of this distribution is denoted by + t. The relationship between the densities t and + t is given by + t (z) =y t t ep( t z) t (y t ep( t z)) (6) where y t = ep(m t ). Note, y t is equal to the median of the income distribution t if log is symmetrically distributed. Furthermore, if income were log-normally distributed (i.e., log has a normal distribution) then + t is the normal distribution with mean 0 and variance. Time-invariance of + t t () = s t then implies for two periods s and t! y s y (s=t), s s (s=t) y s=t t y s=t t It is not hard to show that the transition can also be parametrized by mean income and t ; t () = s = t where m s () = R s ()d.! s=t 0 m s ( t = s ) (s=t), m! s= s( t = s ) t A Thus, the transition from s to t is parametrized by ( ; t ), that is to say, if one knows s, hence s and s, and ; t then t is determined. It follows from (8) that s and t are in the same percentile position of s and t, respectively, if (7) (8) s = '( t ) (9) where the function ' is dened by '() := m s( t= s) s= t. Remark: Naturally, time-invariance of the standardized log income distribution will never hold eactly, even for periods s and t that are close to each other. 2

15 Hypothesis 2 should be considered as an approimation to the actual change in the short-run. It is important to remark that the income distributions can be estimated and therefore one can decide whether the hypothesis satisfactorily captures the main tendency of the actual change. Since we need the income distribution only to compute the mean (integral) it might besucient to know t approimately provided the regression function that we want to integrate, is suciently regular. It might well be that alternative hypotheses will be found that yield a better approimation. The standardized log transformation is particularly simple, yet our approach can be adapted to any other transformation of income distributions that leads to time-invariance. Given s, (8) denes a parametrization of t in terms of ; t. This parametrization of income distribution is \mean-scaled" in the sense of Lewbel (990) and (992). Figures and 2 show kernel density estimates of the standardized log income distribution based on data from the U.K. Family Ependiture Survey Figure : Kernel density estimates of the standardized log income distribution; U.K.-FES, total population. 3

16 Figure 2: Kernel density estimates of the standardized log income distribution; U.K.-FES, subpopulation of full-time employed head of household. 4

17 Net we have to model the change over time of the distribution of household attributes. The modelling approach is formally analogous to the one in section 2.2 in the case of distributions of household characteristics. There is, however, an important dierence; distributions of attributes are observable. For eample, it is a well-established empirical fact that the attribute distribution t j of all households with income depends quite strongly on the income level and the distribution t j is not time-invariant (e.g. see Figures 3, 4 and 5). The heuristic argument preceding the denition of structural stability in section 2.2 suggests the following Hypothesis 3: For two periods s and t that are close to each other the incomeconditioned attribute distribution s j s is \approimately" equal to the incomeconditioned attribute distribution t j t if s and t are in the same percentile position in s and t, respectively. This hypothesis is based on the view that household attributes change relatively slowly as compared with income. We shall consider two versions of Hypothesis 3; we shall assume in section 3 that the dierence s j s, t j t is negligible and in section 4 that the dierence is \small", in a sense to be eplained later. To illustrate the empirical content of Hypothesis 3 we show in Figures 3, 4 and 5 estimates of the age distribution of head of household, the distribution of household size and the distribution of employment status of head of household, respectively, conditioned on four percentile positions: `poor', `lower middle-class', `upper middle-class', and `rich'. 5

18 poor lower middle-class upper middle-class rich Figure 3: Estimates of age distributions of head of household conditioned on four percentile interval positions: \poor" (0-6 %), \lower middle-class" (7-50 %), \upper middle-class" (50-84 %), and \rich" (85-00 %). U.K.-FES, total population. 6

19 poor l m-c u m-c rich poor l m-c u m-c rich Figure 4: Estimates of the distribution of number of persons in household conditioned on four percentile interval positions: \poor", \lower middle-class", \upper middle-class", and \rich" poor l m-c u m-c rich poor l m-c u m-c rich Figure 5: Estimates of the distribution of employment status of head of household conditioned on four percentile interval positions: \poor", \lower middleclass", \upper middle-class", and \rich". Self employed, full-time employed 2. 7

20 3 Aggregation under structural stability of household attributes 3. In this section we eplore the implications of Hypothesis, structural stability of household characteristics, Hypothesis 2, time-invariance of the standardized log income distributions, and a strong version of Hypothesis 3, which is Hypothesis 3 + : Structural stability of household attributes If s and t are in the same percentile position in the income distributions of period s and t, respectively, then s j s = t j t : Hypothesis 3 + is very restrictive; it will be modied later. The hypothesis implies that the distribution of household attributes is time-invariant. Yet the distributions of age, household size etc. that are estimated from time series of cross-section data are not time-invariant, even though they change over time very slowly (see Hildenbrand, Kneip, and Utikal (997)). Proposition : Hypothesis, 2 and 3 + imply C t = IR + c(p t ;r t ; m s ( t = s ) t=s ;)d s =: K s (p t ;r t ; ; t ) (0) that is to say, given the micro-relation c and the distribution s in period s, then mean consumption ependiture C t in period t is a function in p t ;r t ; and t. Proof: By Hypothesis 2 we obtain t () = s = t m s ( t = s )! s=t 0 (s=t), m! s= s( t = s ) t A () Hypothesis, structural stability of household characteristics, and Hypothesis 3 +, structural stability of household attributes, imply t j = s j m s ( t = s )! s= t (2) 8

21 We now substitute () and (2) into the denition of C t and obtain with the notation = t = s C t = IR c(p t ;r t ;;)d s j m s ()! = 3 5 m s () m = The substitution = s(), hence = m s() and d leads to # C t = IR+ " c(p t ;r t ; m s () ;)d s j! = m! = s() s A d d = s ()d: t m, s(), Q.E.D. Proposition shows that C t = K s (p t ;r t ; ; t ) is determined by the microrelation c { which we did not specify up to now { and the distribution s { which is only partially observable. In the following we want to approimate the integral in (0) by a simple epression in the variables p t ;r t ; and t. The simplest approimation of K s (p t ;r t ; ; t ) which comes into mind is a rst-order Taylor epansion in the variables p; r; and at the values p s ;r s ; s and s. However, even for periods s and t which are close to each other, say t = s +, the change, s of mean income (or the price change p t, p s ) will typically not be very small, for eample, mean income might well increase by 0% or even more. Consequently, to obtain a satisfactory approimation of C t we have either to take a higher-order Taylor epansion { which will complicate the analysis { or we look for a suitable non-linear rst-order approimation which is a satisfactory approimation even for values of the variables p; r; ; that are not very near to p s ;r s ; s ; s. The choice of such an approimation requires, of course, some knowledge of the shape of the function that we want to approimate. The particular approimation that is given in Proposition 2 is based on the assumption that the regression 7! w s (p s ;r s ;;a)= c(p s;r s ;;)d s j(; a); a 2A has the following property (that is a well-known empirical w s (p s ;r s ; ; a)j = w s (p s ;r s ;;a) does not depend very sensitively on the income level. To simplify the presentation we replace the comprehensive price system p t by two price indices t and 2 t. We shall write p t =(p t ;p 2 t ), where p t is the price 9

22 system of all commodities for which we consider consumption ependiture and p 2 t is the price system of all other commodities. Let t and t 2 denote a price inde for p t and p 2 t, respectively. We now dene the function G by G( t ; 2 t ;r t ; ; t ):=K s ( t p s; 2 t p 2 s s;r s 2 t ; ; t ) Proposition 2: Hypothesis, 2 and 3 + imply for a smooth micro-relation c G( t ; 2 t ;r t ; ; t ) " Cs = + s log t + s log + s log t s s s + O ma s + s log 2 t 2 s + s log r t r s # ( (log t ) 2 ; (log ) 2 ; (log t ) 2 ; (log t ) 2 ; (log r )! t ) 2 ; (3) s s s s 2 r s the coecients s ; s ; s ; s and s are dened by " s m s () c(p s ;r s ;;)d s j s = s IR + s = s + s = s s = s IR + IR [w(p s ;r s ; ; )] = d s j h c( p s;p 2 s;r s ;;) i = d h c(p s; p 2 s;r s ;;) i = d r [c(p s ;r r s ;;)] r= d s # s ()d s ()d We emphasize that the coecients are dened by the micro-relation and the distribution s in period s; hence they can be interpreted as behavioral parameters that depend on the composition of the population (see section 3.2 for details). Proof: With the notation := t =s; 2 := t 2 =s; 2 := t = s and w(p t ;r t ;;):= c(p t;r t ;;)we obtain from Proposition that " # G( t ; 2 t ;r t ; ; t )= = IR+ w( p m s () s; 2 p 2 t s;r t ; m s () ;)d s j s ()d: = 20

23 Let := = s and r := r t =r s. The integral then becomes IR + m s () w( p s; 2 p 2 s s;rr s ; m s () ; )d s (4) which denes a function f s in ( ; 2 ;r;;); f s (;::: ;) = G( s; 2 s;r s ; s ; s )= s = C s = s and f s ( ; 2 ;r;;)=g( t ; 2 t ;r t ; ; t )=. We now take a rst-order Taylor epansion of the function f s in log =: ~ ; log 2 =: ~ 2 ; log r =: ~r; log =: ~ and log =: ~ at (~ ; ~ 2 ; ~r; ~; ~) = (0;::: ;0), that is to say, we take a usual rst order Taylor epansion of the function (~ ; ~ 2 ; ~r; ~; ~) 7! f s (ep ~ ; ep ~ 2 ; ep ~r; ep ~;ep ~) (5) at (~ ;::: ; ~) =(0;::: ;0). Thus we obtain f s ( ; 2 ;r;;) f s (;::: ;) + s log + s log + s log + s log 2 + s log r which is the claimed approimation. The coecients s ;::: ; s are dened as partial derivatives of the above function (5) at (0;::: ; 0; 0). The coecient s requires a comment. By denition Dene g() = s ~ [f s (; ; ; ; ep ~)] ~=0 [f s (; ; ; ;)] = " IR + m s () w(p s s;r s ; m s () ;)d s #= m s(). Then we obtain s = IR [g()w(p s ;r s ;g() s ;)] = d s We now compute the derivative under the [g()w(p s ;r s ;g() s ;)] = = g 0 ()j = (w(p s ;r s ;;)+@ w(p s ;r s ;;)) Consequently, s epression claimed in Proposition 2. By denition = g 0 ()j c(p s ;r s ;;): = h m c(p s ;r s ;;) i = d s s ~ h fs (; ; ; ep ~;) i ~=0 : 2 which is equal to the

24 f ~ s(; ; ; ep ~;) ~ w(p s ;r s ;ep ~;)d s s inc w(p s ;r s ;ep ~;) ep ~d s ; s inc denotes the partial derivative with respect to income, we obtain for ~ = 0 the epression for s as claimed in Proposition 2. Furthermore, the second 2~ f s(; ; ; ep ~;) can be epected to be quite small for ~ in a large neighbourhood around 0 if on the micro-level for every the inc w(p s ;r s ; ; ) (6) is approimately constant for in a large neighborhood of. Consequently, if one assumes property (6) then it is justied to take a Taylor epansion in log instead of. We remark that property (6) is well supported by empirical evidence if it is applied to the regression function w s (p s ;r s ;;a)= R w(p s ;r s ;;)d s j(; a). By denition s ~ [f s (ep ~ ; ; ; ; )] ~ =0 ~ s w(ep ~ p s;p 2 s;r s ;;)d s = h c( p s;p 2 s;r s ;;) i = d s ~ =0 Analogously for the coecients s and s. Q.E.D. 3.2 Discussion of the coecients The coecient s : By denition of s in Proposition 2 we obtain " s MMP s () s ()d m s () IR + where MMP s () := c(p s ;r s ;;)d s j, i.e., the mean marginal propensity to consume of all households with income. 22 # =

25 The sign of s depends on the form of the function MMP s (). Indeed, it follows from the Lemma in the Appendi that for every density s the coecient s 0( 0) if MMP s () is a decreasing (increasing) function in More generally, s 0 if for all z in a neighborhood of, min MMP s() ma MMP s(): 0z z If the individual household propensity to c(p s ;r s ;;) is decreasing in, i.e. c(p s ;r s ;;) is a concave function in { which is frequently postulated for the micro-model { then it does not necessarily follow that the MMP s () is also decreasing in since the conditional distribution s j depends on. For eample, let s j be equal to with probability () and 2 with probability, (). Let c(p s ;r s ;; i ) be linear in c(p s ;r s ;; ) c(p s ;r s ;; 2 ). Then the MMP () can be decreasing, increasing or not be monotone at all depending on the function (). Figure 6 shows three eamples of the function (). Figure 6 income Figure 7 shows for each function the mean marginal propensity to consume MMP (). 23

26 marginal c(p s ;r s ; ; ) to c(p s ;r s ; ; 2 ) Figure 7 income Thus we have shown that the sign of s does not only depend on suitable properties of the micro-model but also on the composition of the population. In particular, linearity in income on the household level does not necessarily imply that the distribution eect of income is positive or negative nor that it can be neglected. The magnitude of the coecient s depends not only on the form of MMP s (), but also on the income distribution s ; in particular, the magnitude of s depends on the dispersion s. This is best illustrated by an eample: let MMP s () =d s + b s log and s a log normal density with parameters (m s ; s ). Then one can eplicitly compute the coecient s and obtains s =2 2 s b s. How can one estimate the coecient s? If one conditions on household attributes one might epect (or assume) that the conditional distribution s j(; a) does not depend very sensitively on, more c(p s ;r s ;;)d s j(; c(p s;r s ;;)d s j(; a) (?) c s (p s ;r s ;;a) If this condition is satised one can dene the following proi for s : " # s m s () c s (p s ;r s ;;a)d s j s ()d A Indeed, if condition (?) holds with equality, then s = s. The important point now is that s can be estimated from cross-section data in period s. For estimating s one does not need a micro-model c, only the actual consumption decisions in period s are needed. 24 =

27 We have estimated the coecient t (stratication with respect to household size and age in 5 8 disjoint groups) using the data from the U.K. Family Ependiture Survey. The mean value of t for the years is,0:036 for consumption ependiture on non-durables;,0:009 for food and +0:006 for services. The method of estimation and further details can be found in Hildenbrand and Kneip (997). The coecient s : [w(p s ;r s ; ; )] = c(p s ;r s ;;), c(p s;r s ;;) one obtains from the denition of s that s = (@ c(p s ;r s ;;), c(p s ;r s ;;))d s j s IR + c(p s ;r s ;;)d s, C s = s s Hence, the coecient s is negative (positive) if in average is negative (positive). (@ c(p s ;r s ;;), c(p s ;r s ;;)) d s j s ()d Note that the smaller the individual household marginal propensity to c(p s ;r s ;;), the larger is, s. In the etreme case, c(p s ;r s ;;) 0 one obtains s, Cs s. As in the case of the coecient s one can dene a proi s for the coecient s which can be estimated from cross-section data in period s; s := s IR + = s IR + A [w s (p s ;r s ; ; a] = d s c s (p s ;r s ;;a)d s j s ()d s ()d, C s s As before, s is a proi for s provided condition (?) is satised. For estimating s one does not need a micro-relation. Estimates of the coecient s (stratication with respect to household size and age in 5 8 disjoint groups) for the years yield a mean value of,0:242 for consumption ependiture on non-durables;,0:52 for food and +0:022 for services. For details see Hildenbrand and Kneip (997). 25

28 The coecients s ; s and s The interpretation of the coecient s ; s and s can be short. By denition they depend on how, on average, households react to changes in the price level or interest rate. The coecient s = [C s ( p s;p 2 s;r s ; s )] = depends on the price-elasticity of demand. For eample, if, for every income level, demand is zero-elastic { an implausible situation { then one obtains s = C s = s. On the other hand, if an increase in the price level leads, on average, to a reduction of demand, yet to an unchanged ependiture, i.e. unit-elastic demand, then one obtains s =0. The coecient s = [C s (p s; p 2 s;r s ; s )] = depends on the cross-elasticity of demand with respect to an increase in the price-level for all commodities other than those that are included in ependiture. The coecient s = r [C s (p s;p 2 s;r r s ; s )] r= is often considered as negligible. Yet at the level of an individual household a change in the current interest rate might well have a non-negligible eect. For eample, analyzing the standard intertemporal decision problem of a household shows that a change in the interest rate matters; yet the eect is indeterminate. An increase in the current interest rate can lead to either an increase or a decrease in current consumption ependiture depending on the intertemporal utility function, the epectation function for future interest rates and asset holdings. Therefore, it might well happen that on average over the population the eect of a change in the current interest rate is negligible, hence s = 0 even though at the individual level it matters. 26

29 4 Aggregation under slowly changing attribute distributions The last section was based on the hypothesis of structural stability not only of household characteristics (that is, Hypothesis ) but also of household attributes (that is, Hypothesis 3 + ). As mentioned already in section 3 this last hypothesis is in contradiction with empirical facts: indeed, the distributions of household attributes typically change over time; this change, however, is quite slow (see H-K-U (997)). Therefore we base this section on Hypothesis 3 in a less restrictive version than in section 3. The distribution t in period t has been decomposed in the conditional distribution of household characteristics t j(; a) ; the conditional distribution of household attributes t j and the income density t () : Hypothesis 2, i.e., the time-invariance of the standardized log income distribution, implies (see (8) and (9) of section 2) t () =' 0 () s ('()) ; where '() := m s( t= s) s= t. Hypothesis, i.e., structural stability of household characteristics, epresses that t j(; a) = s j('();a) : Instead of assuming t j = s j'() as in section 3, we now assume Hypothesis 3, that is, for periods s and t that are close to each other the dierence t j, s j'() is \small", but not necessarily negligible. We shall eplain in the sequel (see the approimation (8)) in which sense the dierence t j, s j'() should be small. By denition C t = IR+ A c(p t ;r t ;;)d t j(; a) d t j t ()d : 27

30 Since by assumption t j(; a) = s j('();a) and t () =' 0 () s ('()) we obtain C t = IR+ A c(p t;r t ;;)d s j('();a) d t j ' 0 () s ('())d : Substituting t j = s j'()+( t j, s j'()) yields by Proposition where A := IR+ A C t = IR + c(p t ;r t ;', ();)d s + A (7) c(p t ;r t ;;)d s j('();a) d( t j, s j'()) ' 0 () s ('())d : For the rst term on the right hand side of (7) we have given an approimation in Proposition 2 of section 3. We shall now develop an approimation for the second term, that is to say, for the integral which is denoted by A. Substituting = '() yields A = c(p t ;r t ;', ();)d s j(;a) n t j', (), s j o s ()d A = Since ', () = IR+ " A ( t m s () m s( t= s) t=s w(p t ;r t ; we obtain m s () ;)d s j(;a) ) d!# t j m s (), s j s ()d where = t = s and w(p t ;r t ;;)= c(p t;r t ;;). We now want to develop an approimation for A. If t = s ; = s ;p t =p s and r t =r s tend to then tends to ( w(p t ;r t ; m s () ;)d s j(;a) w(p s ;r s ;;)d s j(;a) =c s (p s ;r s ;;a) Note, however, that f t j t m s(), s jg does not necessarily tend to zero. At this point we make use of Hypothesis 3: for periods s and t that are close to each other, f t j t m s(), s jg will be suciently small in order to justify the following approimation: A A := IR + " t s A c s (p s ;r s ;;a) d 28 ) t j m s (), s j!# s ()d (8)

31 Thus, in this section the term \approimate" in Hypothesis 3 has to be interpreted as implying A A. To evaluate A we assume that the set of household attributes is nite A = fa ;::: ;a m g and use the following notation: Furthermore we make the following s jfa i g =: i s() and s fir + a i g =: i s: Hypothesis 4: For every s and t that are in the same percentile position in period s and t, respectively, t( i t ) s( i s ) i t s i Let us recall, Hypothesis 3 is used to justify the approimation A A and Hypothesis 4 is a technical assumption which allows to evaluate A. Indeed, A = s IR + " m i= c s (p s ;r s ;;a i ) t( i m s () ), s() i Since P m i= t() i = and P m i= s() i =we obtain m Hence A = s IR + " m i= Now Thus, " i= ~c s (p s ;r s ;) t( i m s () ), s() i! =0!# s ()d: [c s (p s ;r s ;;a i ), ~c s (p s ;r s ;)][t( i m s () ), s()] i t( i m s () ), s() i # = A = 2 4 i t( t m s() ) i t i t, i s() s i = i s() i s = i s() m i= i s 29 i t, i s t i s i t i s i,!!, : i s 3 5 # s ()d

32 where i s = s IR + [c s (p s ;r s ;;a i ), ~c s (p s ;r s ;)] i s() s ()d: Note that the coecient i s can be estimated from cross-section data. The coecient i s measures to what etent the Engel curve of the subpopulation consisting of all households with attribute a i, i.e., c s (p s ;r s ; ;a i ) diers (on average) from the Engel curve of the whole population, i.e., ~c s (p s ;r s ; ). Consequently, i s t i s i describes the eect (on C t = ) of the changing composition of the population with respect to the attribute a i. In summary, in this section we etended Proposition 2 and derived the following approimation:,! C t =, C s = s change in the aggregate consumption ratio s log t s + s log t s eect of the changing income dispersion eect of mean (nominal) income growth + s log t s + s log 2 t 2 s + s log rt r s eects of price-ination eect of interest rate changes + P m i= i s i t i s, eect of the changing distribution of attributes 30

33 Appendi Lemma: For every continuous and decreasing function v of IR + every density on IR + such that into IR and A() := R v()()d R ()d is dened on an open interval around =if follows [A()] = 0 Proof: In order to prove the assertion of the lemma it suces to show that for all with 0 <, R v()()d R ()d R v()()d R ()d () Let m() := R m(), ()d and z() :=. Then, since <, m() m() m() if 0 z() (2) and Since R m() ()d = R z() 0 " m(), m() m() and by relations (2) and (3), z() 0 m(), m() m() ()d =, one obtains m() " # ()d =, z() ()d =, if z() <; : (3) z() m(), m(), m() # ()d; ; m() ()d : (4) Relation () holds, if and only if for all with 0 < one has R() := m(),! v()()d 0: (5) m() 3

34 However, (5) is an immediate consequence of (4) and the assumption on v. Indeed, z() R() = 0 m(),! v()()d + m() z() m(),! v()()d m() " min v() 0z(), " # z() 0 # ma v() z()< m(), m() ()d z() m(), ()d =0 m() Q.E.D. 32

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