- Email: [email protected]
Distribution movements, macroeconomic regularities and the representative consumer
Ricerche
Economiche
(1993) 47, MS199
Distribution movements, macroeconomic and the representative consumer ARTHUR
regularities
LEWBEL
Department of Economics, MA 02254, U.S.A.
Brandeis
University,
Waltham,
Summary The behaviour of agents and the distribution of agents play equivalent and joint roles in determining macroeconomic reiationships. A general characterization of the joint restrictions on preferences and distributions of consumers that rationalize a representative consumer is provided. Linear Engel curves and social welfare maximizing income distributions are shown to be polar examples of this general characterization. Joint restrictions that lie between these extremes, employing regularities in both consumers’ preferences and the distribution of income, can explain the empirical observation that macroeconomic demands resemble a non-linear representative consumer. Similar joint regularities in behaviour and distributions are likely to account for many observed macroeconomic regularities. J.E.L. Classification: C43, Dll, ElO. Keywords: Aggregation, representative distribution.
1. Behaviour regularities
and distributions
consumer,
in determining
income
aggregate
The literature on aggregation largely focuses on aggregate implications of microeconomic relationships. Representative agent models assume equality of micro and aggregate models, and the exact aggregation literature constructs aggregate models by summing linear or quasi-linear micro-models. In both cases the roles of micro functional forms rather than the distribution of agents is stressed in determining macroeconomic relationships. More attention should be placed on modelling the distribution of agents. In fact, the roles of distribution and micro functional form are equivalent in determining aggregate relationships; and many observed regularities in macroeconomic data may be due more to stability in the relative distributions of agents than to specific micro functional forms. 189 00355054/93/020189
+ 11 sos.oo/o
0 1993 University
of Venice
A. LEWBEL
190
To see the equivalence of micro functional form and agent distribution in determining aggregate relationships, let y =g(x,Z) denote a microeconomic relationship where y and x are vectors of variables relating to individual agents and 2 is a vector of macroeconomic variables. For example, y could be demands for goods by a household, x could be income and demographic composition variables for the household, and 2 could include the vector of prices for goods that all households face, making g a system of household demand equations. Let f(x,Z) denote the density function of the distribution of x in the population, which in any time period can depend on the values of macroeconomic variables 2. In the above example, 2 could include not only prices but mean income in the population and other statistics of the distribution of X. Let Y= E(y) be the mean level of y in the population. Then by definition the macroeconomic relationship Y= G(Z) is given by X, with the integration being over all posY= G(Z) = j&,-WW)d sible values of X. In the example, Y= G(Z) is the aggregate demand system, relating aggregate demands Y to prices, mean income, etc. See Stoker (1984) for further implications of this form of aggregation. x sh ows the equivalent roles that The equation Y = lg(x,Z)f(x,Z)d micro functional form and distribution play, in that if g was the density function and f was the micro functional form, the same aggregate model Y= G(Z) would result. The relevance of this observation is that stable macroeconomic relationships may be due as much to stability (over time) in the distribution function f as to specific microeconomic functional forms g.
2. Why consider
a representative
consumer?
Define a “rationalization” of a representative agent model to be a set of conditions concerning individual behaviours and distributions that, if they held, would result in aggregate data that behaved as if it were derived from a representative agent model. What is the point of rationalizing representative consumer models? Why bother deriving conditions under which aggregate demands in an economy will equal those of a utility maximizing representative consumer? The usual reason given is that macroeconomists often assume a representative consumer in their models, so these conditions provide a foundation or rationalization of the macroeconomists’ models. There are two flaws in this argument. First, macroeconomic models would probably benefit from dropping the representative consumer assumption. Kirman (1992) summarizes many of the arguments for abandoning representative agent assumptions in macro-models. Second, modern macro-models generally deal with
DISTRIBUTION
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dynamic effects and posit representative consumers within the context of dynamic maximization and uncertainty about the future. The vast majority of work that has been done rationalizing (or refuting) representative consumer models employs static consumer allocation analysis without uncertainty. Much of the current aggregation literature is ignored by macroeconomists, and will continue to be ignored as long as the focus of aggregation work remains on static analyses. Stoker’s (1986) finding that aggregation errors can induce apparent dynamics in aggregate data, Granger’s (1980) result that AR(l) individuals can aggregate into long-term memory processes, and Lewbel’s (1992) observation that moments of the distribution of individuals’ dynamic behaviour can be recovered from aggregate dynamics are rare examples of theoretical aggregation results that relate directly to macroeconomic dynamic issues. If rationalizations of static representative consumer models are of limited use for macroeconomists, then why bother generating them? One reason is historical, in that the issue of the representative consumer has a long pedigree. I believe the most compelling reason for rationalizing static representative consumer models, though only rarely stated, is that representative consumer models often empirically fit aggregate (economy-wide) demand data quite well. Some aggregate empirical studies (e.g. Christ,ensen, Jorgenson & Lau, 1975; Gallant, 1981) reject representative consumer attributes like homogeneity and Slutsky symmetry, although the rejections are not as economically significant as would be expected from aggregation theory. In contrast, the non-parametric methods employed by Diewert and Parkan (1978), Varian (1983), and Manser and McDonald (1988) do not reject the restrictions implied by a utility maximizing representative consumer. Lewbel (1988, 1991) shows that the fit of a representative consumer model is only slightly poorer than the fit of exact aggregation-based models using U.S.A. and U.K. data. Representative consumer models fit quite well, especially when they are applied to groups of consumers that are relatively homogeneous (in terms of major demographic attributes like family size) rather than to the entire population of a country, or when a few demographic variables (such as, for instance, percentage of the population that live alone) are included in the model. A good fit also requires a relatively flexible specification of price and income effects, e.g. quasi-homothetic models tend to be rejected. This principal motivation for looking at conditions that rationalize representative agent models has two implications. First, the empirical fit says nothing about welfare, i.e. the utility function of the data-based representative consumer may have no relationship to any sensible measure of welfare in the economy. For example, Jerison (1984) showed that in a choice model framework a utility
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maximizing representative agent could choose an alternative that is not favoured by any of the individuals comprising the economy. The empirical motivation for rationalizing a static representative consumer also suggests that the standard linear (quasi-homothetic) Engel curve aggregation results of Gorman (1953) and variants thereof are not very useful, because empirically both individual consumer and aggregate demands appear to have non-linear Engel curves, as is evidenced by the popularity of nonlinear demand systems like the AIDS and Translog models. Empirical evidence for non-linear Engel curves goes back at least to Leser (1963). Existing empirical results generate a mild paradox. Aggregate demands empirically resemble a representative consumer with non-linear Engel curves, but standard (Gorman type) aggregation theory says that aggregate demands can only resemble a representative consumer if that representative consumer has linear Engel curves. The resolution of this paradox is that theorems requiring linearity do so because they place no restrictions on how the distribution of income may change over time. Theorems that show how income redistributions can produce non-representative consumer behaviour in many settings [Kirman and Koch (1986) is a strong example] are not empirically relevant, unless the kinds of income redistributions they posit actually occur. The relative distribution of income tends to be quite stable over time, at least in industrialized countries, and this regularity helps to explain the paradox of the observed non-linear representative consumer. To illustrate this point, let xi, equal the income or total consumption expenditures of agent i in time t, and let X,=E(x,) equal the average (per capita) income (or consumption expenditures) in the economy in time t, so E denotes averaging across agents. Let In denote the natural logarithm, and define a, by a,= {E(xi, In xi& X,} - In X,. Lewbel(l991) shows that for the U.S.A. and the U.K., a, is very close to constant over time. Therefore, if individual agents have budget shares linear in lnxi, (e.g. the popular AIDS or Translog models), aggregate, macroeconomic budget shares will be approximately linear in lnX,. This result helps to explain the paradox. See Lewbel (1991) for a more complete resolution. Theorems like Gorman’s that hold for all income distributions must require linearity and other excessive restrictions. At the other extreme, Hardle, Hildenbrand and Jerison (1991) Grandmont (1991) and others show that uncompensated aggregate demand curves are downward sloping if agents are sufficiently heterogeneous. Results like these, which depend only on distribution conditions and not on attributes of individual behaviour, are in their own way as inherently limited as results that depend only on linearity of individual behaviour.
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For applied work, the most useful aggregation theorems must exploit observed empirical regularities both in agent behaviour and in the distributions of agents. To do otherwise discards valuable information.
3. Joint
restrictions
on preferences
and distributions
Suppose all consumers in an economy maximize their own utility functions, and all face the same vector of prices P. The aim here is to characterize the most general conditions under which aggregate demands in the economy will equal those of a utility maximizing representative consumer. The conditions consist of joint restrictions on the behaviour of individual consumers and on the distribution of consumers. The idea is to apply aggregation structures of the form described in Section 1 to the representative consumer problem discussed in Section 2. Throughout, sufficient regularity to permit taking derivatives and passing them through integrals is assumed. Let x=c(u,P,a) equal the total expenditures (income for short) required by a consumer having attributes a and prices P to attain utility level U, so c is what is called a cost or expenditure function. Attributes a can include both direct observables like demographic attributes of the consumer (e.g. age, race, or family size) as well as idiosyncratic personal tastes and preferences. The price vector P consists of prices Pj for each good or service j. Let G(x,ai U,P,B) denote the joint distribution function of total expenditures x: and attributes a in the population, given prices P. For now, the scalar U and vector B are just parameters that are assumed to completely parameterize this distribution function G. Define the function F by F(u,al U,P,B) = G[c(u,P,a),ai U,P,B]. The function F is the joint distribution function of attained utility levels u and attributes a in the population, given prices P and parameters U and B. Let X= E(x) denote the average total expenditures in the population. Define the function C(U,P,B) by C(U,P,B)=X=SxdG(x,alU,P,B)=Sc(u,P,a)dF(u,alU,P,B).
(1)
If we define the scalar U to be the attained utility level of the representative consumer, then the function C can be defined as the representative consumer’s cost function, since it equals average total expenditures in the population. The vector of distribution parameters B are interpreted as attributes of the representative consumer’s cost function, just as a denotes attributes of the cost functions of individual consumers. So far, everything presented has been just notation and definitions. No restrictions have yet been placed on the behaviour or
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distribution prices P. those of a terization properties
of agents, apart from existence of X and common The key observation is that aggregate demands equal utility maximizing representative consumer if a parameof F or G exists such that the function C has the of a cost function generating aggregate demands.
LEMMA: Aggregate demands equal those of a utility representative consumer if and only if
maximizing
(1) C is an increasing function of U, (2) C is homogeneous of degree one in P, (3) j c(u,P,a) d@.(u,al U,P,B) =O, where the function Qj is defined by @j(u,al U,P,B) = dF(u,al U,P,B)/aPj for each good or service j. PROOF: conditions (1) and (2) are defining features of cost functions. To derive (3), let q = h(u,P,a) equal the vector of quantities of goods and services purchased by an agent having utility level u, attributes a, and facing prices P, so the function h is the vector of Hicksian compensated demand functions. A property of cost functions is h(u,P,a)= dc(u,P,a)/aP (here the derivative denotes the gradient vector). Let Q = E(q), and define the function H by H( U,P,B) = X( U,P,B)/aP. Taking the gradient of equation (1) with respect to P gives
H(U,P,B) = j h(u,P,a) Wu,al WV) + { c(u,P,a)d@(w4 U,P,B),
(2)
where @(u,aj U,P,B) = aF(u,ai U,P,B)/dP. But note that j h(u,P,a) dF(u,al U,P,B) = E(q) = Q. H aving a utility maximizing representative consumer requires Q= H(U,P,B), and hence requires condition (3). The Lemma provides a characterization of the joint restrictions on individual behaviours c and the distribution of individuals F (or G) required for a representative consumer. Now consider the conditions in the Lemma in more detail. Condition (1) is that C be an increasing function of U. Recall that U is a parameter of the income (total expenditures) distribution and X= C is the mean of this distribution, so increases in the parameter U must correspond to increases in the mean X. U is, equivalently, a parameter of the distribution of utility across agents, via F. If the representative consumer’s utility level is to be used as a summary measure of welfare in the economy, then the Lemma must hold with U defined appropriately. For example, typical social welfare functions are U= inf(u) or U= E(u) across all agents, which corresponds to parameterizing the distribution F in terms of its minimum or mean value, along with other parameters B.
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Condition (2) is that C( U,P,B) be homogeneous of degree one in prices, or equivalently that H(U,P,B) be homogeneous of degree zero in prices. Given that individual consumers maximize utility, by equation (2) a sufficient condition for (2) to hold is that the distribution function F be homogeneous of degree zero in prices. Four alternative representations of condition (3) are: (3a) jxd@j{u(x,P,a),a~
U,P,B} = 0,
(3’)
V(X,P,‘)>P,‘l
Sxd@j{4~,J’,a),aI
=O,
(3~) H( U,P,B) = j h(u,P,a) dF(u,al U,P,B), and (34 ff{ VGWJW’,~~ = j Nu(x,P,a),P,al dG*(vdX,P,J9; where u(x,P,a) is the indirect utility function of individual agents, that is, u is the inverse function of c as a function of U; V(X,P,B) is the indirect utility function corresponding to C(U,P,B), and G*(x,alX,P,B) = G{x,ai V(X,P,B),P,B} is the distribution of x and a parameterized in terms of X, P, and B instead of U, P, 23. The representations (3~) and (3b) are derived by subsituting u and V for u and U directly into condition (3). Representations (3~) and (3d) are, respectively, direct Hicksian and Marshallian representations of the statement Q = E(q). A sufficient condition for both (2) and (3) to hold is that F be independent of P, i.e. that the distribution of utility u (and attributes a) across individuals does not depend on prices. Except in special cases (e.g. the PIGLOG example given later), having F independent of P corresponds to a social planner’s outcome, in which people’s incomes x change so as to keep the distribution of utility essentially fixed. In particular, if U is defined to equal some measure of social welfare and the distribution of income x is determined by maximizing social welfare, then F will be independent of P. This is a polar case in which no restriction other than utility maximization is placed .on the functional form c of individual behaviours, and the representative consumer arises entirely from restrictions on the distribution of x (and a) across consumers. The other polar extreme is Gorman’s linear parallel Engel curves, which yield a representative consumer for any distribution of x. In the framework of the Lemma, Gorman’s case defines U and c by U= E(u) = 1~ dF(u,ul U,P,B), and c(u,P,u) = a(P) +p(P)u for some functions a and j?. In this case
= a(P) a{1 dF(u,al U,P,B)}/dP+/3(P)d{Ju = a(P) a[l]/aP+p(P) a[q/aP=o,
dF(u,ui
U,P,B)}/dP
which directly confirms condition (3). The extension of the Gorman case to c(u,P,u) = u(P,u) +p(P)u is straightforward. More useful applications of the Lemma are cases in between these polar extremes, where empirically plausible restrictions are
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A. LEWBEL
placed on both the behaviour of agents and their distribution. For example, consider PIGLOG demand systems, which are demands having budget shares linear in In x (see Muellbauer, 1975, 1976). PIGLOG demands generally fit demand data better than other rank two models (see Lewbel, 1991), and the AIDS and Translog demand systems are examples of PIGLOG models. Drop attributes a now for simplicity. PIGLOG demands are c(u,P) = exp(a(P) +/?(P)u} f or some functions CI and j?. Define U, B, and the function C by X= C(U,P,B) = exp{B + a(P) +p(P)U} for some scalar B. Let the distribution function F be independent of P, and be such that equation (1) holds with these definitions of c and C. It is straightforward to check (by writing out equation (2) and converting from u and 7J to x and X) that this assumption about F is equivalent to assuming that E(x In x) = E(x) {In E(x) + B}. This is the empirically plausible example given in the previous section. An interesting feature of this example is that having F, the distribution of utility, independent of prices in this case, places restrictions on the distribution of income x that do not depend on prices. Apart from special constructions like these, having F independent of prices will typically place restrictions on the distribution of income that depend on prices. For a more general set of examples, consider Marshallian demands for individual consumers of the form Q =A,(P)+ A,(P)x+ w h ere A,, A,, and A, are vector-valued functions. ~,tOAd-W, Virtually all empirical studies of aggregate data that explicitly consider both aggregation and individual utility maximization have employed special cases of demands of this form. Demands of this form can have rank as high as three (see Lewbel, 1991). Define a vector of parameters B implicitly as the solution to v(X,P,B)= S~(x,P,a)dG*(x,alX,P,B). If a solution B exists then &= A,(P) + A,(P)X+ A,(P)ty(X,P,B), yielding a representative consumer. Lewbel (1989) essentially shows that a solution B, which does not depend on prices, is always possible when cy itself is a function that does not depend on prices. See also Lewbel (1988).
4. Conclusions
Kirman (1992) and others have argued that all representative agent paradigms should be discarded in macroeconomics, and replaced with models of economies composed of interacting agents. A less ambitious suggestion is that representative agent assumptions be replaced with models of individual behaviour combined with models of the distribution of agents. The argument that macro-models would be improved by abandoning representative agent assumptions is valid. Nevertheless, there is a persuasive reason for investigating conditions under
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which aggregate demands in an economy equal those of a utility maximizing representative consumer. This reason is the empirical observation that aggregate demands can in fact be well approximated by appropriate representative consumer models. However, theorems that rationalize an aggregate representative consumer via unrealistic assumptions (e.g. linear parallel Engel curves) are of limited value. Theorems showing why aggregate demands need not resemble a representative consumer also fail to address observed aggregate and distributional regularities. This paper provided a general characterization of the joint conditions on preferences and distributions of individuals required for aggregate demands to resemble those of a representative consumer. The well-known examples of linear Engel curves or presence of a social welfare maximizer are polar cases of this general characterization. Conditions that explain the empirically observed representative consumer lie in-between these polar extremes, in which regularities in the joint distribution of income and attributes across individuals and regularities in individual preferences combine to produce the observed outcome. Some attributes of a representative consumer are more empirically acceptable than others. For example, Slutsky symmetry tends to be empirically violated more often than is homogeneity of degree zero in income and prices (i.e. the absence of money illusion). Just as the results in the previous section describe joint restrictions on distributions and on preferences that yield a full representative consumer, one may seek analogous restrictions that yield just some attributes of a representative consumer. For example, Lewbel (1990) shows that if an individual consumer’s demands are homogeneous of degree zero in income and prices then aggregate demands will also be so if the distribution of income possesses a property called “Mean Scaling”. It is important to realize the limitations of the empirical representative consumer. First, while representative consumer models fit aggregate data reasonably well, models that explicitly consider and incorporate aggregation effects fit better (see Lewbel, 1991). As a result, forecasts based on representative consumer models will be only slightly inferior to those based on exact aggregation models, but only if the distribution regularities observed in the past are maintained in the future. In particular, policy implications drawn from representative consumer models may be badly biased if the contemplated policy affects the income distribution. Also, the utility level of the empirical representative consumer need not correspond, even approximately, to any sensible measure of social welfare, and therefore maximization of the representative consumer’s utility function might not constitute a rational economic policy objective. In summary, macroeconomists should not assume a representat-
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ive consumer. In general, models that are better (in terms of appropriate policy implications, empirical fit, and forecasting), but remain tractable, can be constructed by replacing the representative consumer with a combination of models of individual consumer behaviour and a model of the distribution of consumers. Nevertheless, from pure fit or forecast criteria, the error in an appropriately specified representative consumer (e.g. one that is PIGLOG rather than homothetic as macroeconomists commonly assume for convenience) is not large. This is an empirical regularity that appropriately constructed models should be able to both explain and exploit.
Acknowledgements
This research was supported in part by the National Science Foundation through grant SES-9011806. I would like to thank Chuck Blackorby and Mike Jerison for their comments on preliminary stages in the work in Section 3.
References Christensen, L.R., Jorgenson, D.W. & Lau, L.J. (1975). Transcendental logarithmic utility functions. American Economic Review, 65, 367-383. Diewert, W.E. & Parkan, C. (1978). Tests for the consistency of consumer data and nonparametric index numbers. Working Paper 78-27, University of British Columbia. Gallant, A.R. (1981). On the bias in flexible functional forms and an essentially unbiased form. Journal of Econometrics, 15, 211-245. Gorman, W.M. (1953). Community preference fields. Econometrica, 21, 63380. Grandmont, J.M. (1991). Transformations of the commodity space, behavioural heterogeneity, and the aggregation problem. Unpublished draft, CEPREMAP, Paris. Granger, C. (1980). Long memory relationships and the aggregation of dynamic models. Journal of Econometrics, 14, 227-238. Hardle, W., Hildenbrand, W. & Jerison, M. (1991). Empirical evidence on the law of demand. Econometrica, 59, 152551550. Jerison, M. (1984). Social welfare and the unrepresentative consumer. Discussion paper, State University of New York at Albany. Kirman, A.P. (1992). Whom or what does the representative individual represent? Journal of Economic Perspectiues, 6, 117-136. Kirman, A.P. & Koch, K. (1986). Market excess demand functions in exchange economies: identical preferences and collinear endowments. Review of Economic Studies, 55, 457-463. Leser, C.E.V. (1963). Forms of Engel functions. Econometrica, 31, 694-703. Lewbel, A. (1988). Exact aggregation, distribution parameterizations, and a nonlinear representative consumer. In G.F. Rhodes & T.B. Fomby, Eds. Advances in Econometrics, 7, 253-290. Greenwich: JAI Press. Lewbel, A. (1989). Exact aggregation and a representative consumer. Quarterly Journal of Economics, 104, 622-633.
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Lewbel, A. (1990). Income distribution movements and aggregate money illusion. Journal of Econometrics, 43, 35-42. Lewbel, A. (1991). The rank of demand systems: theory and nonparametric estimation. Econometrica, 59, 711-730. Lewbel, A. (1992). Aggregation and simple dynamics. Working paper, Brandeis University, MA. Manser, M.E. & McDonald, R.J. (1988). An analysis of substitution bias in measuring inflation, 19591985. Econometrica, 56, 909-930. Muellbauer, J. (1975). Aggregation, income distribution, and consumer demand. Review of Economic Studies, 62, 269-283. Meullbauer, J. (1976). Community preferences and the representative consumer, Econometrica, 14, 525-543. Stoker, T.M. (1984). Completeness, distribution restrictions, and the form of aggregate functions. Econometrica, 52, 887-908. Stoker, T.M. (1986). Simple tests of distributional effects on macroeconomic equations. Journal of Political Economy, 94, 763-795. Varian, H.R. (1983). Non-parametric tests of consumer behavior. Review of Economic Studies, 50, 99110.
Economiche
(1993) 47, MS199
Distribution movements, macroeconomic and the representative consumer ARTHUR
regularities
LEWBEL
Department of Economics, MA 02254, U.S.A.
Brandeis
University,
Waltham,
Summary The behaviour of agents and the distribution of agents play equivalent and joint roles in determining macroeconomic reiationships. A general characterization of the joint restrictions on preferences and distributions of consumers that rationalize a representative consumer is provided. Linear Engel curves and social welfare maximizing income distributions are shown to be polar examples of this general characterization. Joint restrictions that lie between these extremes, employing regularities in both consumers’ preferences and the distribution of income, can explain the empirical observation that macroeconomic demands resemble a non-linear representative consumer. Similar joint regularities in behaviour and distributions are likely to account for many observed macroeconomic regularities. J.E.L. Classification: C43, Dll, ElO. Keywords: Aggregation, representative distribution.
1. Behaviour regularities
and distributions
consumer,
in determining
income
aggregate
The literature on aggregation largely focuses on aggregate implications of microeconomic relationships. Representative agent models assume equality of micro and aggregate models, and the exact aggregation literature constructs aggregate models by summing linear or quasi-linear micro-models. In both cases the roles of micro functional forms rather than the distribution of agents is stressed in determining macroeconomic relationships. More attention should be placed on modelling the distribution of agents. In fact, the roles of distribution and micro functional form are equivalent in determining aggregate relationships; and many observed regularities in macroeconomic data may be due more to stability in the relative distributions of agents than to specific micro functional forms. 189 00355054/93/020189
+ 11 sos.oo/o
0 1993 University
of Venice
A. LEWBEL
190
To see the equivalence of micro functional form and agent distribution in determining aggregate relationships, let y =g(x,Z) denote a microeconomic relationship where y and x are vectors of variables relating to individual agents and 2 is a vector of macroeconomic variables. For example, y could be demands for goods by a household, x could be income and demographic composition variables for the household, and 2 could include the vector of prices for goods that all households face, making g a system of household demand equations. Let f(x,Z) denote the density function of the distribution of x in the population, which in any time period can depend on the values of macroeconomic variables 2. In the above example, 2 could include not only prices but mean income in the population and other statistics of the distribution of X. Let Y= E(y) be the mean level of y in the population. Then by definition the macroeconomic relationship Y= G(Z) is given by X, with the integration being over all posY= G(Z) = j&,-WW)d sible values of X. In the example, Y= G(Z) is the aggregate demand system, relating aggregate demands Y to prices, mean income, etc. See Stoker (1984) for further implications of this form of aggregation. x sh ows the equivalent roles that The equation Y = lg(x,Z)f(x,Z)d micro functional form and distribution play, in that if g was the density function and f was the micro functional form, the same aggregate model Y= G(Z) would result. The relevance of this observation is that stable macroeconomic relationships may be due as much to stability (over time) in the distribution function f as to specific microeconomic functional forms g.
2. Why consider
a representative
consumer?
Define a “rationalization” of a representative agent model to be a set of conditions concerning individual behaviours and distributions that, if they held, would result in aggregate data that behaved as if it were derived from a representative agent model. What is the point of rationalizing representative consumer models? Why bother deriving conditions under which aggregate demands in an economy will equal those of a utility maximizing representative consumer? The usual reason given is that macroeconomists often assume a representative consumer in their models, so these conditions provide a foundation or rationalization of the macroeconomists’ models. There are two flaws in this argument. First, macroeconomic models would probably benefit from dropping the representative consumer assumption. Kirman (1992) summarizes many of the arguments for abandoning representative agent assumptions in macro-models. Second, modern macro-models generally deal with
DISTRIBUTION
MOVEMENTS
191
dynamic effects and posit representative consumers within the context of dynamic maximization and uncertainty about the future. The vast majority of work that has been done rationalizing (or refuting) representative consumer models employs static consumer allocation analysis without uncertainty. Much of the current aggregation literature is ignored by macroeconomists, and will continue to be ignored as long as the focus of aggregation work remains on static analyses. Stoker’s (1986) finding that aggregation errors can induce apparent dynamics in aggregate data, Granger’s (1980) result that AR(l) individuals can aggregate into long-term memory processes, and Lewbel’s (1992) observation that moments of the distribution of individuals’ dynamic behaviour can be recovered from aggregate dynamics are rare examples of theoretical aggregation results that relate directly to macroeconomic dynamic issues. If rationalizations of static representative consumer models are of limited use for macroeconomists, then why bother generating them? One reason is historical, in that the issue of the representative consumer has a long pedigree. I believe the most compelling reason for rationalizing static representative consumer models, though only rarely stated, is that representative consumer models often empirically fit aggregate (economy-wide) demand data quite well. Some aggregate empirical studies (e.g. Christ,ensen, Jorgenson & Lau, 1975; Gallant, 1981) reject representative consumer attributes like homogeneity and Slutsky symmetry, although the rejections are not as economically significant as would be expected from aggregation theory. In contrast, the non-parametric methods employed by Diewert and Parkan (1978), Varian (1983), and Manser and McDonald (1988) do not reject the restrictions implied by a utility maximizing representative consumer. Lewbel (1988, 1991) shows that the fit of a representative consumer model is only slightly poorer than the fit of exact aggregation-based models using U.S.A. and U.K. data. Representative consumer models fit quite well, especially when they are applied to groups of consumers that are relatively homogeneous (in terms of major demographic attributes like family size) rather than to the entire population of a country, or when a few demographic variables (such as, for instance, percentage of the population that live alone) are included in the model. A good fit also requires a relatively flexible specification of price and income effects, e.g. quasi-homothetic models tend to be rejected. This principal motivation for looking at conditions that rationalize representative agent models has two implications. First, the empirical fit says nothing about welfare, i.e. the utility function of the data-based representative consumer may have no relationship to any sensible measure of welfare in the economy. For example, Jerison (1984) showed that in a choice model framework a utility
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maximizing representative agent could choose an alternative that is not favoured by any of the individuals comprising the economy. The empirical motivation for rationalizing a static representative consumer also suggests that the standard linear (quasi-homothetic) Engel curve aggregation results of Gorman (1953) and variants thereof are not very useful, because empirically both individual consumer and aggregate demands appear to have non-linear Engel curves, as is evidenced by the popularity of nonlinear demand systems like the AIDS and Translog models. Empirical evidence for non-linear Engel curves goes back at least to Leser (1963). Existing empirical results generate a mild paradox. Aggregate demands empirically resemble a representative consumer with non-linear Engel curves, but standard (Gorman type) aggregation theory says that aggregate demands can only resemble a representative consumer if that representative consumer has linear Engel curves. The resolution of this paradox is that theorems requiring linearity do so because they place no restrictions on how the distribution of income may change over time. Theorems that show how income redistributions can produce non-representative consumer behaviour in many settings [Kirman and Koch (1986) is a strong example] are not empirically relevant, unless the kinds of income redistributions they posit actually occur. The relative distribution of income tends to be quite stable over time, at least in industrialized countries, and this regularity helps to explain the paradox of the observed non-linear representative consumer. To illustrate this point, let xi, equal the income or total consumption expenditures of agent i in time t, and let X,=E(x,) equal the average (per capita) income (or consumption expenditures) in the economy in time t, so E denotes averaging across agents. Let In denote the natural logarithm, and define a, by a,= {E(xi, In xi& X,} - In X,. Lewbel(l991) shows that for the U.S.A. and the U.K., a, is very close to constant over time. Therefore, if individual agents have budget shares linear in lnxi, (e.g. the popular AIDS or Translog models), aggregate, macroeconomic budget shares will be approximately linear in lnX,. This result helps to explain the paradox. See Lewbel (1991) for a more complete resolution. Theorems like Gorman’s that hold for all income distributions must require linearity and other excessive restrictions. At the other extreme, Hardle, Hildenbrand and Jerison (1991) Grandmont (1991) and others show that uncompensated aggregate demand curves are downward sloping if agents are sufficiently heterogeneous. Results like these, which depend only on distribution conditions and not on attributes of individual behaviour, are in their own way as inherently limited as results that depend only on linearity of individual behaviour.
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For applied work, the most useful aggregation theorems must exploit observed empirical regularities both in agent behaviour and in the distributions of agents. To do otherwise discards valuable information.
3. Joint
restrictions
on preferences
and distributions
Suppose all consumers in an economy maximize their own utility functions, and all face the same vector of prices P. The aim here is to characterize the most general conditions under which aggregate demands in the economy will equal those of a utility maximizing representative consumer. The conditions consist of joint restrictions on the behaviour of individual consumers and on the distribution of consumers. The idea is to apply aggregation structures of the form described in Section 1 to the representative consumer problem discussed in Section 2. Throughout, sufficient regularity to permit taking derivatives and passing them through integrals is assumed. Let x=c(u,P,a) equal the total expenditures (income for short) required by a consumer having attributes a and prices P to attain utility level U, so c is what is called a cost or expenditure function. Attributes a can include both direct observables like demographic attributes of the consumer (e.g. age, race, or family size) as well as idiosyncratic personal tastes and preferences. The price vector P consists of prices Pj for each good or service j. Let G(x,ai U,P,B) denote the joint distribution function of total expenditures x: and attributes a in the population, given prices P. For now, the scalar U and vector B are just parameters that are assumed to completely parameterize this distribution function G. Define the function F by F(u,al U,P,B) = G[c(u,P,a),ai U,P,B]. The function F is the joint distribution function of attained utility levels u and attributes a in the population, given prices P and parameters U and B. Let X= E(x) denote the average total expenditures in the population. Define the function C(U,P,B) by C(U,P,B)=X=SxdG(x,alU,P,B)=Sc(u,P,a)dF(u,alU,P,B).
(1)
If we define the scalar U to be the attained utility level of the representative consumer, then the function C can be defined as the representative consumer’s cost function, since it equals average total expenditures in the population. The vector of distribution parameters B are interpreted as attributes of the representative consumer’s cost function, just as a denotes attributes of the cost functions of individual consumers. So far, everything presented has been just notation and definitions. No restrictions have yet been placed on the behaviour or
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distribution prices P. those of a terization properties
of agents, apart from existence of X and common The key observation is that aggregate demands equal utility maximizing representative consumer if a parameof F or G exists such that the function C has the of a cost function generating aggregate demands.
LEMMA: Aggregate demands equal those of a utility representative consumer if and only if
maximizing
(1) C is an increasing function of U, (2) C is homogeneous of degree one in P, (3) j c(u,P,a) d@.(u,al U,P,B) =O, where the function Qj is defined by @j(u,al U,P,B) = dF(u,al U,P,B)/aPj for each good or service j. PROOF: conditions (1) and (2) are defining features of cost functions. To derive (3), let q = h(u,P,a) equal the vector of quantities of goods and services purchased by an agent having utility level u, attributes a, and facing prices P, so the function h is the vector of Hicksian compensated demand functions. A property of cost functions is h(u,P,a)= dc(u,P,a)/aP (here the derivative denotes the gradient vector). Let Q = E(q), and define the function H by H( U,P,B) = X( U,P,B)/aP. Taking the gradient of equation (1) with respect to P gives
H(U,P,B) = j h(u,P,a) Wu,al WV) + { c(u,P,a)d@(w4 U,P,B),
(2)
where @(u,aj U,P,B) = aF(u,ai U,P,B)/dP. But note that j h(u,P,a) dF(u,al U,P,B) = E(q) = Q. H aving a utility maximizing representative consumer requires Q= H(U,P,B), and hence requires condition (3). The Lemma provides a characterization of the joint restrictions on individual behaviours c and the distribution of individuals F (or G) required for a representative consumer. Now consider the conditions in the Lemma in more detail. Condition (1) is that C be an increasing function of U. Recall that U is a parameter of the income (total expenditures) distribution and X= C is the mean of this distribution, so increases in the parameter U must correspond to increases in the mean X. U is, equivalently, a parameter of the distribution of utility across agents, via F. If the representative consumer’s utility level is to be used as a summary measure of welfare in the economy, then the Lemma must hold with U defined appropriately. For example, typical social welfare functions are U= inf(u) or U= E(u) across all agents, which corresponds to parameterizing the distribution F in terms of its minimum or mean value, along with other parameters B.
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Condition (2) is that C( U,P,B) be homogeneous of degree one in prices, or equivalently that H(U,P,B) be homogeneous of degree zero in prices. Given that individual consumers maximize utility, by equation (2) a sufficient condition for (2) to hold is that the distribution function F be homogeneous of degree zero in prices. Four alternative representations of condition (3) are: (3a) jxd@j{u(x,P,a),a~
U,P,B} = 0,
(3’)
V(X,P,‘)>P,‘l
Sxd@j{4~,J’,a),aI
=O,
(3~) H( U,P,B) = j h(u,P,a) dF(u,al U,P,B), and (34 ff{ VGWJW’,~~ = j Nu(x,P,a),P,al dG*(vdX,P,J9; where u(x,P,a) is the indirect utility function of individual agents, that is, u is the inverse function of c as a function of U; V(X,P,B) is the indirect utility function corresponding to C(U,P,B), and G*(x,alX,P,B) = G{x,ai V(X,P,B),P,B} is the distribution of x and a parameterized in terms of X, P, and B instead of U, P, 23. The representations (3~) and (3b) are derived by subsituting u and V for u and U directly into condition (3). Representations (3~) and (3d) are, respectively, direct Hicksian and Marshallian representations of the statement Q = E(q). A sufficient condition for both (2) and (3) to hold is that F be independent of P, i.e. that the distribution of utility u (and attributes a) across individuals does not depend on prices. Except in special cases (e.g. the PIGLOG example given later), having F independent of P corresponds to a social planner’s outcome, in which people’s incomes x change so as to keep the distribution of utility essentially fixed. In particular, if U is defined to equal some measure of social welfare and the distribution of income x is determined by maximizing social welfare, then F will be independent of P. This is a polar case in which no restriction other than utility maximization is placed .on the functional form c of individual behaviours, and the representative consumer arises entirely from restrictions on the distribution of x (and a) across consumers. The other polar extreme is Gorman’s linear parallel Engel curves, which yield a representative consumer for any distribution of x. In the framework of the Lemma, Gorman’s case defines U and c by U= E(u) = 1~ dF(u,ul U,P,B), and c(u,P,u) = a(P) +p(P)u for some functions a and j?. In this case
= a(P) a{1 dF(u,al U,P,B)}/dP+/3(P)d{Ju = a(P) a[l]/aP+p(P) a[q/aP=o,
dF(u,ui
U,P,B)}/dP
which directly confirms condition (3). The extension of the Gorman case to c(u,P,u) = u(P,u) +p(P)u is straightforward. More useful applications of the Lemma are cases in between these polar extremes, where empirically plausible restrictions are
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placed on both the behaviour of agents and their distribution. For example, consider PIGLOG demand systems, which are demands having budget shares linear in In x (see Muellbauer, 1975, 1976). PIGLOG demands generally fit demand data better than other rank two models (see Lewbel, 1991), and the AIDS and Translog demand systems are examples of PIGLOG models. Drop attributes a now for simplicity. PIGLOG demands are c(u,P) = exp(a(P) +/?(P)u} f or some functions CI and j?. Define U, B, and the function C by X= C(U,P,B) = exp{B + a(P) +p(P)U} for some scalar B. Let the distribution function F be independent of P, and be such that equation (1) holds with these definitions of c and C. It is straightforward to check (by writing out equation (2) and converting from u and 7J to x and X) that this assumption about F is equivalent to assuming that E(x In x) = E(x) {In E(x) + B}. This is the empirically plausible example given in the previous section. An interesting feature of this example is that having F, the distribution of utility, independent of prices in this case, places restrictions on the distribution of income x that do not depend on prices. Apart from special constructions like these, having F independent of prices will typically place restrictions on the distribution of income that depend on prices. For a more general set of examples, consider Marshallian demands for individual consumers of the form Q =A,(P)+ A,(P)x+ w h ere A,, A,, and A, are vector-valued functions. ~,tOAd-W, Virtually all empirical studies of aggregate data that explicitly consider both aggregation and individual utility maximization have employed special cases of demands of this form. Demands of this form can have rank as high as three (see Lewbel, 1991). Define a vector of parameters B implicitly as the solution to v(X,P,B)= S~(x,P,a)dG*(x,alX,P,B). If a solution B exists then &= A,(P) + A,(P)X+ A,(P)ty(X,P,B), yielding a representative consumer. Lewbel (1989) essentially shows that a solution B, which does not depend on prices, is always possible when cy itself is a function that does not depend on prices. See also Lewbel (1988).
4. Conclusions
Kirman (1992) and others have argued that all representative agent paradigms should be discarded in macroeconomics, and replaced with models of economies composed of interacting agents. A less ambitious suggestion is that representative agent assumptions be replaced with models of individual behaviour combined with models of the distribution of agents. The argument that macro-models would be improved by abandoning representative agent assumptions is valid. Nevertheless, there is a persuasive reason for investigating conditions under
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which aggregate demands in an economy equal those of a utility maximizing representative consumer. This reason is the empirical observation that aggregate demands can in fact be well approximated by appropriate representative consumer models. However, theorems that rationalize an aggregate representative consumer via unrealistic assumptions (e.g. linear parallel Engel curves) are of limited value. Theorems showing why aggregate demands need not resemble a representative consumer also fail to address observed aggregate and distributional regularities. This paper provided a general characterization of the joint conditions on preferences and distributions of individuals required for aggregate demands to resemble those of a representative consumer. The well-known examples of linear Engel curves or presence of a social welfare maximizer are polar cases of this general characterization. Conditions that explain the empirically observed representative consumer lie in-between these polar extremes, in which regularities in the joint distribution of income and attributes across individuals and regularities in individual preferences combine to produce the observed outcome. Some attributes of a representative consumer are more empirically acceptable than others. For example, Slutsky symmetry tends to be empirically violated more often than is homogeneity of degree zero in income and prices (i.e. the absence of money illusion). Just as the results in the previous section describe joint restrictions on distributions and on preferences that yield a full representative consumer, one may seek analogous restrictions that yield just some attributes of a representative consumer. For example, Lewbel (1990) shows that if an individual consumer’s demands are homogeneous of degree zero in income and prices then aggregate demands will also be so if the distribution of income possesses a property called “Mean Scaling”. It is important to realize the limitations of the empirical representative consumer. First, while representative consumer models fit aggregate data reasonably well, models that explicitly consider and incorporate aggregation effects fit better (see Lewbel, 1991). As a result, forecasts based on representative consumer models will be only slightly inferior to those based on exact aggregation models, but only if the distribution regularities observed in the past are maintained in the future. In particular, policy implications drawn from representative consumer models may be badly biased if the contemplated policy affects the income distribution. Also, the utility level of the empirical representative consumer need not correspond, even approximately, to any sensible measure of social welfare, and therefore maximization of the representative consumer’s utility function might not constitute a rational economic policy objective. In summary, macroeconomists should not assume a representat-
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ive consumer. In general, models that are better (in terms of appropriate policy implications, empirical fit, and forecasting), but remain tractable, can be constructed by replacing the representative consumer with a combination of models of individual consumer behaviour and a model of the distribution of consumers. Nevertheless, from pure fit or forecast criteria, the error in an appropriately specified representative consumer (e.g. one that is PIGLOG rather than homothetic as macroeconomists commonly assume for convenience) is not large. This is an empirical regularity that appropriately constructed models should be able to both explain and exploit.
Acknowledgements
This research was supported in part by the National Science Foundation through grant SES-9011806. I would like to thank Chuck Blackorby and Mike Jerison for their comments on preliminary stages in the work in Section 3.
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