Heterogeneity and Aggregation

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Journal of Economic Literature Vol. XLIII (June 2005), pp. 347–391

∗ RICHARD BLUNDELL and THOMAS M. STOKER

1. Introduction lems at all? One could take the view that economics is mainly about the behavior of ifferent people do, in fact, behave dif- individuals or of individual markets, and ferently. To describe the behavior of a D assert a methodological position that only group, one must come to grips with this het- analysis of such individuals or individual erogeneity. In terms of empirical research in markets makes any real sense. However, economics, this means facing and resolving such a view eliminates the applicability of aggregation problems. the tenets of economic behavior to some of Aggregation problems are more than a the most important questions in economics, cloying annoyance in empirical work. They namely those that concern economic aggre- exist at virtually every level, from the initial gates. Economic policy is most often con- issues of data construction and model speci- cerned with prices, interest rates, aggregate fication to the subsequent issues of how to consumption and savings, market demand usefully summarize and apply results. and supply, total tax revenues, aggregate Because of their broad reach, aggregation wages, unemployment, and so forth. The problems have traditionally been kept clos- valuation and allocation of scarce resources eted within the practice of empirical work, requires that attention be paid to large along with other issues for which there are groups of individuals. no simple answers. But recently this is It is important to study relationships changing, as there has been substantial among economic aggregates, and to bring progress in dealing with aggregation prob- individual economic behavior to bear on lems in applied research. This survey covers those relationships. Addressing aggregation much of this development. problems just means creating a bridge At the outset we must address a basic between those behaving individuals and the question: why consider aggregation prob- economic aggregates. After stating this goal, ∗ Blundell: Institute for Fiscal Studies and Department of however, we immediately meet several vex- Economics, University College London. Stoker: Sloan ing issues as to how to even start to think School of Management, Massachusetts Institute of about linking “individual” and “aggregate.” Technology. We are grateful for comments from numerous colleagues, especially Orazio Attanasio, Martin Browning, At one extreme are the almost philosophi- Jim Heckman, Arthur Lewbel, and the reviewers. Thanks cal issues of where, or at what level, to apply are also due to Howard Reed and Zoe Oldfield who helped the strictures of economic theory. Are we to organize the data for us. Financial support from the ESRC Centre for the Micro-Economic Analysis of Fiscal Policy at assume that regularities associated with IFS is gratefully acknowledged. The usual disclaimer applies. rationality apply to entire economies, to

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“reasonably homogeneous” groups of house- differences, or heterogeneity, are relevant holds, firms, or other types of economic for each application area. As an organizing agents, or to Hicks’s Mr. Brown or Mr. Jones, principle, we consider (1) heterogeneity in as well as any of our own relatives or neigh- individual tastes and incomes, (2) hetero- bors. 1 To assert that there is a “correct” indi- geneity in wealth and income risks faced by vidual level at which to apply a mathematical individuals, and (3) heterogeneity in market model that is in line with rational behavior is participation.4 There is a generic tension to take a stand on those issues; a stand which between the degree of individual hetero- could only be properly validated by experi- geneity accounted for and the ease with mentation or much more extensive empirical which one can draw implications for econom- research than has been performed to date.2 ic aggregates. We point out how different At the other extreme are questions per- types of heterogeneity are accommodated in taining to what the appropriate “aggregate” the different application areas. is. One typically considers sums or averages Our approach is practical, and we hope to as reported in national income accounts as address many of the concerns faced by the relevant aggregates. They are usually the empirical research regarding aggregation. most interpretable numbers and the most We take a “micro-econometric” view of the relevant for pricing or policy analysis. But individual model—namely an econometric with large populations, one could consider model (obeying restrictions of economic the- many other kinds of aggregates or statistics ory) is applicable to individuals or house- from the population.3 holds. We consider model specifications that Once these issues are settled—what is the are typically used in empirical analysis of relevant “individual level” and what is the individual data in each application area. We relevant “aggregate”—then aggregation take the relevant “aggregates” to be either problems become purely practical. For any totaled or per-capita (averaged) values of the application, a model must be specified individual variables of interest, coinciding which captures all important economic with aggregates as typically reported for effects, allows for relevant individual hetero- regions or whole economies. Whether such geneity, and bridges the gap between indi- aggregates are easy to model or not, they are vidual and aggregate, facilitating analysis at the most interpretable and the most useful both levels. types of aggregates for interpretation, policy This survey covers specific solutions and analysis, or other uses of empirical results. related work in primarily three application We are concerned with models that strike a areas: consumer demand analysis, consump- balance between realism (flexibility), adher- tion and saving analysis, and analysis of ence to restrictions from economic theory, wages and labor market participation. A key and connections between individual behavior issue is to identify what kinds of individual and aggregate statistics. We consider several settings where individual models are intrinsi- 1 J. R. Hicks (1956), p. 55. cally nonlinear, and for those we must make 2 This effort is underway, primarily in work on psycho- specific assumptions on the distributions of logical tendencies in economics, finance and marketing. This work examines what is “consistent economic behav- relevant heterogeneous characteristics. We ior,” and whether departures from rational behavior are present results that can be used to explore the systematic or idiosyncratic. impact of heterogeneity in empirical applica- 3 For instance, to study inequality, a relevant aggregate would be the Gini coefficient. The choice of aggregate may tions that assume reasonable (and hopefully even be informed by empirical regularities in individual data. For example, if an individual model is best specified with the logarithm of observed income, the geometric 4 This roughly coincides with the categorization of het- mean of income might be a more natural aggregate than erogeneity discussed in Martin Browning, Lars P. Hansen, total income or average income. and James J. Heckman (1999, chapter 8). jn05_Article 1 6/10/05 1:37 PM Page 349

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plausible) parametrizations of both individual heterogeneity found in consumption rela- equations and distributions of heterogeneity. tionships, as well as various other aspects of We do not go into details about estimation; our modeling, illustrating with empirical but for each application area we will present data. We follow this with a brief discussion of models with empirically plausible equations modeling liquidity constraints and the for individuals and consistent equations for impacts on aggregate consumption. We close the relevant economic aggregates. The point this section with a discussion of recent is to have the ability to address empirical progress in general equilibrium modeling of issues at the individual (micro) level, the consumption, saving, and wealth. aggregate (macro) level, or both. Section 4 covers recent work on labor par- We begin with our coverage of consumer ticipation and aggregate wage rates. The demand models in section 2, the area which main issues here concern how to interpret has seen the most extensive development of observed variations in aggregate wages—are solutions to aggregation problems. The diffi- they due to changes in wages of individuals cult issues in consumer demand include or to changes in the population of participat- clear evidence of nonlinearity in income ing workers? We focus on the issues of het- effects (e.g., Engel’s Law for food) and per- erogeneity in market participation and vasive evidence of variations in demand with develop a paradigm that allows isolation of observable characteristics of households. We the participation structure from the wage discuss each of these problems in turn, and structure. This involves tracking the impacts use the discussion to cover traditional results of selection on the composition of the work- as well as “aggregation factors” as a method ing population, the impacts of weighting of empirically studying aggregation bias. We individual wage rates by hours in the con- cover recent empirical demand models and struction of aggregate wages, and the impact present aggregation factors computed from of observed wage heterogeneity. We show data on British households. That is, we cover how accounting for these features gives a the standard issues faced by aggregating substantively different picture of the wage over heterogeneous households in a static situation in Britain than that suggested by decision-making format and illustrate with observed aggregate wage patterns. Here we application to empirical demand models in have a situation where there is substantial current use. We close with a discussion of heterogeneity and substantial nonlinearity, recent work that studies aggregate demand and we show how to address these issues structure without making specific behavioral and draw conclusions relevant to economic assumptions on individual demands. policy. In section 3, we discuss models of overall Section 5 concludes with some general consumption growth and wealth. Here we observations on the status of work on must consider heterogeneity in tastes, but aggregation in economics. we focus on the issues that arise from het- This paper touches on many of the main erogeneity in income shocks, showing how ideas that arise in addressing aggregation different types of shocks transmit to aggre- problems, but it is by no means a comprehen- gate consumption. We start with a discussion sive survey of all relevant topics or recent of quadratic preferences in order to focus on approaches to such problems. For instance, income and wealth, and then generalize to we limit our remarks on the basic nature of recent empirical models that permit precau- aggregation problems, or how it is senseless to tionary saving. Because of the log-linear ascribe behavioral interpretations to estimated form of these models, we must make explic- relationships among aggregate data without a it distributional assumptions to solve for detailed treatment of the links between indi- aggregate equations. We cover the types of vidual and aggregate levels. It is well known jn05_Article 1 6/10/05 1:37 PM Page 350

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that convenient constructs such as a “repre- cussion away from heterogeneity in individ- sentative agent” have, in fact, no general justi- ual reactions and other behavior. We do fication, we will not further belabor their lack make reference to time series properties of of foundation. See the surveys by Thomas M. income processes as relevant to our discus- Stoker (1993) and Browning, Hansen, and sion of individual and aggregate consump- Heckman (1998) for background on these tion, but do not focus on time series basic problems. It is useful to mention two properties in any general way. Interested related lines of research that we do not cover. readers can pursue Clive W. J. Granger The first is the work on how economic theory (1980, 1987, 1990) and the book by Mario provides few restrictions on market excess Forni and Marco Lippi (1997) for more demands—see Hugo Sonnenschein (1972) comprehensive treatment of this literature.5 and Wayne Shafer and Sonnenschein (1982) among others, and Donald J. Brown and Rosa 2. Consumer Demand Analysis L. Matzkin (1996) for a recent contribution. The second is the work on collective decision We begin with a discussion of aggregation making within households as pionerred by and consumer demand analysis. Here the Pierre-Andre Chiappori (1988) and François empirical problem is to characterize budget Bourguignon and Chiappori (1994). allocation to several categories of commodi- We will also limit our attention to aggrega- ties. The individual level is that of a house- tion over individuals and not discuss the hold, which is traditional in demand analysis. voluminous literature on aggregation over The economic aggregates to be modeled are commodities. This latter literature concerns average (economywide, per household) the construction of aggregate “goods” from expenditures on the categories of commodi- primary commodities, as well as the consis- ties. We are interested in aggregate demand, tency of multistage budgeting and other sim- or how average category expenditures relate plifications of choice processes. While very to prices and the distribution of total budgets important for empirical work, the issues of across the economy. commodity aggregation apply within decision In a bit more detail, we assume that house- processes of individuals and, as such, would holds have a two-stage planning process, take us too far afield of our main themes. See where they set the total budget for the cur- the survey by Richard Blundell (1988) as well rent period using a forward looking plan and then allocate that current budget to the cat- as the book by Charles Blackorby, Daniel 6 Primont, and R. Robert Russell (1978) for egories of nondurable commodities. As background on commodity aggregation and such, we are not concerned with heterogene- multistage budgeting. Moreover, we do not ity in the risks faced by households in income cover the growing literature on hedonic/char- and wealth levels—they have already been acteristics models, which can serve to facili- processed by the household in their choice of tate commodity aggregration or other total budget (and, possibly, in their stocks of simplifications in decision making. durable goods). We consider commodity cat- Finally, we do not cover in great detail egories that are sufficiently broad that work that is associated with time series household expenditures are non-zero (food aggregation. That work studies how the time series properties of aggregate statistics relate 5 See Thomas M. Stoker (1986d, 1993) and Arthur Lewbel (1994) and others for examples of clear problems to the time series processes of associated in inferring behavioral reactions from time series results in data series for individuals, such as stationar- the presence of individual heterogeneity. ity, cointegration, etc. To permit such focus, 6 Provided that intertemporal preferences are additive, this accords with a fairly general intertemporal model of that work relies on strictly linear models for expected utility maximization (see Angus S. Deaton and individual agents, which again turn the dis- John Muellbauer 1980b, among others). jn05_Article 1 6/10/05 1:37 PM Page 351

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categories, clothing categories, etc.), and so 2.1 Aggregation of Consumer Demand we are not concerned with zero responses or Relationships heterogeneity in market participation. We are concerned with heterogeneity in Our framework requires accounting for total budgets and in needs and tastes. It is a individuals (households), goods, and time well-known empirical fact that category periods. In each period t, individual i choos- expenditure allocations vary nonlinearly with es demands qijt (or equivalently, expenditures = total budget size (for instance, Engel’s Law pjtqijt) for j 1,…, J goods by maximizing with regard to food expenditures). Early preferences subject to an income constraint, = applications of exact aggregation demand sys- where i 1,…, nt. Prices pjt are assumed to tems had budget shares in semi-log form be constant across individuals at any point in = (with or without attributes), namely the pop- time, with pt (p1t,…, pJt) summarizing all ular Translog models of Dale W. Jorgenson, prices. Individuals have total expenditure =Σ Lawrence J. Lau, and Stoker (1980, 1982) budget mit j pjtqijt, or income for short, and Almost Ideal models of Angus S. Deaton and are described by a vector of household and John Muellbauer (1980a, 1980b) respec- attributes zit, such as composition and demo- 7 tively. More recent empirical studies have graphic characteristics. The general form shown the need for further nonlinear terms for individual demands is written in certain expenditure share equations. In (1)qgpmz= (),, . particular, evidence suggests that quadratic ijt jt t it it logarithmic income terms are required (see, This model reflects heterogeneity in income for example, Anthony B. Atkinson, Joanna mit and individual attributes zit. Specific Gomulka, and Nicholas H. Stern 1990, empirical models involve the specification Herman J. Bierens and Hettie A. Pott-Buter of these elements, including a parametric 1990, Jerry A. Hausman, Whitney K. Newey, 8 formula for gjt. and James L. Powell 1994, Wolfgang Härdle, Economywide average demands and aver- Werner Hildenbrand, and Michael Jerison age income are 1991, Arthur Lewbel 1991, and Blundell, ∑ ∑ iijtq m Panos Pashardes, and Guglielmo Weber (2),,,jJ= 1… and iit . 1993). This nonlinearity means that aggregate nt nt demands will be affected by total budget size We assume that the population of the econ- as well as the degree of inequality in budgets omy is sufficiently large to ignore sampling across consumers. It is also well known that error, and represent these averages as the category expenditures vary substantially with associated population means demographic composition of households, ()= () such as how many children are present or (3)Eqtijt,,, j1… J and Em t it . whether the head of household is young or Our general framework will utilize various elderly, see Anton P. Barten (1964), Robert A. other aggregates, such as statistics on the Pollak and Terence J. Wales (1981), Ranjan distribution of consumer characteristics z . Ray (1983), and Browning (1992). it Our aim is to understand how behavioral effects for households impinge on price 7 effects and distributional effects on aggre- It is common parlance in the demand literature to refer to “total budget expenditure” as “income,” as we do gate demands. Understanding these effects here. In the later section on consumption, we return to is a key ingredient in understanding how the using “income” more correctly, as current consumption expenditures plus savings. composition of the population affects 8 For most of our discussion, zit can be taken as observ- demand growth over time and relative prices able. When we discuss explicit empirical models, we will across the different commodity categories. include unobserved attributes, random disturbances, etc. jn05_Article 1 6/10/05 1:37 PM Page 352

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2.1.1 Various Approaches: Exact aggregation factors.9 Write aggregate Aggregation and Distributional demand as Restrictions ()= () ( )+ () (8) Eqtijt b01 jt pEm t it b jt p We begin by discussing various approach- × π () ()+ () es to aggregation in general terms. From (1), 1titEmln Embp mit2 j t aggregate demand is given formally as × π ()() 2tititEm Ez , (4)Eq()= ∫ g()() pmz,,dF m , z . tijt jt t it ittitit where () where Ft(mit,zit) is the cross-section distribu- Emln m (9) π = it it and tion of income and attributes at time t. At 1t Em()ln Em () the simplest level, approaches to aggrega- it it tion seek a straightforward relationship Emz() between average demand, average income π = it iit . 2t Em()() Ez and average attribute values it it π π (5)Eq()= G() p,,Em() Ez () . The factors 1t and 2t show how the coeffi- tijt jt t t it t it cients in (7) are adjusted if individual The exact aggregation approach is based demand is evaluated at average income and π on linearity restrictions on individual prefer- average attributes, as in (8). 1t reflects ences/demands gijt that allow the relation- inequality in the income distribution

ship Gjt to be derived in a particularly simple through the entropy term E(mit lnmit) and π way, such that knowledge of Gjt is sufficient 2t reflects the distribution of income of the to identify (the parameters of) the individual elderly, as the ratio of the elderly’s share in / demand model. Take for example, aggregate income E(mit zit) E(mit) to the percentage of elderly E(z ) in the popula- (6) gpmz(),, = bpm() it jt t it it0 j t it tion. Aggregation factors are useful for two + bpm() ln m+ bbpmz() reasons. First, if they are stable, then aggre- 1 jt it it 2 jt itit gate demand has similar structure to indi-

where we suppose zit is a single variable that vidual demand. Second, their average value = = has zit 1 for an elderly household and zit 0 indicates how much bias is introduced in otherwise. Individual demand has a linear estimation using aggregate data alone.10 term in income, a nonlinear term in income, In contrast, the distributional approach and the slope of the linear term is different considers restrictions on the heterogeneity for elderly households. All of these slopes distribution Ft(mit,zit). Suppose the density

can vary with pt. Now, aggregate demand is dFt(mit,zit)isassumed to be an explicit function of E (m ),E(z ) and other parameters, Eq()= b() pEm ( )+ b() p t it it (7) tijt01 jt t it jt such as variances and higher order moments. × ()+ ()( ) Then with a general nonlinear specification Emitln m it bpEmz b2 jt itit , of individual demands gijt,wecould solve (4) which depends on average income E(mit) and two other statistics, E(mit lnmit) and E(mit zit). The coefficients are the same in the individ- 9 The use of aggregation factors was first proposed by Blundell, Panos Pashardes, and Guglielmo Weber (1993). ual and aggregate models, which is the 10 For instance, in (8), b1j(pt) is the coefficient of bridge through which individual preference π mit 1n mit, whereas b1j(pi) 1t is the coefficient of π π = π π parameters manifest in aggregate demands E(mit)1n E(mit). If 1t is stable, 1t 0, then b1j(pi) 1t (and can be recovered using aggregate data). is proportional to b1j(pi). In this sense, the structure of aggregate demand matches that of individual demand, In order to judge the impact of aggrega- but the use of aggregate data alone would estimate the π tion on demand, it is convenient to use individual coefficient with a proportional bias of 0. jn05_Article 1 6/10/05 1:37 PM Page 353

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directly, expressing aggregate demand Et(qijt) demand. Aggregate demand reflects hetero- as a function of those distributional parame- geneity only through variation in the income ters. Here, recovery of individual demand distribution—there is not enough variation

parameters from aggregate demand would in the zit distribution over t to recover the be possible with sufficient variation in the individual effects from aggregate demand. 11 distribution Ft(mit,zit) over t. We discuss various other examples of partial While conceptually different from exact distribution restrictions below. aggregation, the distributional approach 2.1.2 Demand and Budget Share Models should not be thought of as a distinct alternative in empirical modeling. With distri- There has been a substantial amount of bution restrictions, formulating a model via work on the precise structure of individ- direct integration in (4) may be difficult in ual preferences and demands consistent practice. As such, distributional restric- with exact aggregation. The most well- tions are often used together with exact known result of this kind is in the extreme aggregation restrictions, combining simpli- case where the aggregate model simply

fying regularities of the income-attribute relates average demands Et(qijt) to the vec- distribution with linearity restrictions in tor of relative prices pt and average expen- individual demands. diture Et(mit). William M. Gorman (1953) One example is with mean scaling, as dis- showed that this required preferences to cussed in Lewbel (1990), where the distribu- be quasi-homothetic; with individual

tion of income does not change relative demands linear in mit. shape but just scales up or down. Mean scal- Omitting reference to attributes zit for ing can arise with a redistribution mecha- now, the general formulation for exact nism where individual budgets are all scaled aggregation has demands of the form the same, as in m = m (E (m )/E (m )). it it 1 t it t 1 it 1 = ()+ () ( ) This structure allows distributional statistics (12) qapbphmijt000 j t j t it such as those in (7) to be computed from ++ () ( ) mean income only. … bphmMj t M it Another example arises from (distribu- with aggregate demands given as tional) exclusion restrictions. Certain attributes can be excluded from aggregate (13) Eq()= a() p+ b() p demand if their distribution conditional on tijt 00jt jt income is stable over time; if × []()++ (()[] ( ) EhmtitMjt0 … b pEhtM m it. ()= ()()∗ (10) dFtititzit m, z f z mit dF t m it As above, provided there is sufficient variation in the statistics E [h (m )],…,E [h (m )], where ƒ (z |m ) does not vary with t, then t 1 it t M it z it it the coefficients a (p ),b (p ),…,b (p ), and from (4), 0j t 0j t Mj t hence individual demands, can be fully ()= ()()recovered from aggregate data. (11) Eqtijt ∫ gjt pmz t,, it itzit fzmit Lau (1977, 1982) originally proposed the ∗ ∗∗ × dF() m = ∫ gpmdFm()(), . exact aggregation framework, and demon- tit jt t it t it strated that demands of the form (12) were

That is, zit and its distributional statistics are not only sufficient but also necessary for excluded from the equation for aggregate exact aggregation or aggregation without distributional restrictions (c.f. Stoker 1993 and Jorgenson, Lau, and Stoker 1982). 11 Technically, what is necessary for recoverability is completeness of the class of income-attribute distributions; Muellbauer (1975) studied a related prob- see Stoker (1984a). lem, and established results for the special jn05_Article 1 6/10/05 1:38 PM Page 354

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case of (12) with only two income terms.12 The same remarks on recoverability apply They both showed several implications of here: the individual budget share coeffi- applying integrability restrictions to (12). If cients b1j(pt),…,bMj(pt) can be identified with demands are zero at zero total expenditure, aggregate data with sufficient variation in = the distributional terms E (µ h (m )),…, then a0j(pt) 0. The budget constraint t it 1 it = E ( h (m )) over time. As above, aggrega- implies that one can set h0(mit) mit, without t it M it loss of generality. With homogeneity of tion factors can be used to gauge the differ- degree zero in prices and incomes, one can ence between aggregate shares and assert the forms of the remaining income individual shares. We have terms, which include the entropy form Epq() = (16) tjtijt= Ew()µ h1(mit) mitlnmit and the power form () titijt = θ Emtit h1(mit) mit. This theory provides the background requirements for specific exact = ()+ (()π bp01tt bp 1t aggregation demand models, such as those 13 × ()()+ we discuss below. ttithEm1 … The tradition in empirical demand analysis + bp()π hEm() ( ) is to focus on relative allocations and esti- Mt MtMt it mate equations for budget shares. The exact where by construction aggregation form (12) is applied to budget shares for this purpose. In particular, if we Ehm()µ () = = (17) π = titkit,,,kM= 1… set a0j(pt) 0 and h0(mit) mit in (12), then kt hEm()() = / kt it budget shares wijt pjtqijt mit take on a similar linear form. We have are the aggregation factors. These factors

pqjt ijt give a compact representation of the distrib- (14) w ==bp()+ bp() ijt m 01jt jt utional influences that cause the aggregate it model, and the elasticities derived from it, to × hm()++… b()ph ( m ) differ from the individual model. 1 it MjjtM it The budget share form (14) accommodates where b0j(pt),…,bMj(pt) and h1(mit),…,hM(mit) exact aggregation through the separation of are redefined in the obvious way. If we income and price terms in its additive form. denote individual expenditure weights as As before, when integrability restrictions are µ = / it mit Et(mit), then aggregate budget applied to (14), the range of possible model shares are specifications is strongly reduced. A particu- Epq() larly strong result is due to Gorman (1981), (15) tjtijt Ew()µ who showed that homogeneity and symmetry () titijt Emtit restrictions imply that the rank of the × + = ()+ () J (M 1) matrix of coefficients bmj(pt) can be bp01jt b jpt no greater than 3. Lau (1977), Lewbel (1991), × Ehm()µ ()+… and others have characterized the full range of titit1 possible forms for the income functions. + ()()µ ( ) bpEMj t t it hm M it 2.1.3 Aggregation in Rank 2 and Rank 3 Models 12 Muellbauer (1975) studied the conditions under which aggregate budget shares would depend only on a Early exact aggregation models were of single representative income value, which turned out to be rank 2 (for a given value of attributes zit). analogous to the exact aggregation problem with only two 14 expenditure terms. With budget share equations of the form 13 See also Lewbel (1989b, 1991, 1993) and Stoker (1984a, 1984b). 14 This is Muellbauer’s (1975) PIGL form. jn05_Article 1 6/10/05 1:38 PM Page 355

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(18)wbpbphm= ()+ () ( ) , Distribution restrictions can also facilitate ijt011 j t j t it the more modest goal of a stable relationship between aggregate budget shares and aggre- preferences can be specified that give rise to = gate total expenditure. For instance, suppose either the log-form h1(mit) 1nmit and the = θ that the total expenditure distribution obeys power form h1(mit) mit. Typically the for- ()= () mer is adopted and this produces Engel (22) Emtitittitln m cEm1 curves that are the same as those that under- + () () lie the Almost Ideal model and the Translog cE2 tittit mln E m . model (without attributes).15 In this case, aggregate shares have the form Then aggregate budget shares are Epq() () tjtijt Epqtjtijt (19) = Ew()µ (23) = bp() Em() titijt () 0 jt tit Emtit = ()+ ()π ( ) bp01jt bp jttt1 ln Em tit + ()()+ () bpc112jt cln E tmit where the relevant aggregation factor is so that a relationship of the form the following entropy measure for the mit distribution: Epq() tjtijt= ˜ () Em()µ ln Em()ln m (24) b0 jtp π = titit= titit () (20) Emtit 1t ln Em() Em(()ln E () m tit tit tit + ˜ () (() b1 jtpEmln t it µ = / where we have recalled that it mit Et(mit). π The deviation of 1t from unity describes the would describe aggregate data well. degree of bias in recovering (individual) Here, integrability properties from indi- price and income elasticities from aggregate vidual demands can impart similar restric- data alone. tions to the aggregate relationship. Lewbel Distribution restrictions can be used to (1991) shows that if individual shares facilitate computation of the aggregate sta- = ()+ () tistics as well as studying the aggregation fac- (25) wbpbpmijt01 j t j tln it tors. For instance, suppose income is

lognormally distributed, with 1nmit distrib- satisfy symmetry, additivity, and homogeneity µ uted normally with mean mt and variance properties, then so will σ 2 . The aggregation factor (20) can easily be mt = ()+ ()()κ + seen to be (26)wbpbpijt01 j t j tln m it . 1 (21)π =+1 . 1t ()µσ2 + The analogy of (23) and (26) makes clear that 21mt mt = if c2 1, then the aggregate model will satisfy symmetry, additivity, and homogeneity. As To the extent that the log mean and variance π such, some partial integrability restrictions are in stable proportion, 1t will be stable. If 16 π > may be applicable at the aggregate level. the log mean is positive, then 1t 1, indicating positive bias from using 1nEt(mit). 16 = = It is tempting to consider the case of c1 0, c2 1, π = which would imply that the aggregation factor 1t 1 (and 15 It is worthwhile to note that with the power form, no aggregation bias). However, that case appears impossi- estimation of with aggregate data would be complicat- ble, although we do not provide a proof. For instance, if mit = = ed, because the aggregation statistics would depend in a were lognormally distributed, c1 0, c2 1, would only complicated way on . occur if 1n mit had zero variance. jn05_Article 1 6/10/05 1:38 PM Page 356

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As we discuss in section 2.2 below, rank 2 two terms that depend on mit, but there is no models of the form (18) fail on empirical immediate limit on the number of h terms 17 grounds. Evidence points to the need for that depend only on characteristics zit. more extensive income effects (for given Budget share models that incorporate

demographic attributes zit), such as available consumer characteristics in this fashion were from rank 3 exact aggregation specifications. first introduced by Jorgenson, Lau, and In particular, rank 3 budget share systems Stoker (1980, 1982). Aggregation factors 2 that include terms in (lnmit) (as well as indi- arise for attribute terms, that necessarily vidual attributes) seem to do a good job of involve interactions between income and fitting the data, such as the QUAIDS system attributes. The simplest factors arise for of James Banks, Blundell, and Lewbel terms that depend only on characteristics, as = (1997), described further in section 2.2 in hj(mit,zit) zit, namely below. In these cases, corresponding to the ()µ () 2 z Eztitit Emztitit quadratic term (1nmit) , there will be an (29)π = = . t () ()() additional aggregation factor to examine, Eztit Emtitti Ezt 2 Em()µ ()ln π = titit This can be seen as the ratio of the income (27) 2t 2 ()() weighted mean of zit to the unweighted lnEmtit = mean of zit. If zit is an indicator, say zit 1 for households with children and z = 0 for 2 it ()( ) πz Emtititln m households without children, then jt is the = . ()() ()2 percentage of expenditure accounted for by Emtittitln Em / households with children, Et(mitzit) E(mit), In analogy to (22), one can define partial dis- divided by the percentage of households tributional restrictions so that aggregate with children, Et(zit). shares are well approximated as a quadratic More complicated factors terms arise with function of lnE (m ). expenditure-characteristic effects; for exam- t it = ple, if hj(mitzit) zit1nmit, then the relevant 2.1.4 Heterogeneous Attributes aggregation factor is As we noted in our earlier discussion, the Ezm()µ ln empirical analysis of individual level data (30) π z = tititit 1t Ez()ln Em ( ) has uncovered substantial demographic titt it effects on demand. Here we reintroduce Emz()lnm attributes z into the equations to capture = titiit it it ()() (). individual heterogeneity not related to Emtittittit Ezln Em income. Since z varies across consumers, it As before, in analogy to (22), one can derive for exact aggregation, z must be incorporat- it partial distributional restrictions so that ed in a similar fashion to total expenditure aggregate shares are well approximated as a m . The budget share form (14) is extended it function of E (m ) and E (z ). generally to t it t it 2.2 Empirical Evidence and the (28) wbpbphmz= ()+ () (, ) ijt011 j t j t it it Specification of Aggregate Demand

++ () () 17 … bphMj t M mmzit, it . 2.2.1A Whatsimple lineardo Individual transformation Demands will not in Lookgeneral be consistentLike? with consumer optimization. Blundell, Restrictions from integrability theory must Browning, and Ian A. Crawford (2003) show that if budg- apply for each value of the characteristics z . et sharesDemand have abehavior form that is at additive the individual in functions house-of 1n mit it and demographics, then if (1) Slutsky symmetry holds and For instance, Gorman’s rank theory implies hold(2) the leveleffects is of nonlinear.demographics As on webudget have shares men- are that the share model can be rewritten with tioned,unrestricted it thenis theynot haverealistic to be linear to inassume 1n mit. that jn05_Article 1 6/10/05 1:38 PM Page 357

Blundell and Stoker: Heterogeneity and Aggregation 357

0.7 kernel regression polynomial regression 0.6 pointwise confidence bands 0.5

0.4 ood Share F 0.3

0.2

0.1 0 34 567 Log Expenditure

Figure 1. Nonparametric Engel Curve: Food Share

demands are linear in total expenditures and flexible to capture these empirical patterns. relative prices. To illustrate typical shapes of In the QUAIDS model, expenditure shares income structure of budget shares, figures 1 have the form and 2 present estimates of Engel curves of =+αγ′ +− β()() two commodity groups for the demographic (31) wpmapijt j jln t j ln it ln t group of married couples without children, in the British Family Expenditure Survey 2 18 ()lnm − lnap() (FES). Each figure plots the fitted values + λ it t + u j () ijt of a polynomial (quadratic) regression in log cpt total expenditure, together with a nonparametric kernel regression. We see that for where a(pt) and c(pt)are given as 1 food expenditures, an equation that lnap()= α ′ ln p+ ln p′ ln p expressed the food share as a linear function tttt2 , of log expenditure would be roughly correct. ()= β′ For alcohol expenditures, the income struc- lncptt ln p, ture is more complex, requiring quadratic = = λ = terms in log expenditure. Moreover, as one with ( 1,…, N) , ( 1,…, N) , λ λ varies the demographic group, the shapes of ( 1,…, N) and  γ ′  the analogous Engel curves are similar, but 1 they vary in level and slope. =  \    . The QUAIDS model of (Banks, Blundell, γ ′  and Lewbel 1997) seems to be sufficiently N This generalizes the (linear) Almost Ideal λ demand system by allowing nonzero i val- 18 The FES is a random sample of around 7,000 house- ues, with the denominator c(p ) required to holds per year. The commodity groups are nondurable t expenditures grouped into: food-in, food-out, electricity, maintain the integrability restrictions. gas, adult clothing, children’s clothing and footwear, Banks, Blundell, and Lewbel (1997) do household services, personal goods and services, leisure extensive empirical analysis and establish the goods, entertainment, leisure services, fares, motoring and gasoline. More precise definitions and descriptive statistics importance of the quadratic log expenditure are available on request. terms for many commodities. Interestingly, jn05_Article 1 6/10/05 1:38 PM Page 358

358 Journal of Economic Literature, Vol. XLIII (June 2005) ol Share Alcoh

kernel regression polynomial regression pointwise confidence bands

3.5 4 4.5 5 5.5 6 6.5 7 Log Expenditure

Figure 2. Nonparametric Engel Curve: Alcohol Share

they find no evidence of the rejection of 2.2.2 The Implications for Aggregate integrability restrictions associated with Behavior homogeneity or symmetry. To include demographic attributes, an The stability and interpretation of aggre- attractive specification is the “shape invariant” gate relationships can be assessed from exam- specification of Blundell, Alan Duncan, and ining the appropriate aggregation factors. We Krishna Pendakur (1998). Suppose that can compute the empirical counterparts to 0 the factors by replacing expectations with g j(lnmi) denotes a “base” share equation, π then a shape invariant model specifies budget sample averages. For instance, 1t of (20) is shares as estimated as ∑ ()µˆ lnmn =−0 ()φθ()′ + ′ ϕ (32) πˆ = iititt wgmijt jln i z it z it j . 1t ()∑ ln iittmn

π z The shape invariant version of the QUAIDS and t of (29) is estimated as model allows demographic variation in the j ∑ µˆ zn terms. In Banks, Blundell, and Lewbel (1997), (33) πˆ z = iititt λ t ∑ the j, j, and j terms in (31) are allowed to iittzn 19 vary with many attributes zit. Family size, family composition, labor market status, occu- where we recall that the weights have the ˆ µ = / ∑ / pation, and education are all found to be form it mit ( imit nt). Similarly, quadrat- important attributes for many commodities.20 ic terms in lnmit will require the analysis of the empirical counterpart to the term (27). Interactions of the and τ terms with demo- 19 α + α For instance, j jzit is used in place of j, and sim- β λ graphic attributes necessitates examination ilar specifications for j and j terms. 20 Various methods can be used to estimate the of the empirical counterparts of terms of the QUAIDS model, with the iterated moment estimator of form (30). We can also study aggregation Blundell and Jean-Marc Robin (2000) particularly factors computed over different subgroups straightforward. It is worth noting that Banks, Blundell, and Lewbel (1997) also deal with endogeneity of total of the population, to see how aggregate expenditures, using various instruments. demand would vary over those subgroups. jn05_Article 1 6/10/05 1:38 PM Page 359

Blundell and Stoker: Heterogeneity and Aggregation 359

1.3

1.25

1.2

1.15

1.1

1.05

1 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000

Figure 3. Aggregation Factor for Households With Children

π z Figure 3 presents the estimated t term Figure 5 presents the aggregations factor for the impact of children on household for 1nmit term delineated by certain house- demands. This shows a systematic rise in the hold types. The baseline lnm line (“all”) is share of nondurable expenditure and servic- the same as that in figure 4. The other two es attributable to families with children over lines correspond to interactions for couples the 1980s and 1990s. The aggregate bias as a group and for couples with children. associated with using observed percentage While the time pattern of aggregation fac- of households with children (as opposed to tors is similar, they are at different levels, the income distribution across households indicating different levels of bias associated with and without children) varies from 15 with aggregation over these subgroups. π z percent to 25 percent. The path of t also Finally, it is worthwhile to mention some follows the U.K. business cycle and the path calculations we carried out on whether dis- of aggregate expenditure with downturns in tributional restrictions such as (22) are capa- 1981 and 1992. ble of representing the aggregate π π Figure 4 presents the estimated 1t and 2t movements in total expenditure data. Using 2 terms relating to the 1nmit and (1nmit) the time series of distributional statistics expressions in the QUAIDS demand model. from the FES data, we followed Lewbel It is immediately clear that these also display (1991) and implemented each of these systematic time series variation, but in com- approximations as a regression. With demo- π z parison to t above, they increase over the graphic interaction terms, the aggregate first period of our sample and fall toward the model will only simplify if these conditions end. The bias in aggregation exhibited for also apply to each demographic subgroup. In 2 the (1nmit) term is more than double that virtually every case, we found the fit of the exhibited for the 1nmit term. appropriate regressions to be quite close (say jn05_Article 1 6/10/05 1:38 PM Page 360

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1.12

1.1

1.08

1.06

1.04

1.02 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999

log expenditure log expenditure squared

π π Figure 4. Aggregation Factors for Income Structure: 1t and 2t

R2 in the range of .99). This gives support to we are interested in whether aggregate

the idea that aggregate demand relation- demand for good j decreases when price pj ships are reasonably stable empirically. increases (obeying the “Law of Demand”),

However, the evidence on the cj terms and we omit reference to other goods and implies that aggregation factors are substan- time t for simplicity. Denote the conditional

tially different than one, so again, estimates expectation of demand qij for good j, given of the price and income elasticities using income mi and price pj, as gj(pj,mi). aggregate data alone will not be accurate. Aggregate demand for good j is 2.3 Aggregation of Demand without ()= ()= ()() Eqij Gp j∫ gp j , mdFm Individual Structure / < and our interest in whether dG dpj 0. We close with discussion of a nontradi- Form this derivative, applying the Slutsky tional approach given in Hildenbrand decomposition to g (p ,m) as (1994), which is to study specific aspects of j j aggregate demand structure without relying   dG  dg dg  on assumptions on the behavior of individual (34) =−∫ gp(), m dF () m dp  dp j dm  consumers. This work makes heavy use of jj comp  empirical regularities in the observed distri- p bution of consumer expenditures across the dg dg = ∫∫dF() m− g ( p, m )dF () m population. dp j dm We can understand the nature of this j comp approach from a simple example. Suppose = S − A. jn05_Article 1 6/10/05 1:38 PM Page 361

Blundell and Stoker: Heterogeneity and Aggregation 361

1.05

1.04

1.03

1.02

1.01

1979 1981 1983 1985 1987 1989 1991 1993 1995 1997 1999

log expenditure couples with children couples without children

π z Figure 5. Aggregation Factors for Income Structure by Household Type: 1t

The price effect on aggregate demand distribution of demands across the popula- decomposes into the mean compensated tion. Hildenbrand (1998) shows that increas- price effect S and the mean income effect ing spread is a common phenomena in data A. If we take S as negative, which is fairly on British households and it is likely to be uncontroversial, then we know that valid generally. More broadly, this work has / < > dG dpj 0 if the income effect A 0. stimulated extensive study of the distribu- Looking once more at A we can see various tion of household expenditures, with a dif- ways of ascertaining whether A > 0: ferent perspective than traditional demand dg modeling. Using nonparametric methods, (35) Agpm= ∫ (), dF() m j dm Wolfgang Härdle, Hildenbrand, and Jerison (1991) study aggregate income effects across  ()2  1 dgpm j ,  a wide range of goods and conclude that the = ∫ dF() m . 2 dm “law of demand” likely holds quite generally. Hildenbrand and Alois Kneip (1993) obtain Without making any structural assumptions similar findings on income structure by on g(p ,m), one could estimate A with the j directly examining the dimensionality of first expression using nonparametric esti- vectors of individual demands.21 See mates of g(.) and its derivative. Or, we Hildenbrand (1994) for an overview of this could examine directly whether the 2 work, as well as Hildenbrand (1998) for an “spread” g(pj,m) is increasing with m, and if so, conclude that A > 0. This gives the flavor of this work without 21 This is related to transformation modeling structure doing justice to the details. The main con- of Jean-Michel Grandmont (1992). It is clear that the dimensionality of exact aggregation demand systems is tribution is to link up properties of aggre- given by the number of independent income/attribute gate demand directly with aspects of the terms, c.f. W. E. Diewert (1977) and Stoker (1984b). jn05_Article 1 6/10/05 1:38 PM Page 362

362 Journal of Economic Literature, Vol. XLIII (June 2005)

examination of variations in the British is affected by them. We take into account the expenditure distribution within a similar nature of the income and wealth shocks, as framework. well as the nature of the credit markets that provide insurance against negative shocks. 3. Consumption and Wealth We consider four different types of shocks, delineated by whether the effects are We now turn to a discussion of total con- permanent or transitory, and whether they sumption expenditures. Here the empirical are aggregate, affecting all consumers, or problem is to characterize consumption individual in nature. Aggregate permanent expenditures over time periods, including shocks can refer to permanent changes in how they relate to income and wealth. The the productive capability of the economy— individual level is typically that of a house- such as running out of a key natural resource hold (or an individual person, depending on or skill-biased technical change—as well as data source). The economic aggregate to be to permanent changes in taxes or other poli- modeled is average consumption expendi- cies that affect savings. Individual perma- tures over time, and we are interested in nent shocks include permanent changes in how aggregate consumption and saving an individual’s ability to earn income, such as relate to income and wealth across the econ- chronic bad health and long term changes in omy, as well as interest rates. This relation- type and status of employment. Aggregate ship is essential for understanding how transitory shocks refer to temporary aggre- interest rates will evolve as the population gate phenomena, such as exchange rate vari- changes demographically, for instance. ation, bad weather, and so forth. Individual Consumption expenditures are deter- transitory shocks include temporary job lay- mined through a forward looking plan that offs, temporary illnesses, etc. Many different takes into account the needs of individuals situations of uncertainty can be accounted over time, as well as uncertainty in wealth for by combinations of these four different levels. There is substantial evidence of types of shocks. demographic effects and nonlinearities in In terms of risk exposure and markets, consumption at the individual level, so we there are various scenarios to consider. With will need to consider heterogeneity in tastes complete markets, all risks are insured and as before. 22 Accordingly, aggregate con- an individual’s consumption path is unaffect- sumption is affected by the structure of ed by the evolution of the individual’s income households and especially the age distribu- over time.23 When markets are not com- tion, and aggregate consumption will be plete, the extent of available insurance mar- affected by inequality in the distribution of kets becomes important and determines the wealth. We are not concerned here with het- degree to which different individual risks are erogeneity in market participation per se, as important for aggregate consumption behav- everyone has non-zero consumption expen- ior. For example, in the absence of credit ditures. Later we discuss some issues raised market constraints, idiosyncratic risks may by liquidity constraints, which have much in be open to self-insurance. But in that case common with market participation modeling there may be little insurance available for as described in section 4. aggregate shocks or even for permanent idio- Our primary focus is on heterogeneity syncratic shocks. Our discussion takes into with regard to risks in income and wealth levels, and how the forward planning process 23 See Andrew Atkeson and Masao Ogaki (1996) for a model of aggregate expenditure allocation over time and to 22 See Attanasio and Weber (1993) and Attanasio and individual goods based on addilog preferences, assuming Browning (1995), among many others. that complete markets exist. jn05_Article 1 6/10/05 1:38 PM Page 363

Blundell and Stoker: Heterogeneity and Aggregation 363

account the type of income risks and how =+αβ′ (37)azit it . risk exposure affects aggregate consumption. Most of our discussion focuses on individ- With the discount rate equal to the real ual consumption plans and their implications interest rate, maximizing the expected sum for aggregate consumption. Beyond this, we of discounted utilities gives the following can consider the feedback effects on con- optimal plan for the consumer (Robert E. sumption and wealth generated through Hall 1978), general equilibrium. For instance, if a cer- (38)cz=+=αξβ′ + ξ . tain group of consumers systematically save it it it it it Ω more than others, then in equilibrium those Defining i,t1 as the information set for consumers will be wealthier and their sav- individual i in period t − 1, the consumption ings behavior will be a dominant influence innovation it obeys on the evolution of aggregate wealth. The E[]ξ = 0 study of this important topic is in its infancy (39)it i, t−1 . and has been analyzed primarily with cali- In what follows we will use a time super- brated macroeconomic growth models. We script to denote this conditional expectation, include a discussion of some of this work. t1 ≡ Ω namely E E[.| i,t1] to distinguish it from the population average in period t 3.1 Consumption Growth and Income (which uses a time subscript as in Et ). Shocks Notice, the model (38) is linear in the change in attributes ∆z with constant coef- In our framework, in each period t, indi- it ficients , plus the consumption innovation. vidual i chooses consumption expenditures In other words, this model is in exact aggre- c by maximizing expected utility subject it gation form with regard to the attributes z to an asset accumulation constraint. it that affect preferences. Individual i has heterogeneous attributes zit that affect preferences. There is a common, 3.1.1 Idiosyncratic Income Variation and riskless interest rate rt. We assume separa- Aggregate Shocks bility between consumption and labor supply in each time period and separability of When the only uncertainty arises from preferences over time. real income, the consumption innovation it We begin with a discussion of aggregation can be directly related to the stochastic with quadratic preferences. This allows us to process for income. We begin by spelling out focus on the issues of different types of the income process in a meaningful way. income shocks and insurance, without deal- Express income yit as the sum of transitory ing with nonlinearity. In section 3.2, we con- and permanent components sider more realistic preferences that allow (40) yyy=+P T precautionary saving. it it it When individual within-period utilities and assume that the transitory component is are quadratic in current consumption, we serially independent. We assume that the have the familiar certainty-equivalent for- permanent component follows a random mulation in which there is no precautionary walk saving. Within-period utilities are given as P =+P ηP (41) yyit it−1 it 1 2 ()=−() − P (36) Ucit it a it c it where the innovation is serially inde- 2 it pendent. < for cit ait.Wemodel individual heterogene- Next, decompose these two components ity by connecting ait to individual attributes as into a common aggregate effect and an jn05_Article 1 6/10/05 1:38 PM Page 364

364 Journal of Economic Literature, Vol. XLIII (June 2005)

idiosyncratic effect horizon.24 Clearly, expected growth is determined by preference attributes as ηηεP =+ (42)it tit , t−1 ()= () β (47)EcEcit it i, t−1 z it . (43)yuvT =+ . it tit Aggregate consumption has the form Here t is the common aggregate permanent ()= βητ′ ()++ ε (48)Ectit Ez titt t u t . shock, it is the permanent shock at the individual level, ut is the aggregate transitory Thus, the aggregate data is described exactly shock, and vit is the individual transitory by a representative agent model with quad- shock—the four types of income shocks dis- ratic preferences and characteristics Et(zit) cussed above. This mixture of permanent facing a permanent/transitory income and transitory shocks has been found to pro- process.25 vide a good approximation to the panel data For the second scenario, suppose individ- process for log incomes, see Thomas E. ual shocks can be fully insured, either MaCurdy (1982) and Costas Meghir and through informal processes or through cred- Luigi Pistaferri (2004). We assume that the it markets. Now individual consumption individual shocks are normalized to average growth depends only on aggregate shocks to zero across the population, namely ε = = = βητ′ ++ Et( it) 0 and Et(vit) 0. (49)czit it t t u t . The stochastic process for individual income then takes the form Consequently, with (48), we will have =++ηε + = β′()− ()+ () (44)yuvit t it t it . (50)czEzEcit it t it t it . The stochastic process for aggregate income Thus, consumption growth at the individ- has the form ual level equals aggregate consumption growth plus an adjustment for individual (45) Ey()=+η u titt t preferences.

where, again, Et denotes expectation (associ- Finally, the third scenario is where all ated with averaging) across the population of shocks (aggregate and individual) are fully agents at time t. insurable. Now individual consumption β∆ growth will be the planned changes zit 3.1.2 Income Shocks and Insurance only, and aggregate consumption growth will β∆ be the mean of those changes Et(zit). This The first scenario is where individual (and is the most complete “representative agent” aggregate) shocks are not insurable. Here the case, as complete insurance has removed the optimal consumption innovation it for the relevance of all income risks. individual will adjust fully to permanent income shocks but only adjust to the annuity value of transitory shocks. To see this, again 24 If L is the time horizon, then suppose that real interest rates are constant  −−+()Lt1  τ =+rr()111()−+() r  . and equal the discount rate. Under quadratic t τ → → preferences (36), consumption growth can be Clearly t 0 as r 0. Note that for a small interest τ ≈ written (Deaton and Christina Paxson 1994) rate, we have t 0, so that the transitory shocks become irrelevant for consumption growth. 25 ∆ cz= βη′ +++ε τ() uv + Aside from the drift term Et(zit), aggregate (46) it it t it t t it , consumption is a random walk. In particular, the orthogo- t–1 η + τ ∆ = η + τ ∆ Ω = τ nality conditions E ( t t ut) E( t t ut| i,t–1) 0 where t is the annuitization rate for a tran- hold at the individual level and therefore also hold at the sitory shock with planning over a finite aggregate level. jn05_Article 1 6/10/05 1:38 PM Page 365

Blundell and Stoker: Heterogeneity and Aggregation 365

3.1.3 Incomplete Information 3.2.1 Consumption Growth with CRRA Preferences It is interesting to note that, in our simplest framework, incomplete information We assume that within-period utility is can cause aggregate consumption to fail to   1 have random walk structure. In particular,  1−  sit suppose individual shocks are not complete- a  cit  (54) Uc()= eit ly insurable and consumers cannot distin- it it  1  1 −  guish between individual and aggregate  s  shocks. To keep it simple, also assume that it

there are no varying preference attributes zit. where ait permits scaling in marginal utility Following Jorn-Steffen Pischke (1995), indi- levels (or individual subjective discount

vidual i will view the income process (44) as rates), and sit is the intertemporal elasticity an MA(1) process: of substitution, reflecting the willingness of =−ςθς individual i to trade off today’s consumption (51)y − , it it it 1 for future consumption. As before, we will θ where the parameter is a function of the model the heterogeneity in ait and sit via relative variances of the shocks. individual attributes zit. Changes in individual consumption are We now adopt a multiplicative stochastic simply income process, with the decomposition =−()θς cit1 it . expressed in log-form as t–1 ∆ = Note that it is still the case that E ( cit) lnyuv=+++ηε ∆ Ω = (55)it t it t it . E( cit| i,t1) 0. However, from (51) we have that The permanent and transitory error compo- −=−θθ() nents in the income process are decomposed (52)cc− 1 y . it it1 it into aggregate and individual terms, as in ∆ Replacing yit by (44) and averaging over (44). As noted before, this income growth consumers we find specification is closely in accord with the ()= θ () typical panel data models of income or earn- (53) Ectit E t−−11 c it ings, and it will neatly complement our +−()θη()+ 1 ttu equations for consumption growth with so that aggregate consumption is clearly not CRRA preferences. In addition, we assume a random walk. that the interest rate rt is small, for simplicity, and is not subject to unanticipated 3.2 Aggregate Consumption Growth with shocks. Precautionary Saving With precautionary savings, consumption With quadratic preferences, consumption growth depends on the conditional variances growth can be written as linear in individual of the uninsurable components of shocks to attributes—in exact aggregation form—and income. Specifically, with CRRA prefer- we were able to isolate the impacts of differences (54) and log income process (55), we ent kinds of income shocks and insurance have the following log-linear approximation scenarios. To allow for precautionary saving, for consumption growth we must also account for nonlinearity in the =++ρβϕ()′ (56) lncrit t rz t it basic consumption process. For this, we now −− +++kkσσκt 1 t 1 εεκη+ consider the most standard consumption 1 it 2 At 1 it2 t model used in empirical work, that based on σt–1 where it is the conditional variance of Constant Relative Risk Aversion (CRRA) idiosyncratic risk (conditional on t − 1 preferences. jn05_Article 1 6/10/05 1:38 PM Page 366

366 Journal of Economic Literature, Vol. XLIII (June 2005)

Ω σt–1 information i,t–1) and At is the condi- involves dealing with the “log” nonlinearity, to tional variance of aggregate risk.26 The which we now turn.28 attributes z represent the impact of het- it 3.2.2 How is Consumption Distributed? erogeneity in ait, or individual subjective discount rates, and the intertemporal elas- Since the individual consumption growth = ρ + ϕ ticity of substitution sit zit. equations are nonlinear, we must make dis- Typically in empirical applications, zit will tributional assumptions to be able to formu- include levels and changes in observable late an equation for aggregate consumption. attributes and unobserved factors may also In the following, we will assume lognormali- be appropriate.27 As before, ty of various elements of the consumption process. Here we point out that this is moti- ()εε = t−1 ()= (57) EEit i, t−1 it 0 vated by an important empirical regularity— namely, individual consumption does appear ()ηη = t−1 ()= to be lognormally distributed, at least in (58)EEit i, t−1 it 0 . developed countries such as the United To sum up, in contrast to the quadratic pref- States and the United Kingdom. erence case, the growth equation (56) is Figure 6 shows the distribution of log con- nonlinear in consumption and it includes sumption using U.S. consumer expenditure conditional variance terms that capture the data across the last two decades. Consumption importance of precautionary saving. is taken as real expenditure on nondurables A consistent aggregate of the individual and services, and is plotted by five year bands model (56) is given by to achieve a reasonable sample size. Each log-consumption distribution has a striking () =++ρβϕ()′ () (59) Ecrtittln rEz ttit resemblance to a normal density. In the experience of the authors, this result is often repli- + ()σ t−1 +++σκηt−1 kE1 tit k2 At 2 t cated in more disaggregated data by year and various demographic categorizations, such as ∆ where Et( 1ncit) refers to the population birth cohort, and also in other countries mean of the cross-section distribution of including in the Family Expenditure Survey ∆ 1ncit in period t and so on. The t subscript data for the United Kingdom. Given this reg- again refers to averaging across the popula- ularity, one would certainly start with log-nor- tion of consumers, and we have normalized mality assumptions such as those we make ε = ∆ = Et( it) 0 as before. Provided Et( 1ncit–1) below, and any subsequent refinements would ∆ Et–1( 1ncit–1), equation (59) gives a model of need to preserve normality of the marginal changes over time in Et(1ncit) which is a nat- distribution of log consumption. ural aggregate given the log form of the 3.2.3 Insurance and Aggregation with model (56). Precautionary Saving However, Et(1ncit) is not the aggregate typically observed nor is it of much policy interest. As with our previous discussion, we Of central interest is per-capita consumption must consider aggregation under different

Et(cit) or total consumption ntEt(cit). Deriving an equation for the appropriate aggregates 28 If we evaluate the individual model at aggregate values, we get ∆ = + + ϕ + σ t–1 + ω 1nEt(cit) rt ( rt)Et(zit) k2 At t. 26 ω See Blundell and Stoker (1999) for a precise deriva- Here t is a “catch-all” term containing the features tion and discussion of this approximation. that induce aggregation bias, that will not satisfy the 27 t–1 ω = See Banks, Blundell, and Agar Brugiavini (2001) for orthogonality condition E ( t) 0. It is also worthwhile to a detailed empirical specification of consumption growth note that empirical models of aggregate consumption also σ t–1 in this form. typically omit the terms Et(zit) and k2 At. jn05_Article 1 6/10/05 1:38 PM Page 367

Blundell and Stoker: Heterogeneity and Aggregation 367

1981–1985 1986–1990

1991–1995 1996–2001

Figure 6. The Distribution of Log Nondurable Consumption Expenditure: U.S. 1981–2001

scenarios of insurance for income risks. ()=++ ( ρ We again assume that agents have the (62) Ectitt Eexp ln c it−1 r t Ω =Ω same information set, namely i,t1 t1 for all i,t. ′ −  ()βϕ+ rz+ k σt 1 +κη) We begin with the scenario in which there titAt2 2 t  is full insurance for individual risks, or pool- =++()ρσκηt−1 ing of idiosyncratic risk across individuals. exprktAt2 2 t · Here insurance and credit markets are suffi- ′  ()()βϕ+  ciently complete to remove individual risk Ect  iit−1 exp rz t it  terms in individual income and consumption ε = σ t1 = streams, so it 0 and it 0 for all i,t. The individual model (56) becomes with the impact of log-linearity arising in ′ the final term, a weighted average of =++ρβϕ() (60) lncrit t rz t it attribute terms interacted with lagged − ++σκηt 1 consumption cit–1. k2 At 2 t Of primary interest is aggregate consump- t–1 η = with E ( t) 0. The mean-log model (59) is tion growth, or the log-first difference in now written as aggregate consumption ()− ()= ρ (61) EcEcln ln − r tittitt1  Ec() ∆=lnEc ( ) ln  tit  . ++()βϕ ′ ()+ σκηt−1 + tit  () rEzttit k2 At 2 t . Ectit−−11 The relevant aggregate is per-capita con- This is expressed as

sumption Et(cit). Per-capita consumption is ∆=++ρσκηt−1 given by (63) lnEctit ( ) r t k2 At 2 t jn05_Article 1 6/10/05 1:38 PM Page 368

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  ′   ++θµθ( Ec exp()()β + ϕrz (67) lnczN−−~ Ez ( ),  tit −1 tit  it11 t it c t t t it +ln    Ec()  2  tit−1  σθ+ ∑+∑θθ2 ) ct−1, t zz,, t t t z c−1 t  () Ec− and + ln  t iit 1  .  () 2 Ec−− ()µσ tit11 (68)lncNit−−−111~ c t, c, t . Aggregate consumption growth reflects the We can now solve for an explicit solution to interest and risk terms that are common to (63): apply (65), (67). and (68) and rearrange all consumers, a weighted average of attrib- to get ute terms, and the log-difference in the ′ − =++ρβϕ() average of cit–1 at time t versus time t 1. (69) lnEctit ( ) r t r t Notice first that even if zit is normally dis- × ++σκηt−1 Eztit() k2 At 2 t tributed, we cannot conclude that 1ncit is normal. We also need (as a sufficient con-  ′ ++1 ()βϕrr∑+() βϕ dition) that 1ncit–1 is normal at time t to 2  tzztt, make such a claim. This would further  − ++2()βϕr ∑ seem to require normality of 1ncit–2 at t 1, tzct−1,  . and so forth into the distant past. In any case, we cover this situation with the broad This is the aggregate model of interest, assumption: expressing growth in per-capita consumption as a function of the mean of z, the condi- The distribution of c is the same it–1 tional variance terms from income risk, and in period t − 1 and t. the covariances between attributes z and That is, the population could grow or shrink, lagged consumption cit–1. Thus shows how but the distribution of cit–1 is unchanged. individual heterogeneity manifests itself in Under that assumption, we can drop the last aggregate consumption through distribu- term in (63) tional variance terms. These variance terms  Ec() vary with r if the intertemporal elasticity of tit−1 = t (65)ln  () 0 . substitution varies over the population. Ec−− tit11 Now consider the scenario where some Lagging the individual model (60) gives individual risks are uninsurable. This rein- an equation for cit–1, but there is no natural ε σt–1 troduces terms it and it in consumption way to incorporate that structure directly growth at the individual model, and we must into the equation for aggregate current con- 29 be concerned with how those permanent sumption Et(cit). Therefore, we further risks are distributed across the population. assume In particular, we assume in each period that   µ  lnc − each individual draws idiosyncratic risk from it 1  ct−1 (66)   ~N  , θ z  θ Ez() a common conditional distribution, so that tit tt it σt–1 =σt–1 it it for all i. The individual consump- σ 2 ∑ θ  tion growth equation (56) now appears as  ct−1 zc−1, t t   . ′ θ ∑ θθ∑  (70) lncr=++ρβϕ() rz tzct−1, tzztt, it t t it −− = + +++kkσσκt 1 t 1 εεκη+ . where we have set t ( rt). This 1 It 2 At 1 it2 t assumption says that The mechanics for aggregation within this formulation are similar to the previous case, 29 ε = This is because of the potential dependence of cit–1 on including the normalization Et( it) 0, but we the same factors as cit–2, and so forth. need deal explicitly with how the permanent jn05_Article 1 6/10/05 1:38 PM Page 369

Blundell and Stoker: Heterogeneity and Aggregation 369 ε individual shocks it covary with 1ncit. As The first concerns the evidence on full insur- above, we adopt a stability assumption (64). ance of individual risks. How good an approx- β We then extend (66) to assume that (1ncit–1,( imation would such an assumption be? To +ϕ ε rt) zit, it) is joint normally distributed. The settle this we need to examine whether there growth in aggregate average consumption is is evidence of risk pooling across different now given by individuals and different groups in the econ- ′ omy. For example, does an unexpected (71) lnEc()=++ρβϕ r() r Ez() tit t t tit change in pension rights, specific to one − − 1 cohort or generation, get smoothed by trans- ++kkσσt 1 t 1 ++κη () 1 It 2 AAt 1 tt2 fers across generations? Are idiosyncratic where health risks to income fully insured? Even though we may be able to cite individual =+()βϕ′ ∑+() βϕ+ κσ22 ttzztttrr,,1 ε cases where this perfect insurance paradigm clearly fails, is it nonetheless a reasonable ′ ++2 ()βϕr ∑ +2κσ approximation when studying the time series t zzc−−1,, t1 ε c1 t of aggregate consumption? The second aspect of empirical evidence +∑2κβϕ() +r 1 zt, t . concerns the factors in the aggregate model While complex, this formulation underlines (71) that are typically omitted in studies of the importance of the distribution of risk aggregate consumption. From the point of across the population. In contrast to the full view of estimating the intertemporal elastic- σ 2 ity parameter , how important are these information model (69), there is a term ε,t Λ aggregation factors? How well do they cor- in t that reflects the changing variance in σ t–1 relate with typically chosen instruments and consumption growth. The term 1t captures how idiosyncratic risk varies, based on how likely are they to contaminate tests of t − 1 information. excess sensitivity performed with aggregate We have not explicitly considered unantic- data? ipated shocks to the interest rate rt, or het- 3.3.1 Evidence on Full Insurance and Risk 30 erogeneity in rates across individuals. Pooling Across Consumers Unanticipated shocks in interest would If the full insurance paradigm is a good manifest as a correlation between rt and aggregate income shocks, and would need approximation to reality, then aggregation is treatment via instruments in estimation. considerably simplified and aggregate rela- Heterogeneity in rates could, in principle, be tionships satisfying the standard optimality accommodated as with heterogeneous attrib- conditions can be derived with various con- utes. This would be especially complicated if ditions on individual preferences. There is a the overall distributional structure were to reasonably large and expanding empirical lit- shift as interest rates increased or decreased. erature on the validity of the full insurance scenario, as well as complete markets sce- 3.3 Empirical Evidence on Aggregating the nario. This work is well reviewed in Consumption Growth Relationship Attanasio (1998) and Browning, Hansen, and There are two related aspects of empirical Heckman (1998). Here we present evidence research that are relevant for our analysis of directly related to our discussion of con- aggregation in consumption growth models. sumption growth above. Two rather effective ways of analyzing failures of the full insurance paradigm fit neatly with our discussion. 30 Stephen P. Zeldes (1989b) points out how differing One approach to evaluating the full insur- marginal tax rates can cause interest rt to vary across consumers. ance hypothesis is to look directly for evidence jn05_Article 1 6/10/05 1:38 PM Page 370

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that unexpected shocks in income across dif- (1999), Ricardo Caballero (1990), and ferent groups in the economy leads to differ- Jonathan Skinner (1988), the study by ences in consumption patterns (as consistent Banks, Blundell, and Agar Brugiavini (2001) with (56), which assumes no insurance). This presents evidence that differential variances is not a trivial empirical exercise. First, such of income shocks across date-of-birth income shocks have to be identified and cohorts do induce important differences in measured. Second, there has to be a convinc- consumption growth paths. ing argument that they would not be correlat- 3.3.2 Aggregation Factors and ed with unobservable variables entering Consumption Growth marginal utility, or observables such as labor supply (in a nonseparable framework). There are two issues. First, if one estimates Building on the earlier work by John H. a model with aggregate data alone, is there Cochrane (1991), Barbara Mace (1991), likely to be bias in the estimated parameters of Fumio Hayashi, Joseph Altonji, and Laurence interest? Second, will the omission of aggre- Kotlikoff (1992), and Robert M. Townsend gation bias terms result in spurious inference (1994), the study by Attanasio and Steven J. concerning the presence of excess sensitivity Davis (1996) presents rigorous and convinc- of consumption to transitory income shocks. ing evidence against the full insurance With regard to bias, we consider the elas- hypothesis using this approach. Low frequen- ticity of intertemporal substitution , which cy changes in wages across different educa- is normally a focus of studies of aggregate tion and date-of-birth cohorts are shown to be consumption. In figure 7, we plot the correlated positively with systematic differ- aggregation factor ences in consumption growth. More recently, (72) lnEc()− E() ln c Blundell, Pistaferri, and Ian Preston (2003) tit t it use a combination of the Panel Survey for the sample of married couples from the of Income Dynamics (PSID) and the British FES, used to construct the aggrega- Consumers Expenditure Survey (CES) to tion factors for demand of section 2. The fig- investigate insurance of permanent and tran- ure shows a systematic procyclical variation. sitory income shocks at the individual level. We found the correlation coefficient between They find almost complete insurance to tran- the real interest series and this factor to be sitory shocks except among lower income significant. This indicates that there will exist households. They find some insurance to per- an important aggregation bias in the estimat- manent shocks particularly among the ed intertemporal substitution parameter from younger and higher educated. But they aggregate consumption data (with a log-linear strongly reject the complete insurance model. growth model). This is confirmed in the study The second approach to evaluating full by Attanasio and Weber (1993), where aggre- insurance is to assume risk averse prefer- gate data was constructed from micro survey ences and to model the evolution of idiosyn- information.31 They find an elasticity esti- cratic risk terms. In terms of the model (56), mate for aggregate data of around .35, and this approach examines the relevance of the corresponding micro-level estimates were σ t–1 individual risk terms (e.g., it ) once aggre- twice this size. σt–1 gate risk ( At ) has been allowed for. This is The study of excess sensitivity involves the addressed by looking across groups where use of lagged information as instrumental the conditional variance of wealth shocks is variables in the estimation of the consumption likely to differ over time and to see whether this is reflected in differences in consump- 31 Attanasio and Weber (1993) also note a strong impact tion growth. Following earlier work by of omitting the cross-section variance of consumption Karen E. Dynan (1993), Blundell and Stoker growth. jn05_Article 1 6/10/05 1:38 PM Page 371

Blundell and Stoker: Heterogeneity and Aggregation 371

0.025

0.02

0.015

0.01

0.005

1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 -0.005

-0.01

-0.015

-0.02

Figure 7. Aggregation Factor for Consumption Growth

growth relationship. Omitting aggregation for age, labor supply variables, and demo- bias terms can invalidate the instruments typ- graphics are introduced in a flexible way. ically used. For the consumption data used Moreover, these variables explain why excess above, we computed the correlation of the sensitivity appears to vary systematically over aggregation factor with two typically used the cycle. instrumental variables in consumption growth It is an important finding that evidence of equations—lagged real interest rates and excess sensitivity vanishes once we move to lagged aggregate consumption. The estimated individual data and include observable vari- correlation coefficient between these series ables that are likely to impact preferences for and the omitted bias term was found to be the allocation of consumption over time. It strongly significant.32 has important consequences for our under- Together these results suggest that aggre- standing of liquidity constraints and for partial gation problems are likely to lead to serious insurance. It has implications for understand- bias in estimated intertemporal substitution ing the path of consumption growth over the parameters and also to exaggerate the pres- cycle. It also has implications for the retire- ence of excess sensitivity in consumption ment-savings puzzle, or how consumption growth regressions on aggregate data. drops much more at retirement than is pre- Attanasio and Browning (1995) investigate dicted by standard consumption growth this excess sensitivity issue in more detail and equations. Banks, Blundell, and Sarah Tanner find that excess sensitivity still exists at the (1998) find that once demographics and labor micro-data level but disappears once controls supply variables are allowed to affect the marginal utility of consumption, nearly two thirds 32 Detailed regression results available on request. of the retirement-savings puzzle disappears. jn05_Article 1 6/10/05 1:38 PM Page 372

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3.4 Consumption and Liquidity Constraints since constrained consumers will show a stronger reaction than unconstrained con- Our previous discussion has focused on sumers. In this section, we discuss these heterogeneity in wealth and income risk as it basic issues and indicate how a model of impinges on consumption. We now turn to a aggregates can be constructed. Blundell discussion of liquidity constraints on con- and Stoker (2003) works out the details for sumption, which generate a different kind aggregate consumption models of this type. of aggregation structure. The evidence for There is some subtlety in considering liquidity constraints is relatively limited. what population groups are likely to be liq- Most studies of consumption smoothing at uidity constrained. Poor households with a the individual level find it difficult to reject reasonably stable but low expected stream of the standard model once adequate care is income, may have little reason to borrow. taken in allowing for demographic and labor More likely to be constrained are young con- market interactions; see Attanasio and sumers, who have much human capital but Weber (1995) and Blundell, Browning, and little financial wealth—college students or Meghir (1994), for example. Much of the perhaps poor parents of able children. Such excess sensitivity found in aggregate studies individuals may want to borrow against their can be attributed to aggregation bias as doc- future earned incomes but cannot, in part umented in Attanasio and Weber (1993), because their eventual income is higher than Marvin Goodfriend (1992), and Pischke others, and the growth of their income with (1995). However, there is some evidence experience is higher. Clearly such con- that does point to the possibility that a frac- sumers will react more than others to shocks tion of consumers could be liquidity con- in current income and wealth. strained at particular points in the life-cycle We start with the basic consumption model and business cycle. At the micro level, some discussed earlier, with permanent and transi- evidence can be found in the studies by tory shocks to income. As in (55), the change Fumio Hayashi (1987), Stephen P. Zeldes in current income for consumer i at time t is (1989a), Tullio Jappelli (1990), Jappelli and =++ηε + Marco Pagano (1994), Meghir and Weber (73) lnyuvit t it t it (1996), and Rob Alessie, Michael P. where η + ε is the permanent component Devereaux, and Weber (1997). As men- t it and ∆u +∆v is the transitory component. tioned earlier, the Blundell, Pistaferri, and t it To keep things simple, we assume that per- Preston (2003) study shows that the con- manent income shocks are not insurable, sumption of low income households in the with log consumption given as PSID does react to transitory shocks to ′ income, which suggests that such house- (74) lncr=++ρβϕ() rz++ ηε holds do not have access to credit markets to it t t it t it smooth such shocks. where we assume the precautionary risk σt–1 σt–1 For aggregation, liquidity constraints terms ( it , At ) are included with the zit introduce regime structure into the pop- effects. Note that (74) gives the consump- + β + ϕ ulation. Namely, liquidity constrained tion growth plan ( rt ( rt) zit) as well as consumers constitute one regime, uncon- how consumption reacts to permanent η + ε strained consumers constitute another shocks in income (here t it). regime, and aggregate consumption will Liquidity constraints affect the ability of depend upon the relative distribution consumers to finance their desired con- across regimes. This structure is particular- sumption growth path. We follow an ly relevant for the reaction of consumption approach similar to Zeldes (1989a), where growth to increases in current income, the incidence of liquidity constraints jn05_Article 1 6/10/05 1:38 PM Page 373

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depends on the degree of consumption where Ait is (say) accumulated financial growth the consumer is trying to finance and wealth. Consumer i is liquidity constrained in the existing stock of assets. In particular, liq- period t, or cannot maintain the consumption uidity constraints enter the growth plan only growth plan, if if they are binding in planning period t − 1, ′ ρβϕrrzIm++()+ and then the best response will always be to (77) ttitit−1 it increase consumption growth so as to “jump” −−uv >++γδ A ζ back up to the optimal path. If this response tititit is further frustrated by a binding constraint = which we indicate by Iit 1, as above. In this in period t, consumption will simply grow by case we assume that consumption growth is the amount of resources available. as large as possible, namely This response structure is captured by lncy=+++ ln γδ A ζ additional terms in the equation (74). Let Iit (78)it it it it . denote the indicator In terms of permanent and transitory terms = of income growth, (78) may be rewritten (75) Iiit 1[consumer is constrained in period t − 1] =+ηε (79) lncit t it +++++γδ ζ and suppose that a consumer who is con- uvtititit A . strained in period t − 1 needs to increase This is consumption growth for constrained consumption growth by m to return to the it consumers. The constraints have an impact, optimal growth plan.33 Then, consumption as consumption growth clearly depends on growth for unconstrained consumers is transitory income shocks and wealth levels. =++ρβϕ()′ (76) lncrit t rz t it Aggregate consumption growth will clear- +++ηε ly depend on the proportion of consumers Im− . it1 it t it who are constrained and the proportion that We now model the constraints, as well as are not. Consumers who were constrained consumption growth for constrained con- last period will have a boost in their con- ∆ sumers. With growth in income of 1nyit, sumption growth to return to the optimal consumer i needs to finance a growth rate of path. This regime switching structure is non- ρβϕ++()′ +++− ηε linear in character. Therefore, to model rrzIm− ln y ttitit1 it t it it aggregate consumption growth, we would need to specify distributional structure for =+ρ ()ββϕ+ ′ +−− r rz Im− u v t titit1 it t it all the elements that are heterogeneous for consumption at time t to be on the across the population. We then aggregate growth plan. To model liquidity constraints over the population of unconstrained indi- at time t, suppose that consumer i faces a viduals with consumption growth (76) and borrowing constraint that is associated with a the population of constrained individuals maximum rate of increase of consumption of with consumption growth (79). Using log- normality assumptions, we carry out this γδ++A ζ it it development in Blundell and Stoker (2003). It is clear how aggregate consumption is 33 Various approaches have been applied to account for affected by transitory income shocks, as well the jump term mit in studies of micro level data. See Zeldes as the distribution of wealth. (1989a), Tullio Jappelli, Jorn-Steffen Pischke, and Nicholas S. Souleles (1998), Rene Garcia, Annamaria Lusardi, and Serena Ng (1997), Rob Alessie, Bertrand 3.5 Equilibrium Effects Melenberg, and Weber (1988), Alessie, Michael P. Devereux, and Weber (1997), and Attanasio and Weber As we mentioned at the start, one use of (1998). aggregate consumption equations is to study jn05_Article 1 6/10/05 1:38 PM Page 374

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and understand the evolution of aggregate for consumers to keep track of two things, the consumption and saving by themselves. mean of the wealth distribution and the aggre- Another important use is in studying equilib- gate productivity shock. Thus there is an infor- rium price and interest rate paths over time. mational economy afforded in a similar This is an exercise in general equilibrium fashion to a formal aggregation result; once analysis, and every feature that we have dis- mean wealth is known, the information con- cussed above is relevant—consumer hetero- tained in the distribution of wealth does not geneity, heterogeneity in income and wealth appear to improve forecasting very much. This risks, liquidity constraints, and the distribu- is true even with heterogeneity of many types, tion of wealth. Further complicating this including individual and aggregate income effort is the dynamic feedback that occurs shocks (albeit transitory). wherein the level and distribution of wealth The reason for this is clear once the nature evolves as a result of the level and distribu- of equilibrium is examined. Most consumers, tion of savings. These difficulties make it very especially those with lowest discount rates, hard to obtain analytical results on equilibri- save enough to insure their risk to the point um. Nevertheless, it is extremely important where their propensity to save out of wealth to understand the nature of equilibrium here, is essentially constant and unaffected by cur- including implications on prices and interest rent income or output. Those consumers rates. We now discuss some recent progress also account for a large fraction of the that has been made using calibrated stochas- wealth. Therefore, saving is essentially a lin- tic growth models. A leading example of this ear function in wealth, and only the mean of effort is provided by Per Krusell and Anthony wealth matters to how much aggregate sav- A. Smith (1998), although the approach dates ing is done each period. The same is not true from at least S. Rao Aiyagari (1994) and John of aggregate consumption. There are many Heaton and Deborah J. Lucas (1996). low wealth consumers who become unem- The Krusell–Smith setup has the following ployed and encounter liquidity constraints. features. Consumers are infinitely-lived, Their consumption is much more sensitive to with identical (within-period) CRRA prefer- current output than that of wealthier con- ences, but they are heterogeneous with sumers. In essence what is happening here is regard to discount rates. Each consumer has that the dynamics of the savings process con- a probability of being unemployed each peri- centrates wealth in the hands of a group that od, providing transitory, idiosyncratic income behaves in a homogeneous way, with a con- shocks. Production arises from a constant stant marginal propensity to save. This returns-to-scale technology in labor and cap- (endogenous) simplification allows planning ital, and productivity shocks provide transito- to occur on the basis of mean wealth only. ry aggregate shocks. Consumers can insure It is certainly not clear how applicable this by investing in capital only, so that insurance finding is beyond the context of this study. markets are incomplete, and consumers’ This is a computational finding that depends capital holdings cannot be negative (liquidity heavily on the specifics of this particular set- constraint). This setup is rich but in many up.34 Nonetheless, this form of feedback has ways is very simple. Nevertheless, in princi- some appeal as a explanation of the smooth ple, in order to predict future prices, each evolution of wealth distribution, as well as consumer must keep track of the evolution why forecasting equations (that fit well) are of the entire distribution of wealth holdings. so often much simpler than one would expect Krusell and Smith’s simulations show a from the process that underlies the data. The rather remarkable simplification to this forecasting problem. For computing equilibrium 34 Christopher D. Carroll (2000) makes a similar argu- and for consumer planning, it is only necessary ment, with emphasis on the role of precautionary savings. jn05_Article 1 6/10/05 1:38 PM Page 375

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rich are different (and in this model, the dif- wage process from the participation decision. ference makes them rich), but what is impor- To put it very simply, suppose aggregate tant for forecasting is how similar the rich are wages are increasing through time. Is this to one another. With equal saving propensi- because typical wages for workers are ties, it does not matter which group of rich increasing? Or, is it because low wage indi- people hold the most wealth. viduals are becoming unemployed? Do the In any case, the study of equilibrium sources of aggregate wage growth vary other effects is in its infancy and it will certainly the business cycle? The aggregation problem generate many valuable insights. must be addressed to answer these questions. We now turn to our basic model of wages 4. Wages and Labor Participation that permits us to highlight these effects. We Our final topic area is the analysis of wages then show the size of these effects for aggre- and labor participation. Here the empirical gate wages in the United Kingdom, a country problem is to understand the determinants where there has been large and systematic of wages separately from the determinants changes in the composition of the workforce of participation. The individual level is that and in hours of work. A more extensive ver- of an individual worker. The economic sion of this model and the application is aggregates to be modeled are aggregate given in Blundell, Howard Reed, and Stoker wages and the aggregate participation rate, (2003). They also summarize derivations of or one minus the unemployment rate. These all aggregate equations given below. statistics are central indicators for macroeco- 4.1 Individual Wages and Participation nomic policy and for the measurement of economic well-being. We begin with a model of individual wages Our analysis is based on a familiar para- in the style of Andrew D. Roy (1951), where digm from labor supply. Potential wages are wages are based on human capital or skill determined through human capital and levels and any two workers with the same labor participation is determined by com- human capital level are paid the same wage. paring potential wages to a reservation wage Our framework is consistent with the pro- level. Empirically, there is substantial het- portionality hypothesis of Heckman and erogeneity in the determinants of wages, Guilherme Sedlacek (1990), where there is and substantial heterogeneity in the factors no comparative advantage, no sectoral dif- determining labor participation, and both ferences in wages for workers with the same processes are nonlinear. In particular, it is human capital level, and the return to typical to specify wage equations for individ- human capital is not a function of human uals in log-form, and there is much evidence capital endowments. 35 of age and cohort effects in wages and We assume that each worker i possesses a

employment. As with demand and con- human capital (skill) level of Hi. Suppose sumption, we will need to be concerned human capital is nondifferentiated, in that it

with heterogeneity in individual attributes. commands a single price rt in each time peri- To keep things as simple as possible, we do od t. The wage paid to worker i at time t is not consider forward looking aspects of (80)wrH= . employment choice and so are not con- ti t i

cerned with heterogeneity in income and Human capital Hi is distributed across the wealth risks. population with mean Our primary focus is on heterogeneity in market participation. Aggregate wages 35 Heckman and Guilherme Sedlacek (1985) provide depend on the rate of participation, and the an important generalization of this framework to multiple important issues involve separation of the sector. See also Heckman and Bo E. Honore (1990). jn05_Article 1 6/10/05 1:38 PM Page 376

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= Et(1nHi) js education/cohort/time effects to be included (c.f. Blundell, Reed, and Stoker 2003). δ where js is a level that varies with cohort j Because our examples involve log-linear to which i belongs and education level s of equations and participation (or selection), worker i. In other words, the log wage we summarize the basic framework as equation has the additive form (85)lnwx=+ββ′ + ε , =++δε it0 it it (81) lnwrti ln t js it Iz=+10[]αα′ +> ν , ζ 36 it0 it it where it has mean 0. We will connect js to hh=+γα· () + α′ z+ ν observable attributes below. it00 it it . To model participation, we assume that ∗ Here, xit denotes education, demographic reservation wages wit are lognormal (cohort, etc.), and time effects; zit includes ∗ =++αηζ = (82) lnwBit ln it js it out-of-work benefit variables; and Iit 1 denotes participation. It is clear that the scale where js has mean 0 and where Bit is an of is not identified separately from the par- exogenous income (welfare benefit) level + + ν ticipation index 0 zit it, however we that varies with individual characteristics and retain to distinguish between the fixed hours ≥ ∗ time. Participation occurs if wit wit,orwith case = 0 and the variable hours case ≠ 0. Our notation distinguishes two types of (83)lnrB−+−+−≥αδηεζ ln 0 . titjsjsitit individual heterogeneity in (85). The vari- We represent the participation decision by ables xit and zit are observable at the individ- = ≥ ∗ ual level, while ε and ν are unobservable. the indicator Iit 1[wit wit]. it it For aggregation over hours of work, it is Analysis of data on wages and participation useful to make one of two assumptions. One at the individual level requires assumptions is to assume that the distribution of hours is on the distribution of those unobservable fixed over time. The other is to assume that elements, a process familiar from the literature on labor supply and selection bias. We desired hours hit are chosen by utility maximization, where reservation wages are now review some standard selection formu- ∗ = lae here for later comparison with the aggre- defined as hit(w ) h0 and h0 is the minimum number of hours available for full-time gate formulations. Start with the assumption that the unobserved elements are normally work. We assume hit(w) is normal for each w, and approximate desired hours by distributed ∗  ε      σ 2 σ  hh=+γ () w − w it 0 ε ε (84) it0 ln it ln it (86)   ~,N      . ν     σ σ 2  it 0 ε =+hrBγαδηεζ()ln −ln +− +− . 0 titjsjsititThis allows us to apply some well known selection formulae (given in virtually every This is our base level specification. It is textbook of econometrics). The micro par- simple to extend this model to allow differ- ticipation regression, or the proportion of entiated human capital or differential participants given xit and zit,isaprobit cohort effects due to different labor market model; experience, which permits a wide range of αα+ z  (87)EIxz[], =  0 it  . titit  σ 2  36   Clearly, there is an indeterminacy in the scaling of rt

and Ht. Therefore, to study rt, we will normalize rt for some = = The micro log-wage regression for partici- year t 0 (say to r0 1). We could equivalently set one of the s to zero. pants is jn05_Article 1 6/10/05 1:38 PM Page 377

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We now discuss some aggregate analogues (88) EwIxz[]ln = 1,, = t it it it it of the micro regression equations, and then σ αα+ ′ our final equation for the aggregate wage. ε  z  ββ+ ′x + λ  0 it  The aggregate participation (employment) 0 it σ  σ 2  rate is reflecting the typical (Heckman style) selec-  αα+ ′Ez() (91)EI[]=  0 it  , tion term, which adjusts the log-wage equation t  αασ′ + 2  37  ∑  to the group of participating workers. zz using a formula originally due to Daniel 4.2 Aggregate Wages and Employment McFadden and F. Reid (1975). Aggregate The aggregate of interest is average hourly participation has the same form as the micro

earnings, where aggregation occurs over all participation regression (87) with zit replaced σ workers, namely by E(zit) and the spread parameter ν ∑ iI∈=()1 hw it it αασ′ + 2 = = µ replaced by the larger value ∑ , (89) wt ∑ itw it zz ∑ ∈=() iI∈=()1 h it iI1 reflecting the influence of heterogeneity in where i ∈ (I = 1) denotes a participant the individual attributes that affect the par- (worker), h w is the earnings of individual i it it ticipation decision. The mean of log-wages in period t, and µ are the hours-weights it for participating (employed) workers is h µ = it . (92) EwI[]ln= 11=+ββ′ ExI (= ) it ∑ h titit0 it iI∈=()1 it σ  αα+ ′Ez() + ε λ 0 it Modeling the aggregate wage (89)   , αασ′ ∑+2  αασ′ ∑+2  requires dealing with log-nonlinearity of the zz zz basic wage equation, dealing with participa- using a formula originally derived by tion and dealing with the hours-weighting. McCurdy (1987). This matches the micro All of these features require that distribu- log-wage regression (88) with xit replaced by tional assumptions be made for (observable) = E(xit|I 1), zit replaced by E(zit) and the individual heterogeneity. In particular, we spread parameter changed from σν to make the following normality assumption for αασ′∑ + 2 xit and zit: zz . This is an interesting result,  ββ+ ′x    ββ+ ′Ex() but doesn’t deliver an equation for the (90)  0 it  ~N   0 it  , αα+ ′   αα+ ′  aggregate wagewt. 0 zit 0 Ez( iit ) Blundell, Reed, and Stoker (2003) derive  ββαβ′∑ ′∑  such an equation. The aggregate wage is  xx xz   βααα′∑ ′∑  given as xz zz Ehw[]| I = 1 (93) lnw = ln it it it or that the indices determining log-wage and t Eh[]| I = 1 employment are joint normally distributed.38 it it =+ββ′ +++ 0 Ex( iit)[ t t t ] 37 Here Φ(.) is the normal cumulative distribution function, and λ(.) = φ(.)/Φ(.),where φ(.) is the normal density where the aggregation bias is comprised of function. 38 Assuming that the linear indices are normal is much a spread term

weaker than assuming that xit and zit are themselves joint multivariate normal. Such a strong structure would eliminate = 1 ββσ′ + 2 (94)[]∑ ε , many important regressors, such as qualitative variables. t 2 xx jn05_Article 1 6/10/05 1:38 PM Page 378

378 Journal of Economic Literature, Vol. XLIII (June 2005) Ψ Λ plus two terms t and t, which represent sources of variation. First, aggregate wages separate sources of bias but have very increase if the distribution of log wages shifts complicated expressions.39 to the right, which is a the typical “well- What these terms represent can be seen being” interpretation of aggregate wage most easily by the following construction. movements.40 This source is reflected by the + Begin with the individual wage equation mean 0 E(xit) of log wages. Second, + = evaluated at mean attributes, 0 E(xit) because individual wages are given in log Et(lnwit), or overall mean log wage. Adding form, aggregate wages will increase with Ω t adjusts for log-nonlinearity, as increased spread of the log wage distribution, as reflected by the heterogeneity term Ω . ()= ()+ t lnEwtitt E ln w itt. Third, aggregate wages will increase if the benefit threshold increases, causing more Ψ Adding t adjusts for participation, as lower wage individuals to decide not to par- []= = ()+ ticipate. This is reflected in the participation (95)lnEwit I it1 ln E t w it t . Ψ term t. Fourth, aggregate wages will increase if the hours of higher wage individu- Finally, adding Λ adjusts for hours-weight- t als increase relative to lower wage individuals, ing, as which is captured by the hours adjustment Λ (96) lnwEwI== ln []1 + term t. The aggregate model (93) permits tititt estimation of these separate effects. = ()+++ Ewtitttln t . This framework could be relaxed in many Thus, the bias expressions are complicated ways. We can allow all variance terms to be Ω Ψ Λ time varying, as well as many of the basic but the roles of t, t and t are clear. In Ω behavioral parameters. If the normality words, the term t captures the variance of returns, observable and unobservable. The assumption on the overall log wage and par- Ψ ticipation index is not accurate for the whole term t reflects composition changes within the selected sample of workers from which population, the population can be segment- Λ ed, with separate aggregate equations devel- measured wages are recorded. The term t reflects changes in the composition of hours oped for each segment. These variations, and depends on the size of the covariance among others, are discussed in Blundell, between wages and hours. Reed, and Stoker (2003). The formulation (93) of the log aggregate 4.3 Empirical Analysis of British Wages wage lnw thus captures four important t The different sources of aggregate wage variation bear directly on the issue of whether 39 In particular, we have aggregate wages are procyclical or not. In particular, the participation effect works counter  αα+ ′Ez()(+ β′ ∑+ ασ ) =   0 it xz ε  t ln   αασ′ ∑+2  zz 40 – Comparing 1nwit to mean log wage E(1nwit) is in line  αα+ ′Ez() with the tradition of measuring “returns” from coefficients  0 it   , in log wage equations estimated with individual data; c.f.  αασ′ ∑+2  zz Gary Solon, Robert Barsky, and Jonathan A. Parker (1994).  ++γα γα′ + γβ′ ∑+ α Other comparisons are possible, and some may be prefer-  h()0 0 Ezit xz able on economic grounds. For instance, if aggregate pro- t ln  duction in the economy has total human capital (Σ H ) as  hEz++γα γα ′ ()+ i i  00 it an input, then the appropriate price for that input is rt, so – one might want to compare wt to 1nrt for a more effective γσ + γγα′ ∑+ α σ2 λa  ε zz · σ  interpretation. In any case, it is useful to point out that if ε ,t –  . E(1nHi) is constant over time, then comparing 1nrt to wt is γα′ ∑ ασλ+ 2 a  – zzz · t  the same as comparing E(1nwit) to wt. jn05_Article 1 6/10/05 1:38 PM Page 379

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2.2

2.1

2.0 g average wage g average lo 1.9

1.8 78 80 82 84 86 88 90 92 94 96 year

Figure 8. Male Hourly Wages

to a normal cyclical variation of aggregate participating group consists of employees; wages—decreases in participation can lead to the nonparticipating group includes individ- aggregate wage increases when there is uals categorized as searching for work as essentially no change in individual wage levels well as the unoccupied. The hours measure or distribution. We now turn to an analysis of for employees in FES is defined as usual British wages that shows these features. weekly hours including usual overtime Our microeconomic data is again taken hours, and weekly earnings includes over- from the U.K. Family Expenditure Survey time pay. We divide nominal weekly earn- (FES) for the years 1978 to 1996. The FES ings by weekly hours to construct an hourly is a repeated continuous cross-section sur- wage measure, which is deflated by the vey which contains consistently defined quarterly U.K. retail price index to obtain micro data on wages, hours of work, employ- real hourly wages. ment status, and education for each year Individual attributes include education since 1978. Our sample consists of all men level and cohort effects. Individuals are aged between 19 and 59 (inclusive).41 The classified into three educational groups: those who left full-time education at age 16 or lower, those who left aged 17 or 18, and 41 We exclude individuals classified as self-employed. This could introduce some composition bias, given that a those who left aged 19 or over. Dummy significant number of workers moved into self employ- variables capture effects of five date-of- ment in the 1980s. However, given that we have no data on birth cohorts (b.1919–34, b.1935–44, hours and relatively poor data on earnings for this group, there is little alternative but to exclude them. They are also b.1945–54, b.1955–64, and b.1965–77). We typically excluded in aggregate figures. include various trend variables to account jn05_Article 1 6/10/05 1:38 PM Page 380

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0.90

0.85

0.80

0.75 78 80 82 84 86 88 90 92 94 96 year

Figure 9. Male Employment Rate

for a common business cycle effect. Finally, Figure 9 shows the overall male labor our measure of benefit income (income at employment rate for the same period. zero-hours) is constructed for each individ- Clearly there has been a large fall in the par- ual as described in Blundell, Reed, and ticipation rate of men. Figure 10 presents Stoker (2003). After making the sample the employment rate for those with low selections described above, our sample education. For this group, there is a contin- contains 40,988 observations, of which ued and much steeper decline in employ- 33,658 are employed, or 82.1 percent of the ment. This period also included two deep total sample. recessions in which there have been large fluctuations in male employment. 4.3.1 Real Wages and Employment Considering figures 8–10 together, one can understand the basic importance of sort- Figure 8 shows log average wages in ing out wage growth at the individual level Britain from 1978 to 1996. These show a from changes in participation. The strong strong trend increase over the whole period. trend of aggregate wages is suggestive of The trend appears for more disaggregate great progress at increasing the well-being of groups. Blundell, Reed, and Stoker (2003) laborers in general.42 However, great present a more detailed breakdown by cohort, region, and education group and show that the trend holds widely, including 42 In fact, such a conclusion has been trumpeted by for the least educated group. British newspapers. jn05_Article 1 6/10/05 1:38 PM Page 381

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1.01.0

0.90.9

0.8 0.8

0.70.7

0.60.6 78 8080 8282 8484 86 8888 9090 92 94 96 year

Figure 10. Employment Rate for the Low Educated

increases in unemployment are likely associ- 4.3.2 Aggregation Results ated with unemployment of workers with lowest wages or workers from the poorest The Blundell, Reed, and Stoker (2003) groups. It is very important to understand study considers a number of possible specifi- how much of the upward trend in average cations for our individual level wage equa- wages is due to the elimination of low wage tions which relate to the various earners from employment. specifications. In the simplest of our specifi- There have also been well documented cations, the full proportionality hypothesis is changes in real benefit income over time and imposed on the (nondifferentiated) human across different groups of individuals. While capital model, together with trend terms to it is unlikely that variation in real value of reflect the business cycle effects on skill benefit income relative to real earnings can price. This specification was strongly reject- explain all of the variation in participation ed by the data. The preferred model had full rates, the changes in real benefits act as an interactions of cohort, trend, region, and important “instrumental variable” for sepa- education. These additional variables could rating participation decisions from determi- reflect many differences in minimum educa- nants of wages. Again, to the extent that tional standards across cohorts such as the changes in benefit income have discouraged systematic raising of the minimum school (or encouraged) participation, it is essential leaving age over the postwar period in the to learn the size of this impact relative to the United Kingdom. The prices of different other factors driving changes in wages. (education level) skills are allowed to evolve jn05_Article 1 6/10/05 1:38 PM Page 382

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0.40 Log aggregate wage 0.35 Log aggregate wage:

0.30 with corrections Aggregated micro log wage 0.25

0.20

log Wage 0.15

0.10

0.05

0.00

-0.05 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96

Figure 11. Hourly Wages

in different ways by including an interaction downward shift in the trend and an increased between high education and the trend cyclical component in wage growth shown by terms. These coefficients are marginally sig- both the corrected aggregate series and the nificant and show an increasing trend among estimated micro model. groups with higher levels of human capital. This procedure is repeated for the lower The impact of adjusting for participation is education group in Blundell, Reed, and very important.43 To see the impact of these Stoker (2003). Several features of this analy- results on aggregate wages, we turn to sis are worth mentioning here. For instance, graphical analysis. even the direction of movement of the uncor- Figure 11 displays the (raw) log aggregate rected log aggregate wage does not always wage, the log aggregate wage minus the esti- mirror that of the mean micro log wage. mated aggregation bias terms, and the mean There is a reasonably close correspondence of the log wage from the selectivity adjusted between the two in the 1984–88 period, but micro model. We have plotted the return the 1990–93 period is different. In 1990–93, lines from a common point at the start of the log aggregate wages are increasing, but the time series rebased to zero for 1984 to high- mean micro log wage (and the corrected light the changes in trend growth in wages aggregate wage) is decreasing—precisely the indicated by our corrections. There is a clear period where there is a big decline in participation. What is remarkable is that the aggregate data show reasonable growth in real 43 Blundell, Howard Reed, and Stoker (2003) examined wages, but such growth is virtually absent the impact of our normality assumptions by estimating with semiparametric methods. The estimated wage coefficients from the corrected series. We are left with a were hardly affected by this generalization. much more cyclical profile of wages. jn05_Article 1 6/10/05 1:38 PM Page 383

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If the model is exactly correct, the results income, lagged aggregate consumption, etc. from aggregating the selectivity-adjusted Such equations often fit aggregate data micro model estimates should match the cor- extremely well. Unfortunately, such models rected aggregate series. They show a close could not predict future aggregate variables correspondence in figure 11 and a similar with sufficient precision to dictate optimal close correspondence is noted by Blundell, policies. Even with great statistical fit, there Reed, and Stoker (2003) for more disaggre- was too much uncertainty as to what the gated groups.44 In any case, we view the cor- underlying processes were that drove the respondence between the corrected log aggregate data, and for policy prescriptions, aggregate micro wage and the mean micro it is crucial to know something about those log wage as striking validation of the frame- processes.45 work. This model specification, which pro- What economists could get a handle on vides a good and parsimonious specification was how rational individuals and firms would of the evolution of log real wages, also seems behave in various economic environments. to work well in terms of the specification of Problems like how to allocate one’s budget, aggregation factors. how much to save and invest, or whether to work hard or not so hard, are sufficiently 5. Conclusion familiar that their essence could be captured with some mathematics, and economists Macroeconomics is one of the most could describe and prescribe optimal reac- important fields of economics. It has perhaps tions. Economists could settle how someone the grandest goal of all economic study, being really smart and clear-headed would which is to advise policymakers who are try- behave. Notwithstanding the anomalies ing to improve the economic well-being of pointed out recently by behavioral econo- entire populations of people. In the mid- mists, the predictive power of economics twentieth century, say from 1940 to 1970, rests on the notion that people facing a famil- macroeconomics had an orientation toward iar situation will behave in their interests. its role much like an oracle giving advice Foolish, self-destructive, or purely random while peering down from the top of a moun- behavior will not be repeated once it is con- tain. That is, while economists could see peo- sciously seen to be less good than another ple making detailed decisions about buying course. The transformation of economic products, investing their wealth, choosing analysis by mathematics occurred through jobs or career paths, etc., macroeconomic the systematic understanding of rational and models were extremely simple. For instance, learning behavior by individuals and firms, describing the aggregate consumption of an and the overall implications of that for market entire economy could be done with taking interactions. into account just a few variables: aggregate

45 There are many stories told in the economics profes- 44 To get an idea of the precision of these results, sion about what giants of our field thought were the great- Blundell, Reed, and Stoker (2003) present bootstrap 95 est contributions to social science. In this spirit, we relate percent confidence bands for the corrected log wage esti- the following. In the mid-1980s, one of the authors asked mates for various groups. These plots show that the micro Paul Samuelson what he felt was the greatest failure in model prediction and the corrections to the log aggregate economics. Without hesitation, his answer was “macroeco- wage are both quite tightly estimated. In all cases the nomics and econometrics.” The reason for this is that there micro model prediction and the corrections to the aggre- had been an enormous anticipation in the 1940s, 1950s, gate wage plot are significantly different from the raw and 1960s that simple empirical macroeconomic models aggregate wage measure and not significantly different would, in fact, be accurate enough to allow real economies from each other. This gives us confidence that we have to be guided and controlled, much like an automobile or a identified compositional biases in the measured real wage spacecraft. That this turned out to not be possible was a with a reasonable degree of precision. source of great disappointment. jn05_Article 1 6/10/05 1:38 PM Page 384

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The merging of these two bodies of incorporating distributional information into thought—macroeconomics and optimal aggregate relationships, namely exact aggre- behavior of individuals—is among the great- gation models, in the context of how that est development of economics in the last half approach has been applied to the analysis of century. This advance has been recognized consumer demands. Second, we have shown by Nobel prizes to Lucas, Kydland, and how one can incorporate basic nonlinearity, Prescott, and one should expect more prizes insurance and dynamic elements, in our cov- to be awarded to other important develop- erage of aggregate consumption based on ers. Previous “schools of thought” have been CRRA preferences. Third, we have shown replaced by groups differentiated by how how to account for compositional hetero- they settle the tradeoff between realism and geneity, in our coverage of labor participation strict adherence to optimal economic behav- and wages. The latter two topics required ior. The specification of macroeconomic explicit assumptions on the distribution of models, the judgment of whether they are individual heterogeneity, and we have based sensible, and the understanding of the our solutions on normal and lognormal impacts of economic policy, is now more sys- assumptions on individual heterogeneity. tematic because of its embedding in the While these distributional restrictions are rules of optimal individual behavior. specific, they do permit explicit formulations The trouble is, this embedding cannot be of the aggregate relationships of interest to right without taking account of aggregation. be derived, and those formulations capture A one-person or five-person economy is just both location and spread (mean and vari- not realistic. You can simulate a model with ance) of the underlying elements of individ- a few actors and pretend that it is realistic, ual heterogeneity. We view our solutions in but there is nothing in casual observation or these cases as representative and clear, and empirical data or economic theory that sug- good starting points for empirical modeling gests that such a stance is valid. There is in the respective areas. much to be learned from rational individual Whether one dates the beginning of the behavior, but there must be a explicit bridge study of aggregation problems from the to economic aggregates because real people 1940s, 1930s, or perhaps earlier, one can at and their situations are so very heteroge- best describe progress toward solutions as neous. Aggregation is essential, because het- slow. Aggregation problems are among the erogeneity is a pervasive and indisputable most difficult problems faced in either the fact of life. theoretical or empirical study of economics. In this paper, we have covered recent work Heterogeneity across individuals is extreme- on aggregation problems in a style that we ly extensive and its impact is not obviously hope is useful to empirical economists. Our simplified or lessened by the existence of orientation has been to highlight the impor- economic interaction via markets or other tance of different types of individual hetero- institutions. The conditions under which geneity; in particular, heterogeneity in tastes one can ignore a great deal of the evidence and reaction, heterogeneity in market partici- of individual heterogeneity are so severe as pation, and heterogeneity in uninsurable to make them patently unrealistic. With ref- risks. Our approach has been practical; we erence to our introduction, as annoyances have covered recent advances in econometric go, aggregation problems are particularly modeling that address issues in aggregation by cloying. There is no quick, easy, or obvious considering explicit models at the individual fix to dealing with aggregation problems in level and among economic aggregates. general. We have covered a wide range of ideas. Yet we see the situation as hopeful and First, we have detailed the main approach for changing, and offer the solutions discussed jn05_Article 1 6/10/05 1:38 PM Page 385

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in this paper as evidence of that change. The with aggregate relationships. Within the last sources of this change are two-fold, and it is twenty five years (or the professional lives of worth pointing them out as well as pointing both authors), there has been dramatic out how both are necessary. changes in the ability to implement realistic The first source of change is the increas- models. Before this, it was extremely diffi- ing availability of data on individuals cult to implement models that are necessary observed over sequential time periods. To for understanding impacts of individual address questions of what kinds of individual heterogeneity in aggregation. heterogeneity are important for aggregate Aggregation problems remain among the relationships, one must assess what kinds of most vexing in all of applied economics. heterogeneity are relevant to individual While they have not become less difficult in behavior for the problem at hand and assess the past decade, it has become possible to how much the distributions of the relevant study aggregation problems in a meaningful heterogeneity vary over time. To the extent way. As such, there are many reasons to be that this heterogeneity reflects differences in optimistic about the prospects for steady unexpected shocks to individual agents, the progress on aggregation problems in the mechanisms that are available to individuals future. The practice of ignoring or closeting to insure against such shocks will have a aggregation problems as “just too hard” is no strong bearing on the form of the aggregate longer appropriate. relationship. While we have advanced the idea of using REFERENCES aggregation factors (derived from time series Aigner, Dennis J., and Stephen M. Goldfeld. 1974. “Estimation and Prediction from Aggregate Data of individual data) to summarize the impacts when Aggregates are Measured More Accurately than of aggregation, the specific method one uses Their Components.” Econometrica, 42(1): 113–34. is less important than the ability to use all Aiyagari, S. Rao. 1994. “Uninsured Idiosyncratic Risk and Aggregate Saving.” Quarterly Journal of available types of information to study eco- Economics, 109(3): 659–84. nomic relationships. That is, it is important Aiyagari, S. Rao. 1994. “Co-Existence of a to study any relationship among economic Representative Agent Type Equilibrium with a Non- Representative Agent Type Equilibrium.” Journal of aggregates with individual data as well as Economic Theory, 57(1): 230–36. aggregate data, to get as complete a picture Alessie, Rob, Michael P. Devereux, and Guglielmo as possible of the underlying structure. Even Weber. 1997. “Intertemporal Consumption, Durables and Liquidity Constraints: A Cohort though modeling assumptions will always be Analysis.” European Economic Review, 41(1): 37–59. necessary to develop explicit formulations of Alessie, Rob, Bertrand Melenberg, and Guglielmo aggregate relationships, testing those Weber. 1988. “Consumption, Leisure and Earnings- Related Liquidity Constraints: A Note.” Economics assumptions is extremely important and is Letters, 27(1): 101–04. not possible without extensive individual Allenby, Greg M., and Peter E. Rossi. 1991. “There Is data over sequential time periods. Our view No Aggregate Bias: Why Macro Logit Models Work.” Journal of Business and Economic Statistics, is that the prospects for meaningful advance 9(1): 1–14. continue to brighten, as the data situation Anderson, Simon P., Andre de Palma, and Jacques- with regard to individual behavior and François Thisse. 1989. “Demand for Differentiated Products, Discrete Choice Models, and the aggregate economic variables will continue Characteristics Approach.” Review of Economic to improve. Studies, 56(1): 21–35. The second source of change in studying Atkeson, Andrew, and Masao Ogaki. 1996. “Wealth- Varying Intertemporal Elasticities of Substitution: aggregation problems is the recent, rapid Evidence from Panel and Aggregate Data.” Journal rise in computing power. Realistic accom- of Monetary Economics, 38(3): 507–34. modation of individual heterogeneity typi- Atkinson, Anthony B., Joanna Gomulka, and Nicholas H. Stern. 1990. “Spending on Alcohol: Evidence cally requires extensive behavioral models, from the Family Expenditure Survey 1970–1983.” let alone combinations of individual models Economic Journal, 100(402): 808–27. jn05_Article 1 6/10/05 1:38 PM Page 386

386 Journal of Economic Literature, Vol. XLIII (June 2005)

Attanasio, Orazio P. 1999. “Consumption,” in Handbook “Specification of Household Engel Curves by of Macroeconomics. John B. Taylor and Michael Nonparametric Regression.” Econometric Reviews, Woodford, eds. Amsterdam: North Holland, 741–812. 9(2): 123–84. Attanasio, Orazio P., and Martin Browning. 1995. Bils, Mark J. 1985. “Real Wages over the Business “Consumption over the Life Cycle and over the Cycle: Evidence from Panel Data.” Journal of Business Cycle.” American Economic Review, 85(5): Political Economy, 93(4): 666–89. 1118–37. Blackorby, Charles, Richard Boyce, and R. Robert Attanasio, Orazio P., and Steven J. Davis. 1996. Russell. 1978. “Estimation of Demand Systems “Relative Wage Movements and the Distribution of Generated by the Gorman Polar Form: A Consumption.” Journal of Political Economy, 104(6): Generalization of the S-Branch Utility Tree.” 1227–62. Econometrica, 46(2): 345–63. Attanasio, Orazio P., Pinelopi K. Goldberg, and Blackorby, Charles, Daniel Primont, and R. Robert Ekaterini Kyriazidou. In progress. “On the Relevance Russell. 1978. Duality, Separability and Functional of Borrowing Constraints: Evidence from Car Loans.” Structure: Theory and Economic Applications. Attanasio, Orazio P., and Guglielmo Weber. 1993. Amsterdam: North Holland. “Consumption Growth, the Interest Rate and Blinder, Alan S. 1975. “Distribution Effects and the Aggregation.” Review of Economic Studies, 60(3): Aggregate Consumption Function.” Journal of 631–49. Political Economy, 83(3): 447–75. Attanasio, Orazio P., and Guglielmo Weber. 1995. “Is Blundell, Richard. 1988. “Consumer Behaviour: Consumption Growth Consistent with Intertemporal Theory and Empirical Evidence—A Survey.” Optimization? Evidence from the Consumer Economic Journal, 98(389): 16–65. Expenditure Survey.” Journal of Political Economy, Blundell, Richard, Martin Browning, and Ian A. 103(6): 1121–57. Crawford. 2003. “Nonparametric Engel Curves and Banks, James, Richard Blundell, and Agar Brugiavini. Revealed Preference.” Econometrica, 71(1): 205–40. 2001. “Risk Pooling, Precautionary Saving and Blundell, Richard, Martin Browning, and Costas Consumption Growth.” Review of Economic Studies, Meghir. 1994. “Consumer Demand and Life-Cycle 68(4): 757–79. Allocation of Household Expenditures.” Review of Banks, James, Richard Blundell, and Arthur Lewbel. Economic Studies, 61(1): 57–80. 1997. “Quadratic Engel Curves and Consumer Blundell, Richard, Alan Duncan, and Krishna Demand.” Review of Economics and Statistics, 79(4): Pendakur. 1998. “Semiparametric Estimation and 527–39. Consumer Demand.” Journal of Applied Banks, James, Richard Blundell, and Sarah Tanner. Econometrics, 13(5): 435–61. 1998. “Is There a Retirement-Savings Puzzle?” Blundell, Richard, John Ham, and Costas Meghir. American Economic Review, 88(4): 769–88. 1987. “Unemployment and Female Labour Supply.” Barker, Terence S., and M. Hashem Pesaran. 1990. Economic Journal, 97: 44–64. Disaggregation in Econometric Modeling. London: Blundell, Richard, and Thomas MaCurdy. 1999. Routledge. “Labor Supply: A Review of Alternative Barnett, William A. 1979. “Theoretical Foundations for Approaches,” in Handbook of Labor Economics Vol. the Rotterdam Model.” Review of Economic Studies, III(a). O. Ashenfelter and D. Card, eds. Amsterdam: 46(1): 109–30. North Holland, 1560–95. Barten, Anton P. 1964. “Family Composition, Prices and Blundell, Richard, Panos Pashardes, and Guglielmo Expenditure Patterns,” in Econometric Analysis for Weber. 1993. “What Do We Learn About Consumer National Economic Planning. P. E. Hart, F. Mills, and Demand Patterns from Micro Data?” American J. K. Whitaker, eds. London: Butterworths, 277–92. Economic Review, 83(3): 570–97. Bean, Charles R. 1986. “The Estimation of ‘Surprise’ Blundell, Richard, Luigi Pistaferri, and Ian Preston. Models and the ‘Surprise’ Consumption Function.” 2003. “Consumption Inequality and Partial Review of Economic Studies, 53(4): 497–516. Insurance.” University College London Discussion Beaudry, Paul and David Green. 1996. “Cohort Paper, 03/06. Patterns in Canadian Earnings: Assessing the Role of Blundell, Richard, and Ian Preston. 1998. Skill Premia in Inequality Trends.” University of “Consumption Inequality and Income Uncertainty.” British Columbia Working Paper. Quarterly Journal of Economics, 113(2): 603–40. Becker, Gary S. 1962. “Irrational Behavior and Blundell, Richard, Howard Reed, and Thomas M. Economic Theory.” Journal of Political Economy, Stoker. 2003. “Interpreting Aggregate Wage Growth: 70(1): 1–13. The Role of Labor Market Participation.” American Berndt, Ernst R., Masako N. Darrough, and W. E. Economic Review, 93(4): 1114–31. Diewert. 1977. “Flexible Functional Forms and Blundell, Richard, and Jean-Marc Robin. 2000. Expenditure Distributions: An Application to “Latent Separability: Grouping Goods without Weak Canadian Consumer Demand Functions.” Separability.” Econometrica, 68 (1):53-84. International Economic Review, 18(3): 651–75. Blundell, Richard, and Jean-Marc Robin. 1999. Bewley, Truman F. 1977. “The Permanent Income “Estimation in Large and Disaggregated Demand Hypothesis: A Theoretical Formulation.” Journal of Systems: An Estimator for Conditionally Linear Economic Theory, 16(2): 252–92. Systems.” Journal of Applied Econometrics, 14(3): Bierens, Herman J., and Hettie A. Pott-Buter. 1990. 209–32. jn05_Article 1 6/10/05 1:38 PM Page 387

Blundell and Stoker: Heterogeneity and Aggregation 387

Blundell, Richard, and Thomas M. Stoker. 1999. Dynamic Model of Aggregate Output Supply, Factor “Consumption and the Timing of Income Risk.” Demand and Entry and Exit for a Competitive European Economic Review, 43(3): 475–507. Industry with Heterogeneous Plants.” Journal of Blundell, Richard, and Thomas M. Stoker. 2003. Econometrics, 33(1-2): 237–62. “Models of Aggregate Economic Relationships that Chiappori, Pierre-Andre. 1988. “Rational Household Account for Heterogeneity.” Mimeo. Labor Supply.” Econometrica, 56(1): 63–90. Bourguignon, François, and Pierre-André Chiappori. Cochrane, John H. 1991. “A Simple Test of 1994. “The Collective Approach to Household Consumption Insurance.” Journal of Political Behavior,” in The Measurement of Household Economy, 99(5): 957–76. Welfare. R. Blundell, I. Preston, and I. Walker, eds. Cogan, John F. 1981. “Fixed Costs and Labor Supply.” Cambridge: Cambridge University Press, 70–86. Econometrica, 49(4): 945–63. Brown, Donald J., and Rosa L. Matzkin. 1996. Cowing, Thomas G., and Daniel L. McFadden. 1984. “Testable Restrictions on the Equilbrium Manifold.” Microeconomic Modeling and Policy Analysis. New Econometrica, 64(6): 1249–62. York: Academic Press. Browning, Martin. 1992. “Children and Household Deaton, Angus S. 1985. “Panel Data from Time Series Economic Behavior.” Journal of Economic of Cross-Sections.” Journal of Econometrics, 30(1-2): Literature, 30(3): 1434–75. 109–26. Browning, Martin. 1993. “Estimating Micro Parameters Deaton, Angus S. 1991. “Saving and Liquidity from Macro Data Alone: Some Pessimistic Constraints.” Econometrica, 59(5): 1221–48. Evidence.” Ricerche Economiche, 47(3): 253–67. Deaton, Angus S. 1993. Understanding Consumption. Browning, Martin, Lars P. Hansen, and James J. Oxford: Oxford University Press. Heckman. 1999. “Micro Data and General Deaton, Angus S., and John Muellbauer. 1980a. “An Equilibrium Models,” in Handbook of Almost Ideal Demand System.” American Economic Macroeconomics. J. Taylor and M. Woodford, eds. Review, 70(3): 312–26. Amsterdam: Elsevier, 543–633. Deaton, Angus S., and John Muellbauer. 1980b. Browning, Martin, and Annamaria Lusardi. 1996. Economics and Consumer Behavior. Cambridge: “Household Saving: Micro Theories and Micro Cambridge University Press. Facts.” Journal of Economic Literature, 34(4): Deaton, Angus S., and Christina Paxson. 1994. 1797–1855. “Intertemporal Choice and Inequality.” Journal of Browning, Martin, and Costas Meghir. 1991. “The Political Economy, 102(3): 437–67. Effects of Male and Female Labor Supply on Debreu, Gerard. 1974. “Excess Demand Functions.” Commodity Demands.” Econometrica, 59(4): 925–51. Journal of Mathematical Economics. 1(1): 15–23. Buse, Adolph. 1992. “Aggregation, Distribution and Dickens, Richard. 1996. “The Evolution of Individual Dynamics in the Linear and Quadratic Expenditure Male Earnings in Great Britain, 1974–1994.” CEP Systems.” Review of Economics and Statistics, 74(1): Discussion Paper No. 306. 45–53. Diewert, W. E. 1977. “Generalized Slutsky Conditions Caballero, Ricardo J. 1990. “Consumption Puzzles and for Aggregate Consumer Demand Functions.” Precautionary Savings.” Journal of Monetary Journal of Economic Theory, 15(2): 353–62. Economics, 25(1): 113–36. Dumas, Bernard. 1989. “Two-Person Dynamic Caballero, Ricardo J., and Eduardo M. R. A. Engel. Equilibrium in the Capital Market.” Review of 1991. “Dynamic (S,s) Economies.” Econometrica, Financial Studies, 2(2): 157–88. 59(6): 1659–86. Dynan, Karen E. 1993. “How Prudent Are Caballero, Ricardo J., and Eduardo M. R. A. Engel. Consumers?” Journal of Political Economy, 101(6): 1992. “Microeconomic Adjustment Hazards and 1104–13. Aggregate Dynamics.” NBER Working Paper No. Epstein, Larry G., and Stanley E. Zin. 1989. 4090. “Substitution, Risk Aversion, and the Temporal Cameron, A. Colin. 1990. “Aggregation in Discrete Behavior of Consumption and Asset Returns: A Choice Models: An Illustration of Nonlinear Theoretical Framework.” Econometrica, 57(4): Aggregation,” in Disaggregation in Econometric 937–69. Modeling. T. S. Barker and M. H. Pesaran, eds. Epstein, Larry G., and Stanley E. Zin. 1991. London: Routledge, 206–34. “Substitution, Risk Aversion, and the Temporal Caplin, Andrew, and Barry Nalebuff. 1991. Behavior of Consumption and Asset Returns: An “Aggregation and Imperfect Competition: On the Empirical Analysis.” Journal of Political Economy, Existence of Equilibrium.” Econometrica, 59(1): 99(2): 263–86. 25–59. Fair, Ray C., and Kathryn M. Dominguez. 1991. Caplin, Andrew S., and Daniel F. Spulber. 1987. “Effects of the Changing U.S. Age Distribution on “Menu Costs and the Neutrality of Money.” Macroeconomic Equations.” American Economic Quarterly Journal of Economics, 102(4): 703–25. Review, 81(5): 1276–94. Carroll, Christopher D. 2000. “Requiem for the Feenberg, Daniel R. and Harvey S. Rosen. 1983. Representative Consumer? Aggregate Implications “Alternative Tax Treatments of the Family: of Microeconomic Consumption Behavior.” Simulation Methodology and Results,” in Behavioral American Economic Review, 90(2): 110–15. Simulation Methods in Tax Policy Analysis. M. S. Chetty, V. K., and James J. Heckman. 1986. “A Feldstein, ed. Chicago: University of Chicago Press. jn05_Article 1 6/10/05 1:38 PM Page 388

388 Journal of Economic Literature, Vol. XLIII (June 2005)

Forni, Mario and Marco Lippi. 1997. Aggregation and Errors-in-Variables Models.” Journal of the Microfoundations of Dynamic Macroeconomics. Econometrics, 50(3): 273–95. Oxford: Clarendon Press. Hausman, Jerry A., Whitney K. Newey, and James L. Freixas, Xavier, and Andreu Mas-Colell. 1987. “Engel Powell. 1994. “Nonlinear Errors-In-Variables: Curves Leading to the Weak Axiom in the Estimation of Some Engel Curves.” Journal of Aggregate.” Econometrica, 55(3): 515–31. Econometrics, 65(1): 205–33. Garcia, Rene, Annamaria Lusardi, and Serena Ng. Hayashi, Fumio. 1985. “The Effect of Liquidity 1997. “Excess Sensitivity and Asymmetries in Constraints on Consumption: A Cross-sectional Consumption.” Journal of Money, Credit, and Analysis.” Quarterly Journal of Economics, 100(1): Banking, 29(2): 154–76. 183–206. Goodfriend, Marvin. 1992. “Information-Aggregation Hayashi, Fumio. 1987. “Tests for Liquidity Constraints: Bias.” American Economic Review, 82(3): 508–19. A Critical Survey,” in Advances in Econometrics, Gorman, William M. 1953. “Community Preference Fifth World Congress, Vol. 2. T. Bewley, ed. Fields.” Econometrica, 21(1): 63–80. Cambridge: Cambridge University Press, 91–120. Gorman, William M. 1981. “Some Engel Curves,” in Hayashi, Fumio, Joseph Altonji, and Laurence Essays in the Theory and Measurement of Consumer Kotlikoff. 1996. “Risk-Sharing between and within Behaviour. A. S. Deaton, ed. Cambridge: Cambridge Families.” Econometrica, 64(2): 261–94. University Press, 7–29. Heaton, John, and Deborah J. Lucas. 1996. Gosling, Amanda, Stephen Machin, and Costas “Evaluating the Effects of Incomplete Markets on Meghir. 2000. “The Changing Distribution of Male Risk Sharing and Asset Pricing.” Journal of Political Wages in the U.K., 1966–93.” Review of Economic Economy, 104(3): 443–87. Studies, 67(4): 635–66. Heckman, James J. 1974. “Effects of Child-Care Grandmont, Jean-Michel. 1992. “Transformations of Programs on Women’s Work Effort.” Journal of the Commodity Space, Behavioral Heterogeneity, Political Economy, 82(2): S136–163. and the Aggregation Problem.” Journal of Economic Heckman, James J. 1979. “Sample Selection Bias as a Theory, 57(1): 1–35. Specification Error.” Econometrica, 47(1): 153–61. Granger, Clive W. J. 1980. “Long Memory Heckman, James J. 1990. “Varieties of Selection Bias.” Relationships and the Aggregation of Dynamic American Economic Review, 80(2): 313–18. Models.” Journal of Econometrics, 14(2): 227–38. Heckman, James J., and Bo E. Honore. 1990. “The Granger, Clive W. J. 1987. “Implications of Aggregation Empirical Content of the Roy Model.” with Common Factors.” Econometric Theory, 3: Econometrica, 58(5): 1121–49. 208–22. Heckman, James J., and Guilherme Sedlacek. 1985. Granger, Clive W. J. 1990. “Aggregation of Time-Series “Heterogeneity, Aggregation, and Market Wage Variables: A Survey,” in Disaggregation in Functions: An Empirical Model of Self-Selection in Econometric Modelling.T.Barker and M. H. Pesaran, the Labor Market.” Journal of Political Economy, eds. London and New York: Routledge, 17–34. 93(6): 1077–1125. Grunfeld, Yehuda and Zvi Griliches. 1960. “Is Heckman, James J., and Guilherme L. Sedlacek. 1990. Aggregation Necessarily Bad?” Review of Economics “Self-Selection and the Distribution of Hourly and Statistics, 42(1): 1–13. Wages.” Journal of Labor Economics, 8(1): S329–63. Hall, Robert E. 1978. “Stochastic Implications of the Heckman, James J., and James R. Walker. 1989. Life Cycle-Permanent Income Hypothesis: Theory “Forecasting Aggregate Period-Specific Birth Rates: and Evidence.” Journal of Political Economy, 86(6): The Time Series Properties of a Microdynamic 971–87. Neoclassical Model of Fertility.” Journal of the Hall, Robert E. 1988. “Intertemporal Substitution in American Statistical Association, 84(408): 958–65. Consumption.” Journal of Political Economy, 96(2): Heckman, James J., and James R. Walker. 1990. “The 339–57. Relationship between Wages and Income and the Hansen, Gary D. 1985. “Indivisible Labor and the Timing and Spacing of Births: Evidence from Business Cycle.” Journal of Monetary Economics, Swedish Longitudinal Data.” Econometrica, 58(6): 16(3): 309–27. 1411–41. Hansen, Lars Peter, and Kenneth J. Singleton. 1982. Heineke, J. M., and H. M. Shefrin. 1990. “Aggregation “Generalized Instrumental Variables Estimation of and Identification in Consumer Demand Systems.” Nonlinear Rational Expectations Models.” Journal of Econometrics, 44(3): 377–90. Econometrica, 50(5): 1269–86. Hicks, J. R. 1956. A Revision of Demand Theory. Hansen, Lars Peter, and Kenneth J. Singleton. 1983. London: Oxford University Press. “Stochastic Consumption, Risk Aversion, and the Hildenbrand, Werner. 1981. “Short-Run Production Temporal Behavior of Asset Returns.” Journal of Functions Based on Microdata.” Econometrica, Political Economy, 91(2): 249–65. 49(5): 1095–1125. Härdle, Wolfgang, Werner Hildenbrand, and Michael Hildenbrand, Werner. 1983. “On the ‘Law of Jerison. 1991. “Empirical Evidence on the Law of Demand’.” Econometrica, 51(4): 997–1019. Demand.” Econometrica, 59(6): 1525–49. Hildenbrand, Werner. 1992. “Market Demand, Theory Hausman, Jerry A., Whitney K. Newey, Hidehiko and Empirical Evidence.” Universität Bonn Ichimura, and James L. Powell. 1991. Discussion Paper A-359. “Identification and Estimation of Polynomial Hildenbrand, Werner. 1994. Market Demand: Theory jn05_Article 1 6/10/05 1:38 PM Page 389

Blundell and Stoker: Heterogeneity and Aggregation 389

and Empirical Evidence. Princeton: Princeton Krusell, Per, and Anthony A. Smith, Jr. 1998. “Income University Press. and Wealth Heterogeneity in the Macroeconomy.” Hildenbrand, Werner. 1998. “How Relevant are Journal of Political Economy, 106(5): 867–96. Specifications of Behavioral Relations on the Micro- Lam, Pok-Sang. 1991. “Permanent Income, Liquidity, Level for Modelling the Time Path of Population and Adjustments of Automobile Stocks: Evidence Aggregates?” European Economic Review, 42(3-5): from Panel Data.” Quarterly Journal of Economics, 437–58. 106(1): 203–30. Hildenbrand, Werner, and Alois Kneip. 1993. “Family Lau, Lawrence J. 1977. “Existence Conditions for Expenditure Data, Heteroscedasticity and the Law Aggregate Demand Functions.” Institute for of Demand.” Ricerche Economiche, 47(2): 137–65. Mathematical Studies in the Social Sciences, Houthakker, Hendrick S. 1955. “The Pareto Stanford University, Technical Report No. 248. Distribution and the Cobb-Douglas Production Lau, Lawrence J. 1982. “A Note on the Fundamental Function in Activity Analysis.” Review of Economic Theorem of Exact Aggregation.” Economics Letters, Studies, 23(1): 27–31. 9(1): 119–26. Jappelli, Tullio. 1990. “Who Is Credit Constrained in Lee, Kevin C., M. Hashem Pesaran, and Richard G. the U.S. Economy?” Quarterly Journal of Pierse. 1990. “Testing for Aggregation Bias in Linear Economics, 105(1): 219–34. Models.” Economic Journal, 100(400): 137–50. Jappelli, Tullio, and Marco Pagano. 1994. “Saving, Lee, Lung-Fei, and Robert H. Porter. 1984. “Switching Growth, and Liquidity Constraints.” Quarterly Regression Models with Imperfect Sample Journal of Economics, 109(1): 83–109. Separation Information: With an Application on Jappelli, Tullio, Jorn-Steffen Pischke, and Nicholas S. Cartel Stability.” Econometrica, 52(2): 391–418. Souleles. 1998. “Testing for Liquidity Constraints in Leser, Conrad E. V. 1963. “Forms of Engel Functions.” Euler Equations with Complementary Data Sources.” Econometrica, 31(4): 694–703. Review of Economics and Statistics, 80(2): 251–62. Lewbel, Arthur. 1989a. “A Demand System Rank Johansen, Leif. 1972. Production Functions. Theorem.” Econometrica, 57(3): 701–05. Amsterdam: North Holland. Lewbel, Arthur. 1989b. “Exact Aggregation and a Joint Committee on Taxation. 1992. “Discussion of Representative Consumer.” Quarterly Journal of Revenue Estimation Methodology and Process.” JCS Economics, 104(3): 621–33. 14-92, August 13, 1992. Washington, D.C.: U.S. Lewbel, Arthur. 1990. “Income Distribution Government Printing Office. Movements and Aggregate Money Illusion.” Journal Jorgenson, Dale W., Lawrence J. Lau, and Thomas M. of Econometrics, 43(1-2): 35–42. Stoker. 1980. “Welfare Comparison and Exact Lewbel, Arthur. 1991. “The Rank of Demand Aggregation.” American Economic Review, 70(2): Systems: Theory and Nonparametric Estimation.” 268–72. Econometrica, 59(3): 711–30. Jorgenson, Dale W., Lawrence J. Lau, and Thomas M. Lewbel, Arthur. 1992. “Aggregation with Log-Linear Stoker. 1982. “The Transcendental Logarithmic Models.” Review of Economic Studies, 59(3): 635–42. Model of Aggregate Consumer Behavior,” in Lewbel, Arthur. 1993. “Distribution Movements, Advances in Econometrics, Vol. 1. R. L. Basmann Macroeconomic Regularities and the Representative and G. Rhodes, eds. Greenwich: JAI Press, 97–238. Consumer.” Ricerche Economiche 47(2): 189–99. Jorgenson, Dale W. and Daniel T. Slesnick. 1984. Lewbel, Arthur. 1994. “Aggregation and Simple “Aggregate Consumer Behavior and the Dynamics.” American Economic Review, 84(4): Measurement of Inequality.” Review of Economic 905–18. Studies, 51(3): 369–92. Lippi, Marco. 1988. “On the Dynamic Shape of Jorgenson, Dale W., Daniel T. Slesnick, and Thomas Aggregated Error Correction Models.” Journal of M. Stoker. 1988. “Two-Stage Budgeting and Exact Economic Dynamics and Control, 12(2-3): 561–85. Aggregation.” Journal of Business and Economic Lusardi, Annamaria. 1996. “Permanent Income, Statistics, 6(3): 313–25. Current Income, and Consumption: Evidence from Jorgenson, Dale W., and Thomas M. Stoker. 1997. Two Panel Data Sets.” Journal of Business and “Nonlinear Three-Stage Least Squares Pooling of Economic Statistics, 14(1): 81–90. Time Series and Cross-Section Data,” in Advances in Mace, Barbara J. 1991. “Full Insurance in the Presence Statistical Analysis and Statistical Computing, Vol. 1. of Aggregate Uncertainty.” Journal of Political D. W. Jorgenson, ed. Greenwich: JAI Press, 87–116. Economy, 99(5): 928–56. Katz, Lawrence F., and Kevin M. Murphy. 1992. MaCurdy, Thomas E. 1982. “The Use of Time Series “Changes in Relative Wages, 1963–1987: Supply and Processes to Model the Error Structure of Earnings Demand Factors.” Quarterly Journal of Economics, in a Longitudinal Data Analysis.” Journal of 107(1): 35–78. Econometrics, 18(1): 83–114. Kelejian, Harry H. 1980. “Aggregation and MaCurdy, Thomas E. 1987. “A Framework for Disaggregation of Nonlinear Equations,” in Relating Microeconomic and Macroeconomic Evaluation of Econometric Models. J. Kmenta and J. Evidence on Intertemporal Substitution,” in B. Ramsey, eds. New York: Academic Press. Advances in Econometrics, Fifth World Congress, Kirman, Alan P. 1992. “Whom or What Does the Vol. 2. T. F. Bewley, ed. Cambridge: Cambridge Representative Individual Represent?” Journal of University Press, 149–76. Economic Perspectives, 6(2): 117–36. Mankiw, N. Gregory, and David N. Weil. 1989. “The jn05_Article 1 6/10/05 1:38 PM Page 390

390 Journal of Economic Literature, Vol. XLIII (June 2005)

Baby Boom, the Baby Bust, and the Housing An Alternative Approach.” Journal of Public Market.” Regional Science and Urban Economics, Economics, 22(1): 89–102. 19(2): 235–58. Rubinstein, Mark. 1974. “An Aggregation Theorem for Marshak, Jacob. 1939. “Personal and Collective Budget Securities Markets.” Journal of Financial Economics, Functions.” Review of Economic Statistics, 21: 1(3): 225–44. 161–70. Runkle, David E. 1991. “Liquidity Constraints and the McFadden, Daniel, and F. Reid. 1975. “Aggregate Permanent-Income Hypothesis: Evidence from Travel Demand Forecasting from Disaggregated Panel Data.” Journal of Monetary Economics, 27(1): Behavioral Models.” Transportation Research 73–98. Record, No. 534. Roy, Andrew D. 1951. “Some Thoughts on the Meghir, Costas, and Luigi Pistaferri. 2004. “Income Distribution of Earnings.” Oxford Economic Papers, Variance Dynamics and Heterogeneity.” 3: 135–46. Econometrica, 72(1): 1–32. Samuelson, Paul A. 1956. “Social Indifference Curves.” Meghir, Costas, and Guglielmo Weber. 1996. Quarterly Journal of Economics, 70(1): 1–22. “Intertemporal Nonseparability or Borrowing Sato, Kazuo. 1975. Production Functions and Restrictions? A Disaggregate Analysis Using a U.S. Aggregation. Amsterdam: North Holland. Consumption Panel.” Econometrica, 64(5): Scheinkman, Jose A., and Laurence Weiss. 1986. 1151–81. “Borrowing Constraints and Aggregate Economic Meghir, Costas, and Edward Whitehouse. 1996. “The Activity.” Econometrica, 54(1): 23–45. Evolution of Wages in the United Kingdom: Shafer, Wayne and Hugh Sonnenschein. 1982. “Market Evidence from Micro Data.” Journal of Labor Demand and Excess Demand Functions,” in Economics, 14(1): 1–25. Handbook of Mathematical Economics, Vol. 2. K. J. Modigliani, Franco. 1970. “The Life Cycle Hypothesis Arrow and M. D. Intriligator, eds. New York: North of Saving and Intercountry Differences in the Saving Holland, 339–53. Ratio,” in Induction, Growth and Trade. W. Eltis, M. Skinner, Jonathan. 1988. “Risky Income, Life Cycle Scott, and I. Wolfe, eds. Oxford: Clarendon Press, Consumption, and Precautionary Savings.” Journal 197–225. of Monetary Economics, 22(2): 237–55. Moffitt, Robert. 1991. “Identification and Estimation Solon, Gary, Robert Barsky, and Jonathan A. Parker. of Dynamic Models with a Time Series of Repeated 1994. “Measuring the Cyclicality of Real Wages: Cross Sections.” Brown University. Draft. How Important Is Composition Bias?” Quarterly Muellbauer, John. 1975. “Aggregation, Income Journal of Economics, 109(1): 1–25. Distribution and Consumer Demand.” Review of Sonnenschein, Hugo. 1972. “Market Excess Demand Economic Studies, 42(4): 525–43. Functions.” Econometrica, 40(3): 549–63. Muellbauer, John. 1976. “Community Preferences and Stoker, Thomas M. 1978. “The Pooling of Cross the Representative Consumer.” Econometrica, 44(5): Section and Average Time Series Data.” Draft. 979–99. Harvard University. Muellbauer, John. 1981. “Linear Aggregation in Stoker, Thomas M. 1982. “The Use of Cross-Section Neoclassical Labour Supply.” Review of Economic Data to Characterize Macro Functions.” Journal of Studies, 48(1): 21–36. the American Statistical Association, 77(378): Pesaran, M. Hashem, Richard G. Pierse, and Mohan S. 369–80. Kumar. 1989. “Econometric Analysis of Aggregation Stoker, Thomas M. 1984a. “Completeness, in the Context of Linear Prediction Models.” Distribution Restrictions, and the Form of Econometrica, 57(4): 861–88. Aggregate Functions.” Econometrica, 52(4): Pischke, Jorn-Steffen. 1995. “Individual Income, 887–907. Incomplete Information, and Aggregate Stoker, Thomas M. 1984b. “Exact Aggregation and Consumption.” Econometrica, 63(4): 805–40. Generalized Slutsky Conditions.” Journal of Pollak, Robert A. 1985. “A Transaction Cost Approach Economic Theory, 33(2): 368–77. to Families and Households.” Journal of Economic Stoker, Thomas M. 1985. “Aggregation, Structural Literature, 23(2): 581–608. Change, and Cross-Section Estimation.” Journal of Pollak, Robert A., and Terence J. Wales. 1981. the American Statistical Association, 80(391): “Demographic Variables in Demand Analysis.” 720–29. Econometrica, 49(6): 1533–51. Stoker, Thomas M. 1986a. “Aggregation, Efficiency, Powell, James L., and Thomas M. Stoker. 1985. “The and Cross-Section Regression.” Econometrica, 54(1): Estimation of Complete Aggregation Structures.” 171–88. Journal of Econometrics, 30(1–2): 317–44. Stoker, Thomas M. 1986b. “Consistent Estimation of Powell, James L. 1994. “Estimation of Semi- Scaled Coefficients.” Econometrica, 54(6): 1461–81. Parametric Models,” in Handbook of Econometrics, Stoker, Thomas M. 1986c. “The Distributional Welfare Vol. 4. R. F. Engel and D. L. McFadden, eds. Effects of Rising Prices in the United States: The Amsterdam: Elsevier, 2443–2521. 1970’s Experience.” American Economic Review, Prescott, Edward C. 1986. “Theory Ahead of Business 76(3): 335–49. Cycle Measurement.” Federal Reserve Bank of Stoker, Thomas M. 1986d. “Simple Tests of Minneapolis Quarterly Review, 10(4): 9–22. Distributional Effects on Macroeconomic Ray, Ranjan. 1983. “Measuring the Costs of Children: Equations.” Journal of Political Economy, 94(4): jn05_Article 1 6/10/05 1:38 PM Page 391

Blundell and Stoker: Heterogeneity and Aggregation 391

763–95. de Statistique 29: 157–73. Stoker, Thomas M. 1992. Lectures on Semiparametric de Wolff, Pieter. 1941. “Income Elasticity of Demand, Econometrics. CORE Lecture Series, Louvain-la- a Micro-Economic and a Macro-Economic Neuve, CORE Foundation. Interpretation.” Economic Journal, 51: 140–45. Stoker, Thomas M. 1993. “Empirical Approaches to the Working, Harold. 1943. “Statistical Laws of Family Problem of Aggregation Over Individuals.” Journal Expenditure.” Journal of the American Statistical of Economic Literature, 31(4): 1827–74. Association, 38: 43–56. Theil, Henri. 1954. Linear Aggregation of Economic Zeldes, Stephen P. 1989a. “Consumption and Liquidity Relations. Amsterdam: North Holland. Constraints: An Empirical Investigation.” Journal of Theil, Henri. 1975. Theory and Measurement of Political Economy, 97(2): 305–46. Consumer Demand, Vol. 1. Amsterdam: North Zeldes, Stephen P. 1989b. “Optimal Consumption with Holland. Stochastic Income: Deviations from Certainty Townsend, Robert M. 1994. “Risk and Insurance in Equivalence.” Quarterly Journal of Economics, Village India.” Econometrica, 62(3): 539–91. 104(2): 275–98. Van Daal, Jan, and Arnold H. Q. M. Merkies. 1984. Zellner, Arnold. 1969. “On the Aggregation Problem, a Aggregation in Economic Research. Dordrecht: D. New Approach to a Troublesome Problem,” in Reidel. Economic Models, Estimation and Risk Weber, Guglielmo. 1993. “Earnings-Related Programming, Essays in Honor of Gerhard Tintner. Borrowing Restrictions: Empirical Evidence from a K. A. Fox, G. V. L. Narasimhan, and J. K. Sengupta, Pseudo Panel for the U.K.” Annales d’Economie et eds. Berlin: Springer, 365–74.