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Journal of Economic Literature Vol. XXXVI (December 1998), pp. 2108–2165

Slesnick: Empirical Approaches to Measurement of Welfare

Empirical Approaches to theMeasurement of Welfare

DANIEL T. SLESNICK1

1. Introduction

THE MEASUREMENT of welfareforms the foundation of public policy

analysis. A full consideration of taxes,subsidies, transfer programs, health carereform, regulation, environmental policy,the social security system, and educa-tional reform must ultimately addressthe question of how these policies affectthe well-being of individuals. While cen-trally important to many problems ofeconomic analysis, confusion persistsconcerning the relationship betweencommonly used welfare indicators andwell-established theoretical formula-tions. For years it seemed that theoreti-cal and empirical research developedalong parallel lines with little, if any,cross-fertilization.

The gap between theory and applica-tions in welfare economics is closing. Inthe first part of the survey, I examinethe conceptual and empirical issues re-lated to welfare measurement at the mi-cro level. Over the years, analysts haverelied heavily on consumer’s surplus tomeasure the welfare effects of changesin prices and incomes. This largely re-flects the modest data requirements

and the ease with which it is imple-mented. Despite its ubiquity, it is nowwidely accepted that consumer’s surplusshould not be used as a welfare mea-sure, although there is less agreementas to why. The critical issues of the de-bate are summarized in Section 2.1.

The last twenty-five years have seengreat progress in the development ofmeasures that are ordinally equivalentto household utility. These welfare indi-cators, which are described Section 2.2,do not have the same limitations as con-sumer’s surplus, but are easy to imple-ment and have the same data require-ments. Each infers changes in welfarefrom the consumption behavior ofhouseholds, using either econometricmethods or index numbers. As a result,there is substantial overlap between theempirical issues associated with demandmodeling (both static and intertempo-ral) and the general problem of welfaremeasurement. I summarize several im-portant econometric issues in Section2.3, and recent developments using theindex number framework are presentedin Section 2.4. The first part of the sur-vey concludes with a description ofthree applications that illustrate manyof the general empirical issues.

While welfare measurement at themicro level is of independent interest,of greater concern is the well-being ofgroups of households. The second half

2108

1 University of Texas, Austin. I would like tothank Don Fullerton, Dan Hamermesh, ArthurLewbel, Bob Pollak, Max Stinchcombe, and FrankWolak for providing thoughtful comments andsuggestions on previous drafts of this paper. Anyremaining errors are solely my responsibility.

of the survey will examine the aggrega-tion problem in applied welfare eco-nomics. The most common approach isto assume the existence of a repre-sentative consumer and use market de-mands as the basis for the measurementof social welfare. This is unappealingboth because distributional issues areignored and because much evidenceshows that aggregate demands are in-consistent with the behavior of a singlerepresentative agent.

To go beyond this framework re-quires normative judgements concern-ing the measurability and comparabilityof welfare across heterogeneous agents.In most applications, these assumptionsare implicit but no less arbitrary thanthose based on explicit social welfarefunctions. For example, the commonpractice of representing social welfareby per capita income implicitly assumescardinally measurable welfare that isfully comparable across households.Since incomes are simply averagedacross the population, the implicitsocial welfare function is utilitarian andthe marginal utility of income is as-sumed to be constant.

Social choice theory is a naturalframework for examining the normativebasis for aggregate welfare measuresused in practice. How does one getaround Arrow’s Impossibility Theorem?If this is feasible, what additional as-sumptions are required? Beyond themany conceptual issues, what types ofhousehold welfare functions and dataare required to “build” a function thatcan be used to measure aggregatewelfare? Is the aggregation problem anarcane issue of interest to only a hand-ful of social choice theorists? Infact, the measurement of group wel-fare lies at the heart of some of themost pressing issues in applied econom-ics. In some cases, the method of aggre-gation has little effect on our conclu-

sions, while in others it is vitally impor-tant.

As in all surveys of finite length,some noteworthy contributions are ig-nored here or given only cursory con-sideration. I intend to restrict the focusto developments in the “welfarist” tradi-tion in which well-being is derived fromthe consumption of goods and services.This seems to be the natural focus sincemost empirical work is in this frame-work, although limiting the scope hasthe unfortunate consequence of short-changing several important develop-ments in welfare economics. For exam-ple, Sen’s capabilities approach towelfare measurement is not consideredin depth largely because of the relativescarcity of empirical applications. Thesurvey-based or “subjective” approachpioneered by van Praag, Kapteyn andothers is fundamentally different fromwelfare estimates based on households’revealed preferences and is also ex-cluded. A growing body of theoreticaland empirical work suggests that alloca-tions of goods within households haveimportant consequences for welfaremeasurement. While this issue will bediscussed, it will not play a prominentrole in this survey. Finally, let me em-phasize that no attempt is made to sur-vey the voluminous social choice litera-ture. The intention is to use key resultson the possibility of consistent socialchoice to describe a framework for ex-amining alternative approaches to themeasurement of aggregate welfare.

2. Measuring Household Welfare

It is natural to organize the discus-sion of developments in welfare eco-nomics by the level of aggregation atwhich they are applied. I begin withthe issue of welfare measurement forhouseholds and focus on the welfareeffects of price and income changes.

Slesnick: Empirical Approaches to Measurement of Welfare 2109

Infinite divisibility is assumed and welfareis broadly defined as the money neededto maintain a constant level of utility.

2.1 What’s Wrong with Consumer’s Surplus

It has now been over twenty-fiveyears since Arnold Harberger (1971)published his open letter to the profes-sion in which he proposed a set ofguidelines for applied welfare econo-mists. Simply put, consumer’s surplusshould be used to measure individualwelfare, and social welfare should bebased on its unweighted sum over thepopulation. A casual examination of thelarge number of empirical cost-benefitanalyses suggests that many practitio-ners have taken Harberger’s advice toheart. Consumer’s surplus is the over-whelming choice as a welfare indicator,largely, I would guess, because it is anintuitive concept with modest data re-quirements.

From its inception over 150 yearsago, the validity of consumer’s surplusas a welfare measure has been ques-tioned.2 Harberger’s proposal gener-ated a new round of debate that evokedmany of the same arguments that hadcirculated for years. The discussiontook a more productive turn becauseanalysts not only demonstrated whatwas wrong with consumer’s surplus butsuggested reasonable alternatives foractually “doing” welfare economics.

As a point of departure, I summarizethe conceptual problems with con-sumer’s surplus. Chipman and Moore(1976b, 1980) have shown that for awide variety of applications, consumer’ssurplus is a valid measure only underrestrictive conditions on preferences.Define Mk to be the level of expen-diture of the kth household,3 p = (p1,

p2, ..., pn) is a vector of prices, Ak is avector of demographic characteristicsand xi(p,Mk,Ak) is the demand functionfor good i. In the simplest case of achange in the price of a single good(say, commodity 1) from p1

0 to p11, the

change in consumer’s surplus is thenegative of the change in the area un-der the demand curve:

∆CSk = − ∫ p1

0

p11

x1(t, p2, ⋅⋅⋅ , pn, Mk, Ak)dt. (1)

Is consumer’s surplus an exact repre-sentation of the change in welfare? Thatis, is it single valued and ordinallyequivalent to the change in utility? Anecessary condition is that demands aregenerated by a rational consumer whomaximizes utility subject the constraintof limited resources. Unless consumershave optimized and are at the bounda-ries of their budget sets, it is impossibleto assess the welfare implications ofchanges in prices and incomes. Themaintained hypothesis must thereforeinclude the condition that demands are“integrable” and consistent with a well-behaved utility function.4

Under these conditions, an indirectutility function V(p, Mk, Ak) represents themaximum attainable utility at prices pand expenditure Mk. Application of Roy’s

2 An historical perspective on the early debateover the use of consumer’s surplus is provided byJohn Chipman and James Moore (1976b).

3 While income is commonly used as an argu-ment of the demand function, in a static contextthe appropriate variable is total expenditure.Moreover, I’ll take the basic observational unit tobe a household rather than an individual since thisis the way expenditure data are usually reported.Of ultimate concern, of course, is the welfare ofthe individuals who belong to the household,which is an issue that will be considered furtherwhen I examine the intrahousehold allocation ofresources.

4 The integrability conditions are described byLeonid Hurwicz and Hirofumi Uzawa (1971). Ifdemands are integrable and generated by a well-behaved utility function they must be nonnegative,summable, homogeneous of degree zero in pricesand total expenditure, and the Slutsky matrix ofcompensated price effects must be symmetric andnegative semidefinite.

2110 Journal of Economic Literature, Vol. XXXVI (December 1998)

Identity provides the link between the de-mands and the indirect utility function:

x1(p, Mk, Ak) = − ∂V(p, Mk, Ak) ⁄ ∂p1

∂V(p, Mk, Ak) ⁄ ∂Mk. (2)

If the marginal utility of income is con-stant, substitution of (2) into (1) yieldsan explicit expression that is ordinallyequivalent to the change in utility:5

∆CSk = V(p1, Mk, Ak) − V(p0, Mk, Ak)

∂V ⁄ ∂Mk.

For a single price change, constancy ofthe marginal utility of income is suffi-cient to ensure that consumer’s surplusprovides an exact measure of the changein welfare.

While this condition is restrictive, theproblems become more serious withchanges in more than one price. Thechange in consumer’s surplus must beevaluated using a line integral definedover the path of price changes:

∆CSk = − ∫Σp0

p1

xi(p, Mk, Ak)dpi, (3)

where p0 and p1 are the initial and finalprice vectors.

This extension is conceptuallystraightforward but suffers from theproblem that line integrals generally de-pend on the paths over which they areevaluated. Since we only observe theinitial and final price vectors, thechange in consumer’s surplus (3) mustbe independent of the path that, inturn, occurs if the uncompensated priceeffects are symmetric:6

∂xi

∂pj =

∂xj

∂pi for all i ≠ j.

A rational consumer will exhibit compen-sated price effects that are symmetric,but uncompensated demands will havethis property only if the income effectsare always equal. This requires equalityof the income elasticities which can onlyoccur if they are equal to one and prefer-ences are homothetic.7 This is a seriousproblem from a practitioner’s perspec-tive since now almost 150 years of em-pirical evidence demonstrates that de-mand patterns are inconsistent withhomotheticity.8

If we want to measure welfare in themost general circumstance, in whichprices and total expenditure change, wemust have path independence over bothsets of variables:

∆CSk = − ∫ΣZ

xi(p, Mk, Ak)dpi

+ (M1 − M0), (4)

where Z is a path between (p0, M0) and(p1, M1). Chipman and Moore (1976b) haveshown there are no circumstances inwhich (4) is path independent and ordi-nally equivalent to the the change in wel-fare for a well-behaved utility function.9

John Hicks’ (1942) approach to wel-fare measurement does not have thesame shortcomings as consumer’s sur-plus. When both prices and total expen-diture change, the Hicksian surplus

5 The reliance of consumer’s surplus on the as-sumption of constant marginal utility of income isdiscussed by Chipman and Moore (1976b) amongmany others.

6 See, for example, Angus Taylor and RobertMann (1972), pp. 500–504.

7 This follows from “Engel aggregation” whichrequires the share weighted sum of the incomeelasticities to be equal to one. The requirement ofhomotheticity for path independence has beenshown by Eugene Silberberg (1972) and Chipmanand Moore (1976b).

8 See the survey by Hendrik Houthakker (1957)marking the centennial anniversary of Engel’sLaw.

9 Chipman and Moore (1976b) also consider thecase in which income and all prices except one(taken to be the numeraire) change. In this case,consumer’s surplus provides an exact measure ofthe change in welfare if preferences are quasi-lin-ear in the numeraire. This implies that thedemands for all goods other than the numeraireare independent of income which, again, is an em-pirically untenable restriction.


measure is exactly analogous to (4) oncewe substitute compensated for uncom-pensated demand functions:

∆HSk = − ∫ΣZ

xic(p, V, Ak)dpi + (M1 − M0), (5)

where xic(p, V, Ak) is the compensated de-

mand for the ith good evaluated at utilitylevel V. Compensated price effects aresymmetric, so the line integral in (5) ispath independent and the surplus mea-sure is single-valued.

For simple binary comparisons, theutility level at which (5) is evaluated isoften treated as a matter of little impor-tance. If (5) is calculated at the utilityattained at prices p1 and expenditureM1 (denoted V1), we obtain a general-ized version of the equivalent variation:

EVk

= M(p0, V1, Ak) − M(p1, V1, Ak) + (M1 − M0)

= M(p0, V1, Ak) − M(p0, V0, Ak), (6)

where M(p, V, Ak) is the expenditurefunction. Note that if only prices changeand there is no change in total expendi-ture, (6) simplifies to the conventionalrepresentation of the equivalent vari-ation. Not only is the generalized equiva-lent variation (also referred to as theequivalent gain by Mervyn King 1983)single valued, but it is ordinally equiva-lent to the change in utility (i.e. it ispositive if and only if V1 > V0) .

The utility level at which (5) is evalu-ated is important for multiple compari-sons of price-expenditure changes. Suchcomparisons are the rule rather thanthe exception because it is often thecase that several policies are comparedwith the status quo or some otherbenchmark. The equivalent variation (6)will give an ordering of outcomes that isidentical to that based on utility levels.If, however, (5) is calculated atV0 = V(p0, M0, Ak), we obtain the general-

ized compensating variation:

CVk

= M(p0, V0, Ak) − M(p1, V0, Ak) + (M1 − M0)

= M(p1, V1, Ak) − M(p1, V0, Ak). (7)

Because the utility levels are “cardinal-ized” using different prices for each setof binary comparisons, the ordering ofmultiple price-expenditure outcomes us-ing (7) need not match the orderingbased on utility levels. Chipman andMoore (1980) have shown that consistentrankings of outcomes require restrictionson preferences that are the same asthose needed to resurrect consumer’ssurplus.

The generalized equivalent variation(6) resolves the conceptual problem ofwelfare measurement, but it does nothelp the practitioner because (it wasoriginally thought) compensated de-mands and, therefore, the Hicksian sur-plus measures are unobservable. RobertWillig (1976) made the first attempt totackle this problem by demonstratingthat, in the case of a single pricechange, (observable) estimates of con-sumer’s surplus can be used to providea good approximation to the equivalentor compensating variation. Since thesewelfare indicators provide upper andlower bounds for consumer’s surplus,the accuracy of the approximation de-pends on the magnitude of the incomeeffect.

Willig gave a number of examples il-lustrating the accuracy of his approach.It was only a matter of time, though,before counterexamples involving goodswith large income effects began to ap-pear. Jerry Hausman (1981) consideredthe problem of labor supply and foundWillig’s method to be inaccurate.10 It is

10 While his basic point was correct, Hausman’scalculation of the error was overstated by an arith-metic error. See Robert Haveman, Mary Gabay,and James Andreoni (1987). Hausman also pointed


worth reemphasizing, however, thatwith multiple price and expenditurechanges, consumer’s surplus is notsingle-valued and is of no use in provid-ing an approximation to the Hicksiansurplus measures. 11

2.2 Exact Measures of Individual Welfare

If consumer’s surplus cannot be usedas an approximation, is there any hopeof measuring the equivalent or compen-sating variations? Immediately follow-ing the publication of Willig’s paper,empirical procedures were developedthat had the same data requirements asconsumer’s surplus without the concep-tual problems. Each method beginswith assumed forms for the vector ofdemand functions x(p, Mk, Ak). As longas the integrability conditions are satis-fied, they contain all of the informationnecessary to recover the underlyingutility and expenditure functions. Therecovery procedure involves “integrat-ing backwards” from the demandswhich, in practice, can be difficult if thefunctions are the least bit complicated.The complexity diminishes significantlyif we restrict consideration to changesin the price of a single good and assumethat demands are linear in prices andexpenditure.

To illustrate, assume only two goods;the commodity of interest (denoted good1) and a composite commodity with aprice normalized to one. By Roy’s Iden-tity, the demand for the good is relatedto the indirect utility function by:

x1(p1, Mk, Ak) = − ∂V(p1, Mk, Ak) ⁄ ∂p1

∂V(p1, Mk, Ak) ⁄ ∂Mk. (8)

As long as the demands are consistent withutility maximization, equation (8) is a par-tial differential equation that can be solvedto obtain the indirect utility function andcorresponding expenditure function.

Hausman (1981) described an ana-lytic solution to an equation of this typewhen the demand function is linear orlog-linear in price and expenditure.Suppose the demand function is:

x1 = γpp1 + γmMk + γAAk,

where γp, γm, and γA are unknown parame-ters that can be estimated econo-metrically. Equation (8) can be solved toyield an expenditure function of the form:

M(p1, V, Ak) = Veγmp1

− (1 ⁄ γm)[γpp1 + (γp ⁄ γm) + γAAk]. (9)

Given the expenditure function, an exactmeasure of the change in welfare is eas-ily calculated, and approximations areunnecessary. Note also that only linearregression methods are required to esti-mate the unknown parameters, and thedata requirements are identical to thoseneeded to compute consumer’s surplus.

This simplicity does not come with-out costs. Closed form solutions to thepartial differential equation shown in(8) can be obtained for only a limitedclass of demand functions. Hausmanprovides solutions to the linear and log-linear specifications but also showedthat the problem becomes much morecomplicated when quadratic terms inexpenditure are added. For ease of im-plementation, he purposely restrictedconsideration to a partial equilibriumframework in which only one equationis estimated and the welfare effects of asingle price change are evaluated. Hisapproach is of little use in the generalcircumstance in which a vector of pricesand expenditure change because theexpenditure function cannot be recov-ered from a single equation.

out that, even if Willig’s method provides a decentapproximation of the change in utility, it is not agood approximation of the deadweight loss.

11 This point was illustrated by George McKen-zie (1979).


More general approaches have beendeveloped along several different lines.The problem of recovering the underly-ing utility function from a system of de-mand equations can be solved by begin-ning with an assumed form of theindirect utility function. Application ofRoy’s Identity yields a set of demandequations that can be estimated usingdata on demands, prices, and demo-graphic characteristics. Since the formof the utility function is assumed fromthe outset, it is unnecessary to solve acomplex system of partial differentialequations. In one of the first applica-tions of this approach, John Muellbauer(1974) assumed that demands were con-sistent with the Stone-Geary utilityfunction and estimated the unknownparameters by fitting the linear expen-diture system to household budgetdata.12 Given the parameter estimates,the expenditure function was obtaineddirectly and used to measure changes inwelfare that resulted from multipleprice and expenditure changes.

To illustrate further, consider theutility function proposed by Dale Jor-genson, Lawrence Lau and ThomasStoker (1982):

ln V(p, Mk, Ak) = αp′ln p + 1 ⁄ 2ln p′Bppln p

− D(p)ln Mk + ln p′BpAAk, (10)

where D(p) = − 1 + ιBpp′ln p and αp, Bpp

and BpA are unknown parameters. Thesecan be estimated by applying the loga-rithmic version of Roy’s Identity to (10)to obtain a system of share equations:

wk = 1

D(p)(αp + Bppln p

− ιBppln Mk + BpAAk), (11)

where wk is the vector of budget sharesof all goods consumed by the kth house-hold. Using the estimated parameters,

welfare changes can be calculated usingthe implied expenditure function:

ln M(p, V, Ak) = 1

D(p) (αp′ln p

+ 1 ⁄ 2ln p′Bppln p − ln V + ln p′BpAAk).

While this is more general than Haus-man’s approach, it has disadvantages.For an assumed form of the utility func-tion, the functional forms of the de-mands are the same for every good,which may hinder the ability of themodel to fit the data. Moreover, Haus-man argues that sound econometricpractice should begin with the “best”specification of the function to be fit tothe data, which, in this application, isthe set of demand equations.

Is it possible to start with an arbitrarydemand system (rather than a utilityfunction) and measure the welfare ef-fects of multiple price changes? Twoelegant procedures have been proposedthat require more complicated calcula-tions but do not impose restrictions onthe form of the demand functions otherthan the standard integrability condi-tions. The first method is based on anapproximation to the indirect moneymetric utility function, which is definedas:

µ(p, Mk, Ak; p0) = M(p0, V(p, Mk, Ak), Ak).

Developed originally by Lionel McKen-zie (1957), the indirect money metricutility function serves as an exact repre-sentation of utility. In fact, the general-ized version of the equivalent variationdefined in (6) is simply the differencebetween the indirect money-metric util-ity functions evaluated at the initial andfinal price and expenditure vectors.

For an arbitrary, unknown function µ,McKenzie and Ivor Pierce (1976)showed that monetary measures of thechange in utility can be measured usinga Taylor’s series expansion about theinitial equilibrium:

12 Another early example of this approach isHarvey Rosen (1978).


∆µ = ∂µ∂p

′∆p + (1 ⁄ 2)∆p′∂2µ

∂p∂p′∆p +

∂µ∂M

+

∂2µ∂p∂M

′∆p + 1 ⁄ 2∂2µ∂M2

∆M ∆M + R, (12)

where R represents higher order termsin the series. At first glance, (12) doesnot appear to be useful for empirical ap-plications because the derivatives of µcannot be observed.

McKenzie and Pearce noted thatmoney-metric utility is identical to totalexpenditure when evaluated at the ref-erence prices. The marginal utility ofincome is therefore equal to one, andall higher order income derivatives arezero. This fact can be used with Roy’sIdentity to represent (12) as a functionof the Marshallian demands and theirincome and price derivatives:

∆µ = − x′∆p − (1 ⁄ 2)∆p′

∂x∂p

− x∂x∂M

′ ∆p

+ 1 −

∂x∂M

′∆p ∆M + R. (13)

Given knowledge of the demand func-tions and the magnitudes of the priceand expenditure changes, one has all ofthe information necessary to get as accu-rate an estimate of the change in utilityas desired. As a practical matter, though,the analyst has no information as to thenumber of terms in the Taylor’s seriesthat are required to obtain an accurateestimate. If higher order terms are re-quired, the calculations become morecumbersome (but not intractable).

Yrjo Vartia (1983) developed an algo-rithm that estimates the expenditurefunction numerically to any desiredlevel of accuracy. Let p(t) and Mk(t) de-scribe the paths of price and expendi-ture changes for 0 ≤ t ≤ 1. Using Roy’sIdentity, he shows that as prices and ex-penditure change, the movements ofdemands along an indifference curve

are described implicitly by the first or-der differential equation:

dMk(t)dt

= ∑ i = 1

n

xi(p(t), Mk(t), Ak) dpi(t)

dt .

Integrating over t yields an equation thatcan, in principle, be solved forM(p(t), V0, Ak) which is the centerpiece inthe Hicksian surplus calculations:

M(p(t), V0, Ak) − M(p0, V0, Ak) = (14)

∑ i = 1

n

∫ 0

t

xi(p(t), M(p(t), V0, Ak), Ak) dpi(t)

dt dt.

Vartia described several algorithmsthat can be used to solve this equationnumerically over a path p(t) so that,when evaluated at t = 1, we obtainM(p1, V0, Ak). As long as the demandssatisfy the integrability conditions, thesolution to (14) will be independent ofthe price path used in the algorithm.This method is valid for multiple priceand expenditure changes and, because aclosed-form solution is unnecessary, fa-cilitates flexibility in estimating demandpatterns. This is an important advantagealthough it is worth noting that multiplecomparisons of welfare changes requirerepeated application of the algo-rithm.This is more inconvenient thanevaluating an explicit form of the ex-penditure function although, given thecurrent computational power that iswidely available, this is less of an issuetoday than it was when Vartia originallyproposed his approach.

2.3 Empirical Issues

The conceptual problem of how tomeasure household welfare using ordi-nary Marshallian demand functions haslargely been resolved. Since these func-tions are unknown, econometric meth-ods are used to estimate them statisti-cally. Welfare measurement is, as aresult, indelibly linked to empirical


demand analysis and issues related tothe specification, estimation, and datarequirements of the model must be ad-dressed. Comprehensive surveys of thevoluminous literature on demand mod-eling are presented by Angus Deatonand Muellbauer (1980a), Richard Blun-dell (1988), Robert Pollak and TerenceWales (1992), and Arthur Lewbel(forthcoming).

The standard approach is to specify afunctional form for the demand func-tion and estimate the unknown parame-ters using regression methods. But whatfunction should be chosen a priori? Amisspecification will lead to biased esti-mates and erroneous inferences con-cerning the magnitudes and, possibly,even the directions of the changes inwell-being. It is important, therefore, todevelop a framework with enough flexi-bility to allow the data to describe accu-rately the relationships between de-mands and prices and total expenditure.

The specification problem is not lim-ited to the choice of functional form.Households consume many goods, anddata limitations prevent the estimationof every demand function. The modelcan be simplified either by restrictingconsideration to a subset of the goodsor aggregating all items into broad com-modity groups. In many applicationsthis is not a matter of choice but ofnecessity because the data have infor-mation on only a small number of cate-gories. In fact, it may be the case thatwe have data on only a single commod-ity such as labor supply. Under whatconditions is it possible to analyze asubset of all goods in isolation from theothers? If these conditions hold, whatcan be learned about changes in welfarefrom the subsystem of demand func-tions?

Even at the micro level, empirical im-plementation requires the resolution ofa potentially serious aggregation prob-

lem. Expenditure data are reported forentire households by simply adding upthe spending of all members. While wehave the information at the householdlevel, it is the welfare of the individualmembers that we are concerned about.This requires inferences concerning theintrahousehold distribution of consump-tion and the evaluation of welfare de-rived from these goods for each mem-ber. As we shall see, this research hasraised important questions that haveproved difficult to answer and is likelyto be a fruitful area for future work.

Functional Form Specification.13 Un-til the recent development of nonpara-metric statistical models, empirical de-mand analysis began with thespecification of the forms of the de-mand functions (or the correspondingutility function) to be fit to the data.Perhaps because of computational limi-tations, the earliest applications usedspecifications that could be estimatedusing linear or only slightly nonlinearregression methods. In his pioneeringstudy, Richard Stone (1954) developedand implemented the linear expendi-ture system and, much later, Hausman(1981) restricted attention to functionalforms that could be estimated using lin-ear regression. While easy to imple-ment, linear demand functions may notbe sufficiently flexible to measure de-mand responses to price and expendi-ture changes. The estimated elasticitiesmay reflect the functional form assump-tion rather than the demand patternsrevealed by the data.

Out of this concern grew the develop-ment of what have become known as“flexible functional forms”. Any specifi-cation is almost surely incorrect, and thebest one can hope for is an approxima-tion to the demand or utility function.

13 This subsection has benefited greatly fromcomments and suggestions from Arthur Lewbel.


The indirect utility function V(pk∗) pro-

vides a local, second-order approxima-tion to the true function V∗(pk

∗) aboutthe point p– k

∗ if

V∗(p–k∗) = V(p– k

∗)

∂V∗(p– k∗)

∂pk∗ =

∂V(p– k∗)

∂pk∗

∂2V∗(p– k∗)

∂pk∗∂pk

∗′ =

∂2V(p–k∗)

∂pk∗∂pk

∗′ ,

where pk∗ = p ⁄ Mk is the vector of normal-

ized prices. Erwin Diewert (1974)defined functions V that satisfy theseconditions to be (second-order) flexiblefunctional forms.

The translog utility function devel-oped by Laurits Christensen, Jorgensonand Lau (1975) is an example of amodel that satisfies the condition of lo-cal second-order flexibility (a specialcase is shown in equation 10). A criticalfeature is that enough parametersare used to ensure that the demandelasticities are determined, withoutprior constraints, by the data. Othersecond-order flexible functional formsinclude the generalized Leontief formdeveloped by Diewert (1971), WilliamBarnett and Yul Lee’s (1985) minflexLaurent demand system, and the AIDSmodel developed by Deaton and Muell-bauer (1980b).

As mentioned previously, the recov-ery of information on preferences fromdemands requires that they be consis-tent with a rational consumer who maxi-mizes utility subject to a budget con-straint. This requirement imposes cross-equation restrictions on the demandsthat limit their ability to fit the “true”underlying functions. In the context ofwelfare measurement, the challenge is tomaintain the flexibility of the demandsfunctions while imposing the restric-tions necessary to ensure the existence ofa well-behaved indirect utility function.

If demands are consistent with utilitymaximization, the spending on all goodsmust sum to total expenditure Mk (sum-mability) which makes one of the equa-tions superfluous in the estimation pro-cedure. Monotonicity of the utilityfunction implies that the demand foreach good must be nonnegative at allprices and expenditure levels (nonnega-tivity). This is typically difficult to im-pose at all data points, and, for exam-ple, the translog model shown in (11)does not satisfy this restriction globally.The condition that demands be homo-geneous of degree zero in prices andexpenditures (homogeneity) imposesfurther constraints on the price andexpenditure parameters within eachequation. Symmetry of the Slutsky ma-trix of compensated price effects (sym-metry) imposes cross-equation restric-tions on the price parameters, whilequasiconvexity of the utility function re-quires it to be negative semidefinite.The latter condition imposes compli-cated nonlinear inequality restrictionson the demand parameters to ensurethat the eigenvalues of the Slutsky ma-trix are nonpositive. Taken together,the full complement of restrictions im-plied by consumer rationality are quitestrong and can limit the ability of func-tional forms to fit the data well.14

Recent work has focused on flexibledemand models that are linear in func-tions of total expenditure:

xi(p, Mk) = ∑hil

l = 1

L

(p)fl(Mk), (15)

where, for notational simplicity, I havesuppressed the effects of demographic

14 In particular, it is possible that the constraintsreduce the number of free parameters to such anextent that it is impossible to provide a secondorder approximation to an arbitrary function. Therelationship between flexibility and the integrabil-ity conditions has been considered by Diewert andWales (1987) in the production context.


characteristics on the demand functions.This class of models is particularly im-portant because the micro level demandsaggregate exactly to yield macro de-mands that are functions of statistics thatdepend on the level and distribution oftotal expenditure.15 Gorman (1981)showed that if demands have this linear-ity property and are consistent with util-ity maximization, then the maximumrank of the (n x L) matrix H(p) formedfrom the elements hil(p) is three. Of spe-cial interest are the models that Lewbel(1990) describes as “full rank” demandsystems where the rank of H(p) is equalto L. In these models, the econo-metrician achieves the greatest flexibilityin describing the relationship betweendemands and total expenditure with thetightest parametric specification. Lewbelcharacterizes all full rank demandmodels of the form shown in (15) andfinds that many of the commonly used(aggregable) demand systems are of thistype.16

The importance of Gorman’s andLewbel’s results is that they suggestthat many of the demand systems usedin applications are not as general asthey could be. The translog demandsystem (11) and the AIDS model arefull rank demands systems that de-pend on two functions of total expen-diture. Both are functions of Mk andMklnMk and belong to the generalclass of models referred to as “price in-dependent generalized log linear”(PIGLOG) by Muellbauer (1976). Ranktwo models such as these may not ade-quately describe the relationship be-

tween the demands and total expendi-ture.

In fact, recent evidence presented byLewbel (1991) and Blundell, PanosPashardes and Guglielmo Weber (1993)based on Engel curves indicates thatrank three extensions of the PIGLOGmodel provide better descriptions ofexpenditure patterns in both the U.S.and the United Kingdom. In their mod-els, demands depend on three functionsof total expenditure: Mk, MklnMk andMk(lnMk)2.17 Not only does the rankthree extension provide a better de-scription of the data, but it demon-strates that if one restricts considera-tion to exactly aggregable models,higher order approximations that yieldmore than three functions of expendi-ture are superfluous. More importantfor our purposes, James Banks, Blundelland Lewbel (1995) show that a rankthree generalization of the AIDS modelleads to different conclusions concern-ing the welfare effects of commodity taxreform relative to the standard rank twoAIDS model.

While the development of flexiblefunctional forms represented an impor-tant step forward in empirical demandmodeling, several practical problemsemerged. Halbert White (1980) pointedout that the estimated function mightnot provide an accurate approximationto the true underlying utility functioneven in large samples. Part of the prob-lem is that the point of approximation isunknown and little can be said aboutthe behavior of the estimated functionat other data points. Ignoring the re-mainder term of the series, moreover,results in a misspecification that leads15 For further discussion of exactly aggregable

demand models, see Lau (1982), William Gorman(1981), and the survey by Stoker (1993).

16 Lewbel (1989c, 1991) has extended these re-sults for demand specifications that are more gen-eral than (15). For example, if fl is a function of“deflated expenditure” rather than expenditurealone, the maximum possible rank of the demandsystem increases to four.

17 Lewbel (1990) has shown that the other possi-ble forms of the full rank three models are “gener-alized quadratic” and demands that depend ontrigonometric functions. In the former case, thethree functions are Mk, Mk1 − α and Mk1 + α for arbi-trary values of α .


to biased and inconsistent estimates.Beyond these statistical concerns, thequality of the approximation providedby a flexible functional form dependson the behavior of the remainder term.There is no reason a priori to expect thetruncation error to be small evenaround the point of expansion. Globally,very little can be said.18

Some researchers have addressedthese issues by using semiparametric ornonparametric methods to model de-mands. Ronald Gallant (1981) and Ibra-him Elbadawi, Gallant and GeraldoSouza (1983) proposed a hybrid of theparametric and nonparametric tech-niques by appending a Fourier seriesexpansion of V to a second orderflexible functional form such as thetranslog:

V(pk∗) = α′lnpk

∗ + 1 ⁄ 2lnpk∗′Blnpk

∗

+ ∑ a = 1

A

Fa(λa′pk∗).

In this model, Fa is a function of sinesand cosines representing the elements ofthe Fourier expansion and λa is a multi-dimensional index. The model is para-metric in the usual sense but nonpara-metric because the number of terms Aincluded in the Fourier series dependson the sample size. As the sample sizegrows, the semiparametric model is a se-quence of successively more complicatedfunctional forms that, in the limit, pro-vide a global approximation to an arbi-trary utility function and all of its deriva-tives. In this application, the derivativesare particularly important because theyrepresent the demands (the first deriva-

tives) and the elasticities (the second de-rivatives).

The semiparametric approach is at-tractive because the advantages of theparametric models are retained. The in-tegrability conditions are represented asparameter restrictions and the entiresystem of demand equations can be es-timated using the usual multivariate sta-tistical methods. Moreover, an explicitrepresentation of the underlying indi-rect utility function is obtained that canbe used to estimate the welfare effectsof price and expenditure changes.Michael Creel (1997) provides an exam-ple of welfare calculations that arebased on the Fourier model.

These methods, however, have theirown problems. The approximation prop-erties are valid only with infinite sam-ples which, of course, are never foundin practice. With finite samples, thesemiparametric model is essentiallyequivalent to a parametric model, andwe have no reason to expect it to per-form better than any other. For exam-ple, even with appropriately-scaleddata, it is not obvious that the trigono-metric components of the truncatedFourier system would accurately de-scribe consumers’ responses to priceand expenditure changes. While cross-validation and other diagnostics are be-ing used with greater frequency, thebottom line is that the magnitude of thetruncation error induced by finite sam-ples is difficult to assess. As a result, aconflict persists between overparame-terization (as more terms of the seriesare added) and the inaccuracy that re-sults from truncating the series.

There have been several recent ef-forts to estimate demands using a fully-nonparametric specification.19 Assumethe system is given by:

18 White (1980) gives several examples in whichthe biases are large. R.P. Byron and Anil Bera(1983) note, however, that White’s results con-cerning the inaccuracy of the regression functionis overstated because he considers only a first or-der approximation. They demonstrate that secondorder approximations do much better even thoughthe bias persists.

19 See Wolfgang Hardle (1990) for an introduc-tion to these nonparametric statistical methods.


xk = x(p, Mk, Ak).

The demand function for the ith goodxi(.) is estimated using a kernel regres-sion of the form:

xi(.) =

(1 ⁄ K)∑ k = 1

K

xikG(.)

(1 ⁄ K)∑ k = 1

K

G(.)

(i = 1, 2, ..., n),

where G(.) is the kernel used to estimatethe joint density of the regressors.

Nonparametric methods are appeal-ing because they avoid the vexing issueof functional form misspecification andits contaminating effects on statisticalinference. Recent empirical work in de-mand analysis using variants of thesetechniques include Herman Bierensand Hettie Pott-Buter (1990), Banks,Blundell, and Lewbel (1995), Hardle,Werner Hildenbrand and Michael Jeri-son (1991), Hausman and WhitneyNewey (1995) and Lewbel (1991, 1995).This is a promising new approach, but itintroduces additional problems. Whilean explicit functional form for the de-mand function is unnecessary, it re-quires choices for the appropriate ker-nel G(.) and the bandwidth to be used inthe smoothing. Both choices can have asubstantial influence on the estimationresults. Both nonparametric and semi-parametric methods require large num-bers of observations with substantialvariation in the regressors. Whilehousehold budget surveys have enoughobservations, they typically have littleor no price information, which oftenprecludes the use of these techniques.Most applications have focused on therelationship between demand and totalexpenditure, and they have ignoredprice effects.

Another impediment to the applica-tion of these estimators is the “curse of

dimensionality” in which the speed ofconvergence of the estimators to thetrue values is inversely related to thenumber of regressors. Out of this con-cern, demand applications typically useonly one or two explanatory variablesand have modeled the demand using asingle equation. As we shall see below,this requires separability assumptions,which likely result in a misspecificationof the model. This form of specificationerror must be weighed against the ad-vantages of the nonparametric estima-tor in discerning the higher order de-rivatives of the functional form of thedemand function.20 Analyzing a singleequation also avoids the issue of crossequation restrictions implied by thetheory of consumer behavior. It is un-clear how tractable or practically usefulnonparametric methods are in the con-text of modeling a complete, multivari-ate demand system with the impositionof the full set of integrability restric-tions.21

While nonparametric methods arenot uncommon in the analysis of Engelcurves, relatively few studies have usedthese techniques to draw inferencesconcerning changes in welfare. Haus-man and Newey (1995) exploited crosssectional price variation to obtain anonparametric estimator of a single de-mand function x(p, Mk) (gasoline), andsolved numerically for the underlyingexpenditure function using Vartia’smethod described in (14). They ana-lyzed the welfare effects of gasolinetaxes and, after all was said and done,found little difference between the non-parametric estimates and those ob-tained using a simple linear parametric

20 I thank Arthur Lewbel for this point.21 Some progress has been made on this front.

Hausman and Newey (1995) and Lewbel (1995)describe nonparametric tests of the integrabilityconditions although the tests are quite cumber-some and difficult to implement.


model. Deaton (1989) summarized thewelfare effects of differential pricechanges by a single variable, and usedkernel regression methods to examinehow they changed for households withdifferent incomes.

Separability over Goods and Time.How many demand equations should beused to measure welfare? Individualsconsume literally hundreds of goods(thousands if one distinguishes goods bythe date of consumption) so that, withthe typical constraints imposed by lim-ited sample sizes, there is no hope ofestimating a complete demand systemwith any degree of precision. Parsimo-nious specifications can be achievedthrough some form of aggregation overcommodities, which is justified by sev-eral layers of assumptions of separabil-ity of preferences over time and withinperiods.

The welfare measures described inthe previous section are based on a sin-gle period framework in which the ef-fects of changes in the prices and ex-penditures in other periods are ignored.The welfare effect within a period is in-dependent of other time periods ifthere is intertemporal separability. Thisis usually assumed to take the form oftwo-stage budgeting, in which thehousehold decides the total amount thatwill be consumed in each period andthen, conditional on this choice, the al-location across goods within a period.In this context, it is appropriate to mea-sure the single-period welfare effect us-ing only the prices and expenditures ob-served in that time period.22

The two-stage budgeting frameworkhas several implications for the staticwelfare indicators such as (6) and (7).First, it influences the functional struc-ture of both the within-period and in-tertemporal utility functions. Gorman(1959) has shown that spending pat-terns are consistent with two-stagebudgeting if either the lifetime utilityfunction is homothetically separable orthe top-level utility function is additiveand the within-period functions are ofthe Gorman polar form.23 Since ho-mothetic preferences have no empiricalsupport, additivity is the more palatableassumption. Note that even in the ab-sence of two-stage budgeting, the in-tertemporal separability assumptionalone precludes the inclusion of laggedvalues of the demands as arguments ofthe demand functions. This is restric-tive because dynamic specifications ofthis type have been found to be empiri-cally important in explaining consump-tion behavior in the context of habitformation, addiction and durabilitymodels.24

A second implication of two-stagebudgeting is that within-period de-mands must be functions of total expen-diture (or full expenditure if labor sup-ply is endogenous) rather than income.Total expenditure (along with the mar-ginal utility of wealth) serves as a “suf-ficient statistic” that links the within-period model to the intertemporalallocation of consumption. The use ofincome in static demand models, whichis very common, results in a potentiallyserious misspecification because of itssystematic understatement of expendi-ture for low-income households and the22 A separate question is how one should mea-

sure the intertemporal change in well-being. If theprice and expenditure changes cause a rearrange-ment of consumption over time, the “snapshot”welfare indicators alone will give a biased pictureof the change in lifetime utility. This point will beelaborated later when the model is extended tomeasure changes in intertemporal welfare in Sec-tion 2.5.

23 For further discussion, see Charles Blackorby,Dan Primont, and Robert Russell (1978).

24 As examples, see Gordon Anderson and Blun-dell (1982), Gary Becker, Michael Grossman, andKevin Murphy (1994) and Costas Meghir andWeber (1996) among others.


reverse for those with high incomes.This turns out to be particularly impor-tant when welfare measures are used toexamine distributional issues such as in-equality or poverty.

Given intertemporal separability,commodity aggregation for the within-period demands is an additional prob-lem. The analyst often has informationon only a subset of the demands; some-times a single commodity. In measuringthe welfare effects of a change inwages, for example, it may be that onlylabor supply data are available but com-prehensive consumption data are not.This is the situation if one is using, say,the Current Population Surveys in theUnited States. Under what conditionscan we measure the within-periodchange in welfare from a single demandequation?

In general, what is required is the ag-gregation of consumption into a com-posite commodity. One way to do this isto invoke the Hicks-Leontief compositecommodity theorem, which states that agroup of goods can be treated as a sin-gle aggregate if their prices move inparallel. To apply this result to the la-bor supply example, the real prices ofall consumption goods must be per-fectly collinear, which is inconsistentwith price movements in the UnitedStates. Lewbel (1996) has describedsomewhat weaker conditions underwhich this form of commodity aggrega-tion can occur, but it remains to beseen whether his result can be used todefine a single aggregate consumptiongood.

In the absence of Hicks-Leontief ag-gregation, an alternative approach is toassume separability of the commodityfor which there are data from all othergoods. In the labor supply example, thestructure of the indirect utility function(suppressing the demographic effects)must be of the form:

Vk = V(wk, Vc(p, Mkc), Fk), (16)where wk is the wage, p is the vector ofprices for consumption goods, Mkc is thetotal expenditure on consumption goods,and Fk is the level of full expenditure. Inthis model Vc is the indirect subutilityfunction defined over goods and can beinterpreted as the price index of thecomposite commodity. Preferences ofthis type can be recovered from a singlelabor supply equation because totalspending on the two goods (leisure andconsumption) must equal full expendi-ture.

This suggests that changes in welfarecan be measured using a single demandequation as long as demand patterns areconsistent with the separable structureof the utility function shown in (16). Isthis a reasonable description of prefer-ences? In general, preferences of thistype require strong elasticity equalityrestrictions among the goods that com-prise the composite commodity. Testsof weaker forms of separability usuallyreject it.25 Note also that this form ofseparability does not actually solve thedata problem that often motivates sin-gle equation methods. That is, (16)shows that the “income” variable to beincluded in the labor supply equation isfull expenditure, but this requires infor-mation on the quantity of leisure con-sumed as well as the spending on goodsand services.

Since single-equation methods re-quire strong aggregation assumptions,multiple equation models are some-times used. These are frequently basedon the demands for goods and services,but the effects of leisure, public goods,and other nonmarket commodities are

25 Recent examples include Diewert and Wales(1995) (in the production context) and Lewbel(1996). Barnett and Seungmook Choi (1989) pointout that many tests of separability have low powerand they illustrate the problem using Monte Carlosimulations.


typically ignored.26 The welfare effectsderived from these models measurechanges in the subutility function definedover goods (Vc in equation (16)), ratherthan changes in the overall within-pe-riod utility level. If Mc(p, Vc) is the ex-penditure function defined over goodsconsumption alone, a monetary measureof the change in the subutility level is:

EVkc = Mc(p0, Vkc1 ) − Mc(p0, Vkc

0 ), (17)

where Vkc1 = Vc(p1, Mk

1) and Vkc0 = Vc(p0, Mk

0) .

The accuracy of (17) as a welfaremeasure depends on the consistency ofthe assumption of separability. Evi-dence presented by Michael Abbott andOrley Ashenfelter (1976), Barnett(1979), and Martin Browning andMeghir (1991) does not support the as-sumption of separability of leisure fromconsumption. This suggests the impor-tance of exploiting relatively new datasets that have information on both con-sumption and leisure in examining thewelfare effects of policy changes.27

Even if the separability assumption isreasonable, what is the relationship be-tween (17) and the change in within-period utility? Michael Hanemann andEdward Morey (1992) have shown that(17) will not accurately measure theoverall change in utility, becausechanges in the prices and total expendi-ture will lead to reallocations between(in this particular application) leisureand consumption.28 Jeffrey LaFrance

(1993) has found the empirical magni-tudes of such biases can be quite large, sothat seemingly innocuous assumptionsconcerning commodity aggregation canlead to misleading conclusions concerningthe impact of price changes on welfare.

Welfare of Whom? In practice, thedata largely dictate the units of observa-tion for which the welfare effects arecalculated. If expenditure data areused, spending is almost universally ag-gregated over the individuals who makejoint consumption decisions, and wel-fare is measured for the household. Ifinferences concerning well-being arebased on labor supply information, theobservational unit is taken to be the in-dividual (and the influences of othermembers of the household are often ig-nored). Neither of these approaches areentirely satisfactory, and recent re-search has demonstrated the impor-tance of analyzing the intrahouseholdinteractions of individuals.

Theoretical justification for modelingthe household as the decision unit isattributed to both Paul Samuelson(1956) and Becker (1981). While bothapproaches lead to a representativeagent treatment of the family, the as-sumptions concerning the interactionsamong members are quite different.Shelly Lundberg and Robert Pollak(1996) refer to Samuelson’s frameworkas a consensus model in which themembers of the household pool re-sources and make spending decisionsthat are consistent with an agreed uponsocial welfare function. Becker’s modelyields the same empirical implications,but he assumes that the household iscontrolled by an altruistic parent whoallocates resources within the house-hold so as to maximize utility subject toa common budget constraint.

Because individual-level consumptiondata are unavailable, the Samuelson-Becker representative agent assumption

26 As examples, see Muellbauer (1974), Jorgen-son, Lau and Stoker (1980, 1982), and, more re-cently, Frank Wolak (1996).

27 This information is now available in the Con-sumer Expenditure Surveys in the U.S. and theFamily Expenditure Surveys in the United King-dom.

28 A condition under which it is unbiased is thatthe consumer is constrained to consume the sameamount of leisure before and after the policychange. In this case, the change in the utility de-fined over the consumption goods is the same asthe overall change in utility under the rationedequilibrium.


is invoked, and welfare calculations aremade for the household. Research overthe last decade has suggested that thisassumption may be inappropriate.Using data from low-income house-holds, Lundberg (1988) found that theassumption of joint utility maximizationon the part of households does not de-scribe the labor supply behavior of hus-bands and wives. While she found littleempirical support for the joint utilitymodel, the specification that treatedmen and women as autonomous agentsdid not perform much better for thesubsample of families with children.29

A growing body of evidence suggeststhat the joint utility model may be inap-propriate for modeling the allocation ofconsumption across goods and services.An implication of the representativeconsumer model is that the source ofincome (such as that attributed to thehusband or wife) should have no influ-ence on expenditure patterns, since allincome is pooled and spending deci-sions are made on the basis of a com-mon utility function. Using Braziliandata, Duncan Thomas (1990) has foundthat the share of income contributed bythe mother or father influences the nu-trient intakes, survival probabilities andanthropometric outcomes of children.30 Browning, et. al. (1994) find that theshare of income contributed by womeninfluences the share of expenditures ongoods consumed exclusively by them.

If the joint utility model is inconsis-tent with spending patterns, how are in-dividuals making these decisions andwhat are the implications for appliedwelfare economics? Marilyn Manser

and Murray Brown (1980) and MarjorieMcElroy and Mary Horney (1981) havecharacterized household decisions asthe outcome of a bargaining processamong members, although one modelhas little empirical support over an-other.31 Extending the work of PierreChiappori (1988, 1992), Browning et.al. (1994) give conditions under whichspending decisions can be modeled asthe outcome of a household “sharingrule”. This rule can be described as atwo-stage allocation in which resourcesare initially divided between householdpublic goods and the total spending ofeach member. In the second stage, eachindividual decides on the division of hisor her own allocation across the variousgoods. On the basis of this sharing rule,Browning et. al. find that allocationsamong couples depend on their relativeages and incomes, as well as their levelof wealth. Household demand behavioris, therefore, inconsistent with a modelbased on a representative consumer.

What are the prospects for measuringthe welfare of individuals within thehousehold? Given the lack of individ-ual-level consumption data, the intra-household distribution must be in-ferred, and it is difficult to assess theconsistency of the estimated distri-bution with the actual distribution.Even if the distribution were known,welfare measurement requires an as-sessment of individuals’ valuations ofthe goods. For private goods, this canbe done in the usual way. Many goodsconsumed by the household, however,are public so that welfare measurementrequires knowledge of each individual’swillingness to pay. This is obviouslydifficult to assess. It is also undoubt-edly the case that members of a familyor household have interdependentpreferences, which adds an additional

29 Paul Schultz (1990) and Bernard Fortin andGuy Lacroix (1997) also present evidence againstthe joint utility model of labor supply using, re-spectively, data from Thailand and Canada.

30 He also reports (Thomas 1994) the effects ofdifferences in the education levels of parents onthe anthropometric outcomes of children forGhana, Brazil, and the U.S. 31 See also Lundberg and Pollak (1993).


complication to the calculations. Re-gardless, recent work has made impor-tant progress, and an analysis of the in-trahousehold distribution of welfare isan important area for future research.

Data Requirements. Implementationof any of the methods used to measurewelfare requires comprehensive microlevel expenditure data. This is true re-gardless of the number of goods or thedegree of separability that is assumed.If labor supply is modeled, full expendi-ture is needed but it is not reported inlabor surveys such as the CPS. If only asingle consumption good is considered(e.g. gasoline in Hausman and Newey1995) and labor supply is exogenous,total expenditure must be reported.Often, income is used as a proxy forconsumption, which generally results inbiased estimates of the demand parame-ters and the associated welfare effects.

Estimation of the demand equationsand the Hicksian surplus measures alsorequires price variation across the sam-ple. As mentioned previously, this is aproblem for most applications becausecross sectional budget studies reportthe expenditures on commodities butnot the prices paid for them. As a re-sult, prices are obtained from othersources, and variation is introducedthrough regional or temporal move-ments. The Consumer Expenditure Sur-veys provide quarterly data from 1980that can be linked with price series suchas the components of the ConsumerPrice Index (CPI) or the implicit pricedeflators of personal consumption ex-penditures in the National Income andProduct Accounts. Price variation canalso be incorporated by linking microdemand equations with an exactly ag-gregated set of macro demand func-tions.32 Heterogeneity in demand pat-

terns (and the associated welfare ef-fects) is identified using variation in thecross section data, while the price ef-fects are estimated using time seriesvariation in the aggregate model. Appli-cation of this approach, though, re-quires the assumption that every house-hold faces the same prices. Regardlessof the method, it is an understatementto suggest that the prices used in de-mand analysis typically embody mea-surement error.

Finally, to reiterate another pointmade previously, the nonparametric andsemiparametric estimates of the welfareeffects of policy changes are plagued byslow rates of convergence to the trueunderlying function. Precise estimatescan be obtained only with large num-bers of observations and substantialvariation in the independent variables.Overparameterization is a similar con-cern with the flexible series expansionsthat have been proposed. This problemcan only be circumvented with a suffi-ciently large number of observations onhousehold demands that include infor-mation on the prices paid for the goods.I’m unaware of the existence of anysuch data which suggests that furtherapplication of these methods will re-quire the parallel development of alter-native data sources.

2.4 Measuring Welfare Using Index Numbers

The overlap between consumer’s sur-plus (and the Hicksian variations) andindex number theory has been noticedfor some time (Hicks (1942)). The gen-eralized equivalent variation (6) evalu-ates the change in utility as a monetarymeasure of the difference in utilitywhile A. A. Konus’ (1939) quantity in-dex represents it as a ratio:

Qk(p, V0, V1) = M(p, V1, Ak)M(p, V0, Ak)

.32 This approach was followed by Jorgenson,Lau, and Stoker (1982).


This can be estimated using the econo-metric methods described above, but in-dex number proponents have typicallyfollowed a different empirical strategy.33

The econometric approach assumes aparametric form for the utility (or de-mand) function that is the same for allhouseholds. Observations on expendi-ture patterns are used to recover theutility function and measure changes inwelfare that result from actual or simu-lated policies. Heterogeneity is ac-counted for by allowing preferences anddemands to be functions of householdcharacteristics while unobserved differ-ences can be accommodated throughthe stochastic specification of theeconometric model.

The index number approach makesweaker assumptions on preferences, butthe data requirements are more strin-gent. The basic strategy is to avoidfunctional form assumptions on prefer-ences and evaluate the relative levels ofwelfare using Samuelson’s (1948) prin-ciple of revealed preference. Not onlyis this method nonparametric, but italso makes unnecessary the assumptionthat individuals have identical prefer-ences. Let xk

0 and p0 be the quantity andprice vectors in the reference period,and define xk

1 and p1 to be the corre-sponding vectors in the current period.If preferences are internally consistent,the individual is at least as well off inthe base period if p0′xk

0 ≥ p0′xk1 and simi-

larly in period 1 if p1′xk1 ≥ p1′xk

0. If nei-ther condition holds, nothing can besaid about the relative welfare levels inthe two periods. This incompleteness isan obvious impediment to practical ap-plications of this fully nonparametricapproach to welfare economics.

Beyond a definitive statement as tothe direction of the change in welfare,

the magnitude is often of interest. Forthis purpose, an explicit form of the in-dex needs to be specified. Rather thanenumerate the advantages and disad-vantages of the numerous possibilities, Iillustrate the key issues using an indexproposed by Malmquist (1953):

QMk(xk1, xk

0, u) = d(u, xk

1)d(u, xk

0),

where U(x) is the direct utility functionand d is a distance function defined as:34

d(u, x) = max λ λ : U(x ⁄ λ) ≥ u, λ > 0 .

This index has the correct ordinal prop-erties of being greater than unity if andonly if utility is higher in period 1 but,because U is unknown, is inherently un-observable.

While a point estimate of QM is notpossible without further assumptions onthe form of the distance function (and,therefore, preferences), upper andlower bounds on the index can be calcu-lated. These bounds represent indexnumber analogs to the regions of inde-terminacy associated with revealedpreference methods. Diewert (1981)has shown that if the consumer is ra-tional, the bounds on the Laspeyres ver-sion of the Malmquist index are:

minixik1 ⁄ xik

0 ≤ QMk (xk1, xk

0, u0) ≤ p0′xk

1

p0′xk0 ,

where u0 = U(xk0). These bounds need not

be tight. In fact, the lower bound can beless than one and the upper boundgreater than one, implying that it is im-possible to say whether welfare has in-creased or decreased.

With stronger assumptions on prefer-ences, it is possible to make more de-finitive statements concerning the di-rection of the change in welfare using

33 Comprehensive surveys of index numbers areprovided by Diewert (1981, 1990) and AmartyaSen (1979).

34 Deaton (1979) presents a detailed discussionof the distance function and its application to con-sumer theory.


index numbers. In a seminal paper,Diewert (1976a) has shown that a num-ber of commonly-used indexes are “ex-act” in the sense of being ordinallyequivalent to the underlying utilityfunction for some representation ofpreferences. A Laspeyres quantity in-dex, for example, is exact if preferencesare of the fixed coefficient or Leontiefform. Diewert defined indexes to be“superlative” if they are exact for a util-ity function that is second order flex-ible. Such indexes are appealing be-cause the assumed form of the utilityfunction can be interpreted as a localsecond-order approximation to anywell-behaved function. While not fullynonparametric, this approach repre-sents an attractive middle ground.

The Malmquist index is generally un-observable but under certain circum-stances it can be calculated using onlythe observed prices and quantities inthe base and reference periods. Assumethe distance function is of the translogform and the reference indifferencecontour is a geometric average of thebase and reference utility levelsu∗ = (u0u1)1⁄2. Diewert has shown that thefollowing quantity index is exact (andalso superlative) and can be calculatedusing only observations on prices andquantities in the two periods:35

ln QMk(xk1, xk

0, u∗) = 12∑(

n = 1

N

wnk0 + wnk

1 )ln xnk

1

xnk0 ,

where wnk = pnxnk ⁄ Mk is the budget shareof the nth good for the kth household.For a particular assumption on prefer-ences, therefore, it is possible to esti-mate the change in welfare without usingeconometrics. This procedure hasgreater data requirements, however, be-

cause the demands in both periods mustbe observed to calculate the index.

Given the correspondence betweenquantity indexes and the generalizedequivalent variation, it is perhaps notsurprising that Harberger’s (1971) rep-resentation of consumer’s surplus alsohas an index number analog. Using aTaylor’s series approximation to an arbi-trary utility function, he showed thatconsumer’s surplus is approximatelyequal to:

∆CSk = ∑ i

pi0∆xik + (1 ⁄ 2)∑∆

i

pi∆xik, (18)

where ∆pi = pi1 − pi

0 and ∆xik = xik1 − xik

0 .With repeated observations on house-hold demands, this representation ofconsumer’s surplus is easily calculated.The question is whether it is ordinallyequivalent to the underlying utility as is,say, the Malmquist or Konus indexes.

Diewert (1976b) demonstrated that(18) identifies the direction of thechange in utility as long as demands arein the region for which the weak axiomof revealed preference is conclusive.The problem arises in the region of in-determinacy where (18) can be madepositive or negative by simply scalingthe prices in period 1. MartinWeitzman (1988) attempted to resur-rect Harberger’s index by suitably nor-malizing the prices in period 1 butDiewert (1992) pointed out that hiswelfare measure requires the assump-tion of homothetic preferences. It is notsurprising that this is essentially thesame condition that was derived byChipman and Moore (1976b).

The index numbers described to thispoint require only two sets of observa-tions on households’ demands. In theabsence of assumptions on the form ofthe utility function, it is only possible toobtain bounds on the index that neednot be particularly informative. S.N. Af-riat (1967) and Hal Varian (1982) have

35 See Diewert (1976a, pp. 123–24). This par-ticular index number formula is attributed to LeoTornqvist (1936) and can be interpreted as a dis-crete approximation to a Divisia index.


presented methods of welfare analysisthat are fully nonparametric but aremore likely to provide definitive conclu-sions concerning changes in welfare.

Using multiple observations on de-mands, it is possible to describe the setof commodity bundles that are strictlypreferred to the base period demandsxk

0. This is the convex hull of the set ofall points revealed preferred to xk

0. Theminimum cost of attaining a consump-tion bundle in this set (at specificprices) provides an upper bound to theexpenditure function. Conversely, wecan determine the set of all points thatare revealed to be worse than xk

0. Theminimum cost of attaining a point in thecomplement of this set provides a lowerbound to the expenditure function.Given the over- and under-estimates, itis possible to create interval estimatesof the change in welfare in either dif-ference or ratio form.36

The most obvious advantage of theindex number approach is the fact thatfunctional form assumptions on prefer-ences are unnecessary. This precludesthe possibility of erroneous inferencesbeing drawn because of the misspecifi-cation of the form of the utility func-tion. They are also fairly easy to com-pute (at least with two observations)and do not require the implementationof complicated optimization algorithmsas are often required with econometricmethods. The region of indeterminacythat results, however, is a consequenceof these weaker assumptions. Even ifthe bounds on the indexes are tight, orone uses indexes that are exact for aspecific functional form, other prob-lems arise. Multiple observations on ex-penditures are required for each house-hold, and these data are rarelyavailable. In fact, no micro data of this

type are available in the United States,so almost all applications use aggregatedata in conjunction with a representa-tive agent assumption.37 Moreover, it isimpossible to evaluate the effect ofsimulated policies since the demandsunder the alternative scenarios are notobserved. This precludes the use ofthese methods to an important applica-tion of welfare economics. Note alsothat individual demands are assumed tobe measured without error, which, giventhe condition of most budget studies, isan overly-strong assumption.38

2.5 Extensions

While welfare measurement has beenexamined in the context of changes inprices and total expenditure, thesemethods have general applicability andhave been extended in several direc-tions. In this section I take a brief lookat three important applications to illus-trate some of the additional empiricaland conceptual issues.

Equivalence Scales. The model de-scribed to this point has focused on theeffects of price and expenditurechanges on the welfare of a givenhousehold. I now consider the impact ofchanges in household characteristics,which is an issue with several importantpublic policy applications. To deter-mine the appropriate benefit levels oftransfers from the Food Stamp programor Aid to Families with DependentChildren (AFDC), we need to assessthe extent to which “needs” change withhousehold composition. The goal ofsuch adjustments is to ensure that allpoor women with children receive

36 See, for example, Manser and RichardMcDonald (1988), who used this approach to esti-mate bounds on cost-of-living indexes.

37 Some researchers circumvent this problem byassuming that all individuals within narrowly de-fined demographic groups have identical prefer-ences. See, for example, Melissa Famulari (1995).

38 See, for example, the discussion of the Con-sumer Expenditure Surveys by Daniel Slesnick(1992) and how these survey correspond to otherconsumption data.


benefits sufficient to attain the samesubsistence standard of living. Assess-ments of this type are also needed tomake the poverty line comparableacross households, to measure the costsof children and, in some countries, toadjust long term payments to unem-ployed workers and their families.39 Ineach application, a fundamental welfareeconomic question is being asked: Howmuch additional income is required fora household with certain demographiccharacteristics to attain the same wel-fare as a reference household?

The answer can be represented aseither the difference or ratio of expen-diture functions. If AR is the vector ofcharacteristics of the reference house-hold, the additional expenditure re-quired for a household with attributesAk to attain utility VR is:

∆Wk = M(p, VR, Ak) − M(p, VR, AR), (19)

which is analogous to the equivalent vari-ation. In index form, household equiva-lence scales are defined by:

m0(p, VR, Ak) = M(p, VR, Ak)M(p, VR, AR)

. (20)

Because of the policy implications, anumber of studies attempt to measurethe welfare effects of changes in house-hold characteristics such as family size.Many of these propose ad hoc adjust-ments that measure the “needs” ofhouseholds in arbitrary ways. Somestudies focus on the nutritional or foodrequirements of the family members,while others define the scale to be anarbitrary function of the number of per-sons.40 For the purposes of welfare

comparisons, such scales have no obvi-ous interpretation since they are devel-oped outside of a welfare theoreticframework. Rather than summarize thevast literature on equivalence scales(instead, see the surveys by Browning1992 and Lewbel forthcoming), I re-strict consideration to the empirical is-sues that arise in this particular applica-tion of welfare economics.

In measuring the welfare effects ofchanges in prices and expenditures, ef-forts have been directed towards incor-porating price and expenditure effectsflexibly in the demand functions.Demographic variables are oftentreated as an afterthought in an effortto account for heterogeneity. Withequivalence scales, the issue is how tomodel demographic effects in a waythat does not overly restrict prefer-ences. Among the simplest methods ofincorporating demographic variables isto deflate the demands and expenditureby a general equivalence scale, so thatthe Engel curve is:

xk ⁄ m0(Ak) = x(Mk ⁄ m0(Ak)).

This method is attributed to Engel(1895) and is widely used because it iseasily estimated using a single cross sec-tion (see the discussion by Deaton andMuellbauer (1986)). The demographicvariables have the same effect on allcommodities, which seems overly restric-tive since the addition of a child is un-likely to have the same effect on alcoholconsumption as it does on the demandfor milk.

S.J. Prais and Houthakker (1955) gen-eralized this approach in their seminalempirical study by allowing for differ-ent demographic effects across goodswhich yields Engel curves of the form:

39 Lewbel and Richard Weckstein (forthcoming)point out that welfare comparisons of this type arealso made frequently in the legal system in wrong-ful death cases.

40 The equivalence scales implicit in the officialpoverty lines in the U.S. are based on the nutri-tional requirements of households and are de-scribed by Mollie Orshansky (1965). Brigitte Buh-

mann et. al. (1988) define equivalence scales to bea power function of family size, where the powercoefficient indicates the economies of scale in con-sumption.


xik ⁄ mi(Ak) = xi(Mk ⁄ m0(Ak)) (i = 1, 2, ..., n),where mi(Ak) is the commodity-specificequivalence scale for the ith good. Thisapproach is plagued by identificationproblems because the n scales are beingestimated with information on (n-1)freely varying commodity demands (withsummability). Moreover, Muellbauer(1980) has shown that this method of in-corporating demographic effects is con-sistent with rational consumers only ifpreferences are of the Leontief form andsubstitution effects are zero.

These problems can be resolved ifone specifies the direct utility function(rather than the demands) to depend onper equivalent quantities. In this model,developed by A.P. Barten (1964), theindirect utility function depends on theprices scaled by the commodity-specificequivalence scales:

V(p, Mk, Ak) = V(p∗, Mk),where p∗ = (p1m1(Ak), ..., pnmn(Ak)) is thevector of “effective prices” for the kthhousehold. This yields demands of theform:

xik ⁄ mi(Ak) = xi(p∗, Mk) (i = 1, 2, ..., n). (21)

This implies a direct correspondencebetween household compositional ef-fects and price movements. A change incomposition, such as the addition of achild, changes the relative (effective)prices of goods and services and in-duces substitution away from “childgoods”. There is also a scale effect dueto the increased resources needed tomaintain the standard of living. Notethat it is possible for the substitution ef-fect to outweigh the scale effect forsome goods so that an additional childleads to a decrease in demand. This iswidely regarded as an implausible fea-ture of Barten’s model and Muellbauer(1977) shows that data from the U.K.are inconsistent with this method of in-corporating demographic variables in

the demand functions. The equivalencebetween price and compositional move-ments is particularly useful, though, be-cause it facilitates identification of thecommodity-specific scales as long as thedata provide sufficient price variation.

Pollak and Wales (1981) have distin-guished demand models that introducedemographic effects by scaling prices(as in the Barten model) from those inwhich the demands (or budget shareequations) are simply translated by afunction of the demographic charac-teristics. In this latter case the transla-tion parameter γi is interpreted as afixed cost or subsistence level of con-sumption of the good:

xik = γi(Ak) + xi(p, Mk − ∑ j = 1

n

pjγj(Ak))

(i = 1, 2, ..., n). (22)

Gorman (1976) proposed preferencesthat incorporate both translation as wellas scaling, and these preferences havethe attractive feature that the excessivesubstitution found in scaling models isless likely.41

In many ways, the measurement ofthe welfare effects of changes in house-hold characteristics appears exactlyanalogous to the evaluation of the im-pact of price and expenditure changes.In each case we are making an assess-ment of the change in utility as key vari-ables change using observed demandpatterns. While superficially similar, anumber of aspects of equivalence scaleestimation distinguish it from theequivalent or compensating variations.

Measuring the welfare effects ofdemographic changes introduces anormative element to the analysis thatwas previously absent. It is necessary toaddress questions such as how much

41 Generalizations of these models have beenproposed by Lewbel (1985).


more expenditure a family of five needsto be as well off as a married couple.This requires assumptions of interper-sonal comparisons of well-being that arenot empirically refutable. While arbi-trary, interpersonal comparisons areunavoidable if one is to make any state-ment concerning the relative levels ofwell-being of different types of house-holds.42

While economic theory providesguidance concerning the arguments ofthe demand function in measuring thewelfare effects of price and expenditurechanges, it is of little help in determin-ing what equivalence scales should de-pend on. In most applications, attentionhas focused exclusively on the effect offamily size on well-being. Since equiva-lence scales will generally depend onany set of characteristics (observed orunobserved) which influence demandpatterns, omitting the influence ofother demographic variables results inbiased estimates of the parameters ofthe demand functions and the scales. Inthe United States, demand patterns dif-fer significantly across households dis-tinguished by gender, race, age of themembers, and region of residence sothat any assessment of needs based onfamily size alone is likely to be mislead-ing.

Finally, Pollak and Wales (1979),elaborated further by Pollak (1991),contend that information on householdexpenditure patterns cannot be used tomeasure the welfare effects of changesin household characteristics. Demandmodels are estimated conditional onthe existing demographic composition,while welfare comparisons require theevaluation of the change in the uncon-

ditional utility level. Consider a couplethat has been trying to have children foryears and is finally successful. With nochange in total expenditure, the condi-tional preferences will indicate a de-crease in utility because of the increasein consumption requirements. Themeasured welfare effect of this demo-graphic change does not take into ac-count how the couple feels about havingchildren. If the unconditional utilityfunction could be identified, it wouldindicate an increase in well-being.43

More formally, let V(p, Mk, Ak) bethe preferences conditional on thecharacteristics Ak. The unconditionalpreferences incorporate how the fam-ily feels about having these character istics and can be represented asV∗(V(p, Mk, Ak), Ak) . Both descriptions ofpreferences have different implicationsfor equivalence scales but yield thesame demand system. This observa-tional equivalence suggests that, as Pol-lak and Wales argue, household equiva-lence scales cannot be identified fromdemand data without further informa-tion. Blundell and Lewbel (1991) dem-onstrate that with price variation thebest one can do is estimate “relative”equivalence scales.

This problem has several possibleresolutions. It has been suggested thatthe solution is to model the choice ofthose characteristics that are endogen-ous. If we assume that fertility out-comes always reflect choices made byindividuals (a strong assumption), wemight be able to measure how couplesfeel about having children by explicitlyanalyzing their rational choice of familysize. While this identifies the uncondi-tional welfare effect of additional chil-dren, it does not solve the identificationproblem for characteristics that are42 A detailed analysis of the degree of interper-

sonal comparability required for the developmentof equivalence scales has been considered byBlackorby and David Donaldson (1988b) and Lew-bel (1989a).

43 The Pollak and Wales critique has been exam-ined in detail by Browning (1992) and Blundelland Lewbel (1991).


exogenous such as age, race or gender.For these characteristics, one mightidentify the welfare effect using psy-chometric data that asks individualshow they feel about being a particularage or race. While possible, it is notpractical since such information is notreadily available.

An alternative approach is to imposeidentification restrictions on the model.As an example, Blackorby and Donald-son (1988b) and Lewbel (1989a) haveconsidered equivalence scales that areindependent of the reference utilitylevel. This condition holds if the expen-diture function is of the form:

M(p, Vk, Ak) = B(p, Ak)C(p, Vk).

Blundell and Lewbel (1991) have shownthat this restriction on preferences is suf-ficient to identify the equivalence scalesusing demand data alone, although theyfind little empirical support for it in datafrom the United Kingdom. Equivalencescales of this type have been estimatedfor the United States by Jorgenson andSlesnick (1987). Browning (1992) hasshown that the intertemporal allocationof consumption can also be used to iden-tify equivalence scales at least partially.These approaches, along with the devel-opment and integration of alternativedata sources, are promising avenues forresolving the identification problem.

Intertemporal Welfare Measurement.The emphasis to this point has been onthe measurement of welfare in a static,single-period framework. Even if in-tertemporal separability holds, a snap-shot of the household’s welfare couldgive a misleading picture since the ef-fects of policies often last more thanone period. The bias associated withstatic welfare analysis has been demon-strated empirically by James Poterba(1989,1991) in his examination of theincidence of indirect taxes. He showedthat when the distribution of the tax

burden is measured using current in-come as a proxy for household welfare,indirect taxes appear more regressivethan when welfare is measured usinghousehold consumption (which approxi-mates permanent income). This sug-gests that significant errors can bemade by ignoring the “lifetime inci-dence” of policies.44

A more formal description of the bi-ases associated with using the staticequivalent variation to measure in-tertemporal welfare changes is pre-sented by Blackorby, Donaldson andDavid Moloney (1984) and MichaelKeen (1990). Intertemporal welfare ef-fects are often represented as the dis-counted sum of the within-periodequivalent variations. This will differfrom the lifetime equivalent variation tothe extent that households have theability to substitute intertemporally.Let VL be the maximum level of life-time utility of a household that lives Tperiods when the profiles of prices andinterest rates are

pt

and

rt

respec-

tively. If the (optimal) time path of util-ity corresponding to VL at these pricesand interest rates is

Vkt

, the lifetime

expenditure function can be repre-sented as:

ΩL(pt

,

rt

, VL) = ∑

t

gt M(pt, Vkt, Akt),

where:

gt = ∏(1s = 0

t

+ rs)−1,

and r0 is equal to zero. As in the staticframework, the lifetime expenditurefunction can be used to represent an ex-act measure of the change in lifetimewelfare.

Let VL1 be the maximum level of life-

44 See also the evaluation of a value added taxconsidered by Erik Casperson and Gilbert Metcalf(1994).


time welfare when the profile of pricesand interest rates are

pt

1 and

rt

1 and

denote the corresponding time path ofutility as

Vkt

1 . In the reference scenario,

prices and interest rates are pt

0 and

rt

0

resulting in a lifetime utility level of VL0

and within period utilities Vkt

0 . An exact

measure of the change in lifetime wel-fare, evaluated at the reference prices,is:

∆WL = ΩL(pt

0,

rt

0, VL

1 ) − ΩL(

pt

0,

rt

0, VL

0 ). (23)

This is a straightforward extension of thestatic version of the generalized equiva-lent variation (6).

The change in lifetime welfare is notequal to the discounted sum of withinperiod changes in utility. Equation (23)can be rewritten as:

∆WL = ∑ t

gt0[M(pt

0, Vkt1 , Akt)

− M(pt0, Vkt

0 , Akt)] + ∑ t

gt0[M(pt

0, Vkt3 , Akt)

− M(pt0, Vkt

1 , Akt)], (24)

where Vkt

3 is the optimal time path of

utility required to attain lifetime welfareVL

1 when prices and interest rates are pt

0

and rt

0. The first term in (24) is the dis-

counted sum of the within periodchanges in welfare. The second term isthe gain to the household that resultsfrom its ability to adjust the time path ofconsumption as the profile of prices andinterest rates change. This is nonpositivebecause the rational household mini-mizes the cost of attaining a given levelof lifetime welfare, implying that the dis-counted sum of the intraperiod changesin utility overstates the change in life-time welfare. The magnitude of the biasis related to the ability of the householdto adjust consumption intertemporally.

Since the present value of the within-period changes results in a distorted as-

sessment of lifetime well-being, an al-ternative empirical model is required tomeasure the change in welfare accu-rately. Jorgenson, Slesnick and PeterWilcoxen (1991) describe a model inwhich households are assumed to be in-finitely-lived and linked to similarhouseholds in the future through pref-erences that are consistent with inter-generational altruism. The infinitely-lived household, referred to as a“dynasty”, is assumed to maximize anadditive intertemporal utility functionof the form:

VL = ∑ t = 0

∞

δt ln Vkt, (25)

where δ = 1 ⁄ (1 + ρ) and ρ is the subjec-tive rate of time preference. The withinperiod indirect utility function Vkt is rep-resented by:

ln Vkt = αp′ln pt + 12

ln pt ⁄ Bpp ln pt

− D(pt) ln (Mkt ⁄ m0(pt, Akt)), (26)

where m0(pt, Akt) is the number of house-hold equivalent members in period t.The unknown parameters of this utilityfunction can be estimated in the usualway using information on household ex-penditure patterns.

Lifetime utility is maximized over thetime path of utility

Vkt

subject to the

nonlinear budget constraint:

∑ t=0

∞

gt Mkt(pt, Vkt, Akt) = Ωk,

where Ωk is the wealth of the household.This optimization implies a discrete timeEuler equation of the type:

ln Vkt = Dt

Dt − 1 ln Vkt − 1

+ Dt ln

Dt − 1gtm0(pt, Akt)Pt

δDtgt − 1m0(pt − 1, Akt − 1)Pt − 1

, (27)


where Dt denotes D(pt) and Pt is a priceindex. The Euler equation can be usedto obtain a closed form representation ofthe maximum level of lifetime utility andthe intertemporal expenditure function.The change in well-being from a policychange is then easily calculated using(23).45

The evaluation of changes in in-tertemporal well-being introduces sev-eral additional empirical issues. As inthe static model, the parameters of thewithin-period utility function can be re-covered from demand equations thatare estimated using household expendi-ture data. The intertemporal allocationintroduces additional parameters re-lated to the first stage allocation of totalexpenditure in each period. It is neces-sary to estimate the subjective rate oftime preference as well as a parameter(or set of parameters) related to the in-tertemporal elasticity of substitution.

In the model described above, theEuler equation (27) can be used in con-junction with the within-period alloca-tion of expenditures across goods to es-timate the unknown parameters. Thisincreases the data requirements be-cause it requires a panel of householdswhose expenditures are recorded overtime. Since such data are rare, in gen-eral, and nonexistent in the UnitedStates and the United Kingdom, re-peated cross sections have been used tocreate synthetic cohorts to identify theintertemporal preference parameters.The time variation in the cross sectionsfacilitates the estimation of the expen-diture price and demographic parame-ters related to the within-period alloca-tion of goods and services. These typesof models are becoming more commonin investigations of consumption behav-

ior (see, for example, Blundell, Brown-ing and Meghir 1994) among others)but have only rarely been used to mea-sure welfare. A notable exception is theestimation of the intertemporal welfareeffects of children by Banks, Blundelland Ian Preston (1994).46

Another issue that arises in thesemodels is the appropriate treatment ofdemographic characteristics. As de-scribed in the previous subsection, thecharacteristics that affect demands areusually assumed to be exogenous andonly their conditional effects are evalu-ated. In a life-cycle context, exogeneityis no longer a reasonable assumptionand the fact that these householdschoose some of their characteristicsneeds to be accounted for in a frame-work that measures lifetime well-being.Note also that the preferences de-scribed above assume time separability.Further work is necessary to generalizethese models to accommodate habit for-mation and durability and analyze theresulting implications for welfare analy-sis.

Labor Supply: Kinks, Corners andNotches. A common empirical applica-tion of welfare economics is the analysisof tax-induced changes in wages. Ratherthan measure the change in utility perse, the goal is to evaluate the dead-weight loss or excess burden of thetaxes.47 Assume initially that budgetconstraints are linear and individualsface prices p, have a wage wk, full ex-penditure Fk, and that the indirect util-ity function is V(wk, p, Fk, Ak). If p0 andwk

0 are the (fixed) pre-tax prices andwage, a measure of the efficiency loss

45 The intertemporal equivalent variation hasalso been estimated by Don Fullerton and DianeRogers (1993) in their examination of the lifetimeincidence of taxes.

46 See also the paper by Pashardes (1991) whouses somewhat different methods to measure in-tertemporal equivalence scales.

47 Rosen (1978) was among the first to use microdata on labor supply to measure the deadweightloss due to labor taxation. A more recent exampleis Blundell, Meghir, Elisabeth Symons, and IanWalker (1988).


due to distortionary taxation using theequivalent variation is:48

DWLk = F(wk1, p1, V1, Ak)

− F(wk0, p0, V1, Ak) − R(wk

1, p1, Fk, Ak),

where R is the tax revenue collected, F( )is the full expenditure function, h( ) isthe labor supply function, pi

1 = pi0 + ti (i =

1, 2, ..., n), wk1 = (1 − t0)wk

0, and V0 and V1

are the pre- and post-tax utility levels.This measure of the deadweight loss at-tributable to the taxation of goods andleisure (i.e. labor supply) is the amount,in excess of the tax revenue collected,the consumer would be willing to pay (atexisting prices) to forgo all taxes. Put an-other way, it is additional revenue thatcan be collected, with no loss in utility,using lump sum rather than distortionarytaxation:

V(wk0, p0, (Fk − (DWLk + R)), Ak)

= V(wk1, p1, Fk, Ak).

To measure the deadweight loss, it isnecessary to recover the full expendi-ture function so the empirical issues areessentially the same as those addressedpreviously. However, several aspects ofmodeling labor supply distinguish itfrom the demand models that havebeen considered to this point. Whilethe consumption of goods and servicesare aggregated over the members of thehousehold, we frequently have disaggre-gated information on their individual la-bor supplies. This suggests the impor-tance of modeling the interaction ofmembers’ labor supply decisions as inHausman and Paul Ruud (1984), Mi-chael Ransom (1987), Lundberg (1988)and Schultz (1990).

Labor supply also introduces the

problem that only market labor is ob-served in most data sets. Householdproduction issues are usually ignored,as are other uses of time (other than lei-sure) that could influence well-being.49

This is particularly important in examin-ing changes in welfare over time in theU.S., because of the increase in laborforce participation of women in the1970s. At first glance it may appear thatwell-being declined over this period be-cause of the decrease in the consump-tion of leisure. This conclusion wouldobviously change if women were simplysubstituting market for nonmarket la-bor.

Models that incorporate labor supplyare also distinguished by the fact thatbehavior appears discrete rather thancontinuous because the budget con-straint facing most workers has a num-ber of kinks and corners. Some workers,for example, choose not to participatein the labor market and their well-beingmust be evaluated at the corner corre-sponding to zero hours of work. Work-ing has fixed costs such as child care,commuting costs and other occupa-tional expenses, which not only intro-duce kinks to the budget constraint butnonconvexities as well. These costs in-duce individuals either to stay out ofthe labor market or to work enoughhours to cover the associated expenses.The dependence of the tax rates on thelevel of earnings also has the effect ofintroducing additional nonlinearities.

There are two related empirical is-sues. First, one must initially decide onthe appropriate econometric model toestimate the labor supply equations.The second problem is the recovery ofpreferences from these equations andthe subsequent measurement of48 Alan Auerbach (1985) attributes this particu-

lar measure to Herbert Mohring (1971) and pro-vides a general survey of the measurement of ex-cess burden. See also the discussion by PeterDiamond and Daniel McFadden (1974) and J.A.Kay (1980).

49 A notable exception is Patricia Apps and RayRees (1996) who employ time-of-use data and anexplicit household production model to examinethe within family distribution of welfare.


changes in welfare. Hausman (1985)(drawing on his previous work) ad-dressed the first issue by developingmethods that can be used to estimateunknown parameters when choices aremade subject to nonlinear budget con-straints. This requires the determina-tion of the desired hours of work h∗ foreach worker including the kinks andcorners. Hausman’s algorithm exploitsthe feature that individuals are maxi-mizing utility over budget constraintsthat are the union of convex sets.

Once the desired level of labor sup-ply is determined, the next issue is todetermine the wage and expenditure atwhich h∗ should be evaluated. If the op-timum occurs over the interior of a seg-ment of the budget constraint, h∗ isevaluated at the wage and expenditurelevel that describes that segment. If theoptimum occurs at a corner or kink, thefunction must be evaluated at the vir-tual expenditure and prices that ration-alize this quantity of labor supply. If thedesired labor supply occurs at a kinkcorresponding to H1 hours of labor sup-ply and l1 = T − H1 hours of leisure, J.P.Neary and Kevin Roberts (1980) haveshown that the corresponding virtualwage wk

v and expenditure Fkv are defined

implicitly by:

H1 = h∗(wkv, p, Fk

v, Ak), (28)

where Fkv = Fk + (wk

v − wk)l1. If the opti-mum occurs at an interior point of a linesegment, rather than a corner, the virtualwage is the observed after-tax wage. Oncethe desired labor supply is representedas a function of the relevant wage and ex-penditure, the unknown parameters of thelabor supply equations can be estimatedin the usual way using minimum distanceor maximum likelihood methods.

The utility level of the individual,whether at a corner or not, is evaluatedat the virtual wage and expenditure:

Vk = V(wkv, p, Fk

v, Ak).With the representation of utility as afunction of the virtual wage and expendi-ture levels, a monetary measure of thechange in well-being can be calculated inthe usual way using the equivalent vari-ation. Blundell, Meghir, Symons andWalker (1988) have used these methodsto examine the welfare effects of changesto the United Kingdom tax code.

Although labor supply is the mostcommon application, the methods usedto deal with nonlinear budget con-straints have general applicability. Theycan be used to analyze the welfare ef-fects of any policies that are made inthe presence of kinks or corners in thebudget constraints. Bruno De Borger(1989), Manser (1987), Robert Schwab(1985) and Slesnick (1996) use similareconometric models to examine thewelfare effect of in-kind transfers. Theprovision of these benefits produces abinding kink in the budget constraintfor recipients who, with an equivalentcash transfer, would consume less ofthe good than is transferred in-kind.Similar issues arise in models in whichdemand behavior is discrete (such aswith consumer durables or housing) orwhere a large number of individuals de-cide not to make a purchase.

3. Measuring Social Welfare

The conceptual issues underlying themeasurement of household welfare are,for the most part, well understood.Consumption behavior reveals consum-ers’ willingness to pay so that demandequations can be used to recover theirexpenditure functions and measure theequivalent or compensating variations.The focus of attention is not on whatconcept should be used to measurewell-being but the empirical methodsused to model demand behavior.

What emerges from this analysis are


estimates of welfare for households thatare distinguished by their demographiccharacteristics. While these welfare es-timates are of independent interest, thegoal of empirical welfare analysis is oftenan assessment of the effects of policieson groups of households. We want toknow whether tax reform hurts the poormore than the rich or if changes intransfer programs affect women andchildren adversely. Estimates of house-hold welfare are an essential first step,but for welfare economics to be usefulto practitioners, a method of aggrega-tion must be developed.

Any effort to develop an index ofgroup welfare must inevitably makenormative judgements in which thegains to some are weighed against thelosses to others. Should a policy be im-plemented if the welfare of the poor in-creases slightly but the welfare of therich decreases dramatically? The an-swer ultimately depends on the extentto which well-being can be comparedacross the population and on theweights assigned to individual agents.

Due largely to economists’ reticenceto confront such normative questions,measures of group welfare used bypractitioners have been developed in anad hoc manner with little or no welfareeconomic justification. Rather thanavoid the issue, a more appealing strat-egy is to confront it head on and makeexplicit the subjective judgements thatare required to aggregate householdwelfare measures consistently into mea-sures of social welfare. These assump-tions take the form of choosing a repre-sentation for the household welfarefunction, stating the degree of measur-ability and comparability across house-holds, and deciding the form of thesocial welfare function. A criticalempirical issue is the extent to whichsocial welfare judgements vary across theseempirically unverifiable assumptions.

To illustrate the judgements thatoften remain implicit in policy analysis,consider per capita income as an indica-tor of social welfare. Income is used torepresent individual welfare eventhough it bears little relation to any ofthe welfare functions described in Sec-tion 2. It is further assumed that in-come can be used to measure and com-pare well-being and that the socialwelfare function is a utilitarian index inwhich each member of the population isgiven equal weight. Although widelyused, per capita income is no less arbi-trary than an explicitly specified socialwelfare function. Over the remainder ofthis survey, I consider the issues relatedto social welfare measurement.

3.1 The State of Practice

Given a well-defined approach to themeasurement of welfare at the microlevel, how can the resulting indicators beaggregated to yield consistent rankings ofsocial outcomes? There are four meth-ods that are used to varying degrees.

Representative Agent Model. Theeasiest approach to finessing the aggre-gation problem is to assume that marketdemands are generated by a repre-sentative consumer. Under this assump-tion, the preferences of the consumer arerevealed by aggregate demand patternsas long as the integrability conditionshold. The techniques described in Sec-tion 2 are applied to market data to ob-tain monetary measures of the changein utility for the representative agent.

This approach to social welfare analy-sis has several fundamental problems.Market demands need not be consis-tent with the assumption of a rationalrepresentative consumer. Hugo Sonnen-schein (1972) has shown that even ifevery individual has demands that areconsistent with utility maximization, ag-gregate demands need not satisfy any ofthe integrability conditions other than


homogeneity of degree zero in pricesand income and Walras’ Law. Withoutfurther restrictions, there is no hope ofusing market demands to recover thepreferences of the representative agentand measure social welfare.

What restrictions are necessary?Roughly speaking, the marginal incomeeffects must be invariant across thepopulation. Samuelson (1956) demon-strated that aggregate demands can berepresented as those of a rational con-sumer if income is distributed so as tomaximize a social welfare function.Given the disparities in the empiricaldistribution, this assumption can berejected out of hand. Gorman (1953)described alternative functional formconditions that yield a representativeagent. Consider a model in which ag-gregate demands are functions of pricesand aggregate (or average per capita)expenditure:

∑ k = 1

K

xik(p, Mk) = Xi(p, ΣMk). (29)

Gorman showed that if individual con-sumers maximize utility, (29) holds if themicro-level demands are of the form:

xik(p, Mk) = αik(p) + βi(p)Mk, (k = 1, 2, ..., K).

As with Samuelson’s condition, the mar-ginal income effects must be constantacross all consumers. If demands arezero when there is no income, prefer-ences must be identical and homothetic.As mentioned previously, this restrictionis inconsistent with longstanding empiri-cal evidence.50

Aside from these empirical issues, ad-ditional conceptual problems are associ-

ated with using the representative agentmodel to measure social welfare. Distri-butional considerations are ignoredfrom the outset, and these are oftenof paramount concern in examining theeffects of different policies. Moreover,it’s unclear what the utility functionof the representative agent actuallyrepresents. Kirman (1992) presents anexample in which the representativeconsumer prefers (aggregate) marketbasket A to B even though everybody inthe economy prefers their allocation inB to that in A. This violation of themost basic principle of social choicesuggests that the utility function of therepresentative consumer should not beused for policy analysis even in the un-likely event that market demands areconsistent with the aggregation condi-tions.

Pareto Principle. In the early devel-opment of welfare economics, utilitywas considered to be both measurableand comparable across individuals. Bythe end of the 1930s, though, econo-mists began to reject welfare judge-ments founded on interpersonal com-parisons of utility. Lionel Robbins’(1932,1938) critique of this practice hasbeen taken to be the watershed eventthat shifted the focus from utilitarian-ism to an exclusive reliance on thePareto principle that requires only or-dinally noncomparable utilities.51 Ifeveryone is better off under policy 1relative to 2 then, under the Paretoprinciple, it is socially preferred as well.Such unanimity is the exception ratherthan the rule, and this method of wel-fare analysis has little practical useful-ness.

Despite the fact that the ordering ofsocial states is incomplete, the strict

50 See Alan Kirman (1992) and Stoker (1993) fordiscussions of the evidence against the repre-sentative agent assumption. Less restrictive ver-sions of the representative consumer model thatincorporate additional features of the distributionof income have been presented by Muellbauer(1976) and Lewbel (1989b).

51 Robert Cooter and Peter Rappoport (1984)provide an historical review of the debate of theappropriate conceptual basis for welfare econom-ics.


application of the Pareto principle hasremained a central element of welfareeconomics due, in part, to the paralleldevelopment of Kenneth Arrow’s (1951)impossibility theorem. This famous re-sult states that, under a set of “reason-able” conditions, the only social welfarefunction that ranks outcomes consis-tently is a dictatorship. This starklynegative finding became a major stum-bling block to the empirical implemen-tation of an explicit social welfare func-tion. How is it possible to developbenevolent public policy if the only wayto rank alternatives is to use the prefer-ences of one person?

The Arrow impossibility theoremspawned a subfield of economic theorythat focused on its robustness to modifi-cations of the assumptions. Theorem af-ter theorem demonstrated that modestchanges to the axiomatic structure leftthe dictatorial result intact.52 The in-escapable conclusion was that if oneprecludes interpersonal comparisons ofwelfare, the only logically consistentfoundation for welfare analysis is thePareto principle. For economic theo-rists, welfare economics became com-pletely summarized by the two Funda-mental Theorems of Welfare Economicsrelating competitive equilibria to Paretooptimality.

Of course, this was bad news forpractitioners. The Pareto principle canonly rarely be applied in practice, andmost would agree that a dictatorship isan inappropriate basis for the formationand evaluation of public policy. Thisleft a gap between what theorists feltwas a suitable framework for appliedwelfare economics and what policy ana-lysts needed to provide a complete andconsistent assessment of policies.53

Compensation Principles. As early as

the 1930s, efforts were made to makethe Pareto principle operational usingcompensation mechanisms. A typicalcriterion is for policy 1 to be judged animprovement over policy 2 if it is possi-ble to reallocate goods in 1 to yield anallocation that is Pareto superior to 2.While this criterion has been used ex-tensively, it introduces both conceptualand practical problems. It is possible toobtain what Blackorby and Donaldson(1990) refer to as “Scitovsky reversals”in which policy 1 is judged to be pre-ferred to policy 2 and vice versa. Evenif one eliminates from consideration al-locations for which these reversals canoccur, Gorman (1955) has shown thatthe ordering of policies under this crite-rion need not be transitive. These areserious shortcomings that limit the use-fulness of this approach to ordering so-cial states.

The Kaldor-Hicks-Samuelson ap-proach described by Chipman andMoore (1971) provides a more stringentcriterion for comparing outcomes. Pol-icy 2 is preferred to policy 1 if, for anyallocation of aggregate goods under pol-icy 1, it is possible to find an allocationunder policy 2 that is Pareto superior toit. Chipman and Moore have shown thatthis compensation principle does notsuffer from the preference reversalsand intransitivities that plague manycompensation criteria. The stringencyof the requirement for a welfare im-provement, though, results in an order-ing of social states that is often incom-plete.

Ignoring the incompleteness, an im-portant practical issue is how to makethis specific compensation principle

52 See Sen (1995) for a synthesis and review ofthe axiomatic basis for Arrow’s famous theorem.

53 It should be noted that while most of the pro-fession rejected the feasibility of social welfarefunctions, Abram Bergson (1938) and Samuelson(1947) maintained the position that explicit socialwelfare functions were an appropriate basis forpolicy analysis.


operational. Clearly, it is not feasible toexamine all possible lump-sum redis-tributions of goods across the popula-tion under each scenario. Efforts fo-cused on aggregate index numbers asindicators of changes in “potential wel-fare” as revealed by the Kaldor-Hicks-Samuelson criterion. Let X2 be the ag-gregate demands under policy 2 anddefine X1 to be the corresponding de-mands under policy 1. Valuing the de-mands using the prices of policy 1, realaggregate expenditure has increased un-der 2 if:

p1′X2 ≥ p1′X1. (30)

The Paasche version of the increase inreal expenditure is given by:

p2′X2 ≥ p2′X1. (31)

If either or both of these conditions re-veal an increase in potential welfare, wehave a relatively simple method of meas-uring changes in aggregate welfare.

Unfortunately, conditions (30) and(31) reveal increases in potential wel-fare (as indicated by the Kaldor-Hicks-Samuelson criterion) only under restric-tive conditions. Chipman and Moore(1973, 1976a) have shown that (30) isonly a necessary condition for an in-crease in potential welfare under policy2. Both the Paasche and the Laspeyresindicators of the increase in real aggre-gate expenditure are sufficient condi-tions for an increase in potential wel-fare if and only if preferences areidentical and homothetic.

Given these negative results, recentefforts have used summary statisticsother than real aggregate expenditureto describe changes in potential wel-fare. Blackorby and Donaldson (1990)and Javier Ruiz-Castillo (1987) showthat the sums of the individual equiva-lent or compensating variations revealthe direction of change in potential wel-fare under conditions similar to those

derived by Chipman and Moore. Thissuggests that even if one accepts theKaldor-Hicks-Samuelson criterion asthe basis for measuring the change inwelfare, there is no obvious method ofimplementing it without imposing verystrong assumptions on preferences. Per-haps most important, Sen (1979) pointsout that the “New Welfare Economics”is, in a sense, irrelevant. If the com-pensation is not provided, or, evenstronger, cannot be provided due to theunavailability of lump sum taxes andtransfers, its impact on welfare is neverrealized and should have no influenceon public policy. If it is provided, wecan simply apply the Pareto principle.

Aggregate Surplus Measures. Ratherthan use the sum of the compensatingor equivalent variations as indicators ofpotential welfare, an alternative ap-proach is to define a function over theindividual surplus measures as an ex-plicit representation of the change insocial welfare. Such an approach to ag-gregation was advocated by Harberger(1971), among others, in his effort tomake consumer’s surplus the standardtool for applied welfare analysis. At firstglance, this appears to be a reasonableway to proceed. The equivalent vari-ation provides an exact representationof the change in welfare for each indi-vidual, and would seem to be an idealcandidate for the arguments of a socialwelfare function.

At a conceptual level, such measuresof social welfare are represented asnatural extensions of the positive analy-sis of welfare measurement at the microlevel. This is obviously not the case be-cause aggregation necessitates norma-tive judgements related to the treat-ment of unequally situated individuals.If a policy increases the welfare of arich person by $1000 and lowers thewelfare of a poor person by $500, hassocial welfare increased or decreased?


Simply summing the surplus measures,as is common, embodies a version ofutilitarianism, ignores distributionalconcerns, and would indicate an in-crease in aggregate welfare. This andother approaches to aggregation re-quires assumptions concerning distri-butive ethics and there is no wayaround this issue. The differences inthe various approaches lies in whetherthe assumptions are implicit or explicit.

Even if the ethical basis of theaggregation is explicit, there are reasonsto avoid using surplus measures asarguments of social welfare functions.To see why, let m(p, v) = (M1(p, V1),M2(p, V2), ..., MK(p, VK)) be the vector ofindividual expenditure functions anddefine a social welfare function Ψ overthe distribution of expenditure func-tions as:

E(p, v) = Ψ(m(p, v)),

where v = (V1, V2, ⋅⋅⋅, VK) is the vector ofutility functions. This appears to be areasonable approach to aggregation sincethe expenditure functions are exact rep-resentations of individuals’ underlyingpreferences and Ψ can be specified tosatisfy the fundamental principles of so-cial choice.

The problem is that the expenditurefunction provides an exact repre-sentation of individual preferences for afixed set of reference prices p. Thechoice of the prices used to “cardinal-ize” preferences is arbitrary, but the or-dinal ranking of social states needs tobe invariant to this choice. That is, thesocial welfare function should satisfy:

E(p, v) ≥ E(p, v′) if and only if E(p′, v) ≥ E(p′, v′),

for all possible v, v′, p and p′. This is theproperty of “price independence” intro-duced by Roberts (1980a) who has shownthat it will hold only under restrictiveconditions. In the absence of restrictions

of individual preferences, the social wel-fare function must be dictatorial. Withno restrictions on the social welfarefunction, individual preferences must byidentical and homothetic.54 In addition,Blackorby and Donaldson (1988a) haveshown that social welfare functions thatuse money metric utility functions as ar-guments need not be quasiconcave and,therefore, exhibit undesirable ethicalproperties.

3.2 The Aggregation Framework: Social Choice Theory

Since any method of aggregation nec-essarily involves subjective judgements,a reasonable way to proceed is to stateexplicitly the underlying ethical basisfor the social ordering, and Arrow’s(1951) axiomatic framework provides anideal setting for doing this. Using thenotation of Sen (1977) and Roberts(1980c), let R be a complete, reflexiveand transitive ordering over the set ofsocial states X. A social welfare func-tional f is a mapping from the set of in-dividual welfare functions to the set ofsocial orderings so that f(u) = f(u′) im-plies R = R′ where u = (W1, W2, … , WK)is a vector of individual welfare func-tions. Using this notation, the axiomsunderlying Arrow’s impossibility resultare:

1. Unrestricted Domain(U): The so-cial welfare functional f is defined forall possible vectors of the individualwelfare functions u.

This seems like an innocuous assump-tion although it is worth noting that ifthe individual welfare functions are re-lated to utility functions, the domainneed not be unrestricted. In this case,the feasible set is limited to welfarefunctions that are consistent with theintegrability conditions.

54 These results have been extended by Black-orby and Donaldson (1985, 1990) and Slesnick(1991a).


2. Independence of Irrelevant Alter-natives (IIA): For any subset A con-tained in the set of all social states X, ifu(x) = u′(x) for all x in A, then R:A = R′:A,where R = f(u) and R′ = f(u′) and R:A de-notes the social ordering over A.

Roughly speaking, the IIA assump-tion implies that the ranking of any twosocial states must be independent of athird. This axiom has been the subjectof controversy and precludes social wel-fare functions that are based on, for ex-ample, rank-order voting. It also elimi-nates the possibility of incorporatingprinciples of horizontal equity in whichoutcomes are evaluated relative to a ref-erence social state such as the statusquo.

3. Weak Pareto Principle (WP): Forany x, y in X, if Wi(x) > Wi(y) for all indi-viduals i, then xPy where P indicatesstrict social preference.

This form of the Pareto principleimplies that if everybody is better offin one state relative to another, it issocially preferred as well.

Perhaps the most important assump-tion underlying social choice is theassumption of the measurability andcomparability of individual welfarefunctions. Motivated by the fact thatdemands are invariant to monotonictransformations of the utility functionsthat can differ across individuals, Ar-row assumed that welfare functions areordinally noncomparable. Any other as-sumption concerning the extent towhich welfare can be measured or com-pared cannot be refuted by empiricalevidence. To describe this conditionformally, let Lk be the set of admissiblewelfare functions for the kth individual,and define L to be the Cartesian prod-uct of all such sets. Let Λ be the parti-tion of L with elements that yield thesame social ordering.

4. Ordinal Noncomparability (ONC):The set of welfare functions, Λ, that

yield the same social ordering is de-fined by:

Λ = u′: Wk′ = ϕk(Wk)

,

where ϕk is increasing and f(u) = f(u′) forall u in Λ .

Relative to alternative assumptionsrelated to interpersonal comparability,ONC requires the least information onindividual preferences, but imposes themost stringent invariance requirementon the social welfare functional.

If one accepts these seemingly rea-sonable conditions on social prefer-ences, Arrow proved that the only possi-ble social welfare functional is adictatorship. This raises the question ofthe additional information that is re-quired for a more inclusive approachpolicy analysis. The breakthrough oc-curred with a series of seminal booksand papers by Sen (1970, 1973, 1977)who approached the problem of collec-tive choice in a less nihilistic light. Heviewed the Arrovian axioms as restric-tive, rather than reasonable, becausethey rule out by assumption the abilityof the analyst to weigh the gains of thewinners against the losses of the losers.He demonstrated that relaxing the mea-surability and comparability assump-tions expands the spectrum of possiblesocial welfare functionals dramatically.

Sen’s work changed the focus of re-search from demonstrations of impossi-bility to descriptions of the conditionsunder which a nondictatorial social wel-fare function is feasible. Others realizedthat practical application of welfareeconomics requires information based,either implicitly or explicitly, on theanalyst’s assessment of the relative im-pacts of policies on different individu-als. The greater the (assumed) ability tomeasure and compare welfare levels,the wider the set of possible socialwelfare functions. Peter Hammond(1976) considered the case in which


policy makers are able to rank welfarelevels ordinally across individuals butcannot measure the magnitudes of thedifferences in well-being:

Ordinal Level Comparability (OLC).The set of welfare functions, Λ, thatyield the same social ordering is de-fined by:

Λ = u′: Wk′ = ϕ(Wk)

,

where ϕ is increasing and f(u) = f(u′) forall u in Λ .

Under OLC, welfare levels areunique up to monotonic transforma-tions that are the same for every indi-vidual. Relative to ordinal noncom-parability, this assumption imposesweaker invariance restrictions on thesocial ordering but requires more infor-mation since the analyst must now rankthe relative standing of each member ofthe population. Maintaining variants ofthe other Arrovian assumptions, Roberts(1980b) showed that the set of possiblesocial welfare functionals are “posi-tional” dictatorships such as the case inwhich the worst-off individual is deci-sive or its lexicographic extension.55

If the analyst is unable to rank wel-fare levels but can determine that achange in policy hurts individual Atwice as much as individual B, welfarelevels are cardinal unit comparable andthe social ordering must satisfy the con-dition:

Cardinal Unit Comparability. The setof welfare functions, Λ, that yield thesame social ordering is defined by:

Λ = u′: Wk′ = αk + β(Wk)

,

where β > 0 and f(u) = f(u′) for all u in Λ. Under the assumption that changes in

welfare are comparable, the social wel-fare functional is utilitarian and can be

represented as the sum or weightedsum of welfare functions.56 The as-sumption that changes in welfare arecomparable has radically different im-plications for the ranking of outcomescompared to ordinal level comparabil-ity. While the utilitarian framework hasno distributional concern, it is of para-mount importance with the Rawlsian so-cial welfare functional obtained underlevel comparability.

As the assumptions of comparabilityget stronger, the invariance restrictionson the social welfare functional getweaker, and the set of feasible socialwelfare functionals expands. Ordinalnoncomparability assumes the leastabout the analyst’s ability to measureand compare welfare levels but imposesthe most stringent invariance require-ments. If, at the opposite extreme, it isassumed that both levels and changes ofwelfare are comparable, the set of feasi-ble social welfare functionals expands tothose of the following canonical form:57

W(u) = W__

+ g(W1 − W__

,

W2 − W__

, …, WK − W__

),

where

W__

=

∑Wk

k = 1

K

K,

and g(.) is a linearly homogeneous func-tion. Social outcomes are ordered on thebasis of the sum of the mean and an in-dex of deviations from the mean, so thesocial welfare functional incorporatesconcern for both efficiency and equity.

55 That is, the worst-off, nonindifferent individ-ual is decisive. See also Hammond (1976) and Sen(1977) for alternative characterizations of socialwelfare functionals related to the Rawlsian Differ-ence Principle.

56 For further details, see Roberts (1980c,p. 429). Eric Maskin (1978) provides an alternativecharacterization of utilitarianism.

57 This is the assumption of cardinal full compa-rability and is similar to cardinal unit comparabil-ity except that the social ordering must be invari-ant to affine transformations that are the same forevery individual. See Roberts (1980c) for furtherdetails.


The conclusion is that changing themeasurability and comparability as-sumptions expands the set of possiblesocial welfare functionals dramati-cally.58 “Possibility” theorems weresynthesized and extended in a seriesof papers by Roberts (1980b,c,d). Bythe early 1980s, theoretical results dem-onstrating the existence of logically con-sistent, nondictatorial social welfarefunctions had been established but re-mained unexploited empirically.

3.3 Implementation Issues in the Measurement of Social Welfare

Just as the theory of consumer behav-ior provides a framework for measuringindividual welfare, social choice theoryserves a similar function in measuringaggregate well-being. In evaluating al-ternative realized outcomes, the socialwelfare functionals suggest specificforms for the Bergson-Samuelson socialwelfare functions that can be used.59 This requires that a number of addi-tional practical issues be resolved be-fore empirical implementation of thesocial welfare measures is possible.

As a way of illustrating the practicalchoices that need to be made in aggre-gating individual welfare functions, con-sider a widely used social welfare func-tion that was originally proposed byAnthony Atkinson (1970):

W(u) =

11 − ρ

∑ k = 1

K

yk1 − ρ, (32)

where yk is the income of individual k,and the individual welfare function istaken to be Wk = yk

1 − ρ ⁄ (1 − ρ). In addi-tion to the conditions of unlimited do-main, independence of irrelevant alter-natives and the weak Pareto principle,this utilitarian social welfare function re-quires the assumption of cardinal unitcomparability.

The Units of Observation. Whose wel-fare functions should serve as argu-ments of the social welfare function?For a comprehensive assessment ofaggregate well-being, (32) should beevaluated using all members of thepopulation. The problem is that a largesegment of the population (e.g. childrenand other dependents) has no sourcesof income but access to resources thatenable them to maintain a reasonablestandard of living. Using individual in-comes or restricting the sample to thosewho receive income, as is typical in ex-aminations of earnings distributions,undoubtedly distorts assessments of so-cial welfare.

This problem could be resolved if weknew the resources available to eachperson or, better yet, their consump-tion. As a practical matter this type ofinformation is almost never available.As a second best alternative, the unit ofobservation is typically taken to be thehousehold and the distribution of re-sources among members is inferred. Inthe simplest version of this story, in-come or consumption is assumed to bedivided equally among the members (aper capita model) or an optimal intra-household allocation is assumed. In thelatter case, each member attains thesame welfare and the household can betreated as a single autonomous agent.Both assumptions are obviously gross

58 Of course, it is also possible to have a nondic-tatorial social welfare functional without interper-sonal comparability if other axioms are relaxed.For example, if preferences are defined over a sin-gle dimension and are single peaked (violation ofunlimited domain), the social welfare function ismajority rule and aggregate preferences are thoseof the median voter. Sen (1977) and Roberts(1980c) describe in greater detail the possibilitytheorems that result when axioms other than thoserelated to interpersonal comparability are relaxed.

59 For further discussion of the link between themultiple profile framework of social choice de-scribed above and the single profile framework de-scribed by Bergson (1938) and Samuelson (1947),see Roberts (1980d).


simplifications and there is evidence(Lawrence Haddad and Ravi Kanbur1990, Apps and Rees 1996, and others)that shows that such inferences can dis-tort inequality and poverty assessments.

In light of the existence of householdpublic goods and the absence of com-prehensive data on within-householdtransactions, welfare measurement atthe lowest level of aggregation (i.e. in-dividuals rather than households) is farfrom straightforward. Until we learnmore about within-household alloca-tions of resources, the household islikely to remain the primary unit ofobservation in most empirical welfarestudies. Note, however, that treatingthe household as the basic observationalunit greatly complicates the problem ofwelfare comparisons because of the het-erogeneity that arises from differencesin composition. This issue is of centralconcern in the choice of the householdwelfare function.

Choosing a Household Welfare Func-tion. In the absence of changes in theaxioms of unlimited domain, indepen-dence of irrelevant alternatives or theweak Pareto principle, the social choiceresults described in the previous sectionsuggest that some form of measurabilityor comparability is necessary to over-come Arrow’s impossibility theorem.Where does this additional informationcome from? Can this requirement bemet by choosing a particular welfarefunction? At first glance, it appears thatthe use of income as an argument of thesocial welfare function, as in (32), re-solves the problem of measuring andcomparing welfare levels. Income iscardinally measurable and is meaning-fully compared across households. It isimportant to remember, though, that inthis context income is to be interpretedas a measure of welfare. Without addi-tional assumptions, it orders statesuniquely up to monotonic transforma-

tions that may differ across households.Any representation of the welfare func-tion, including income, money metricindexes or the indirect utility function,requires assumptions related to theirmeasurability and comparability acrossthe population. The statement that ahousehold with twice the income of an-other is twice as well off is no less nor-mative than a similar statement basedon utility levels. The fact that income ismeasured in familiar monetary unitsdoesn’t diminish the difficulty of the ex-ercise.

Given its necessity, is it possible toinfer interpersonal comparability fromhousehold behavior? The answer isclearly no. We’ve known since the Ordi-nalist Revolution that demand functionsenable us to identify utility functionsthat are unique up to monotonic trans-formations that can differ across house-holds. For many economists, the failureto provide a “scientific basis” for inter-personal comparisons suggests that theyshouldn’t be invoked at all. As Sen(1995) points out, to do so limits thepossibility of meaningful social choiceand returns us to Arrow’s impossibilityresult with its limited scope for policyanalysis.

Social choice theory treats the house-hold welfare function as an abstractioneven though implementation of any so-cial welfare function requires an ex-plicit representation of Wk. Given thevolume of work devoted to the measure-ment of Hicksian surpluses at the microlevel, it is surprising that nominal in-come remains the most popular choiceas a measure of well-being. In the nota-tion introduced in the previous section,the welfare of the kth household is:

Wk = yk. (33)

This choice is implicit in the use of aver-age household income as a measure ofsocial welfare as well as in the majority


of inequality and poverty studies that arebased on the distribution of householdincome.

The assumption implicit in (33) isthat a ten-person household is as welloff as a couple with the same income.Given the disparity in the consumptionrequirements of these two households,this assumption may not be reasonable.To address this concern, a widely usedadjustment is to deflate income by thenumber of persons in the household, sayNk, so that welfare is represented by:

Wk = yk

Nk. (34)

Consumption requirements are assumedto increase linearly with family size,which may be true for some goods butcertainly is not true for household-levelpublic goods such as housing. To accountfor economies of scale in consumption,Buhmann et. al. (1988) suggested aslightly different welfare function:60

Wk = yk

(Nk)η, (35)

where the parameter η determines thescale economies. If it is equal to one,then the household has no economies ofscale (as in the per capita model) while avalue of zero implies complete econo-mies (as in the household model). Thisincome-based welfare function has theadvantage of simplicity but is ad hoc andhas no apparent welfare-theoretic foun-dation. Note also that in accounting forheterogeneity, the effects of other familycharacteristics (such as the age and gen-der of the family members) are ig-nored.61

The widespread use of income as awelfare measure is related to severalpractical considerations. Income data

are readily available in most countriesand, at least in the United States, areprovided at annual frequencies. Its usealso obviates the need to estimate amodel and recover the underlying util-ity or expenditure function. These ad-vantages come at great cost, however,because income-based methods producea number of distortions in both thelevels and distributions of well-being.

One source of bias arises from thefact that judgements are being made inthe absence of information on prices. Ifwelfare is related to the utility function,a social ordering defined over incomewill coincide with another that is basedon the indirect utility function (and,therefore, is a function of prices) only ifhousehold preferences or the form ofthe social welfare function are severelyrestricted.62 Moreover, in a static con-text, the appropriate welfare indicatorshould be a function of total expendi-ture rather than income. The perma-nent income and life cycle hypothesessuggest that (annual) income is likely tobe an inaccurate proxy for total expen-diture because of intertemporal con-sumption smoothing. Households withtemporarily low incomes will maintaintheir standard of living through borrow-ing or dissaving and exhibit high con-sumption to income ratios. The reversewill hold for households with unusuallyhigh incomes. For both types of house-holds, income will typically be an inac-curate reflection of their standard ofliving.

Many of these problems can be re-solved if welfare is represented by amoney-metric utility function such as:

Wk = Mk

Pk(p0, p, Vk, Ak)m0(p0, Vk, Ak) (36)

= M(p0, Vk, AR), 60 See also Fiona Coulter, Frank Cowell and

Stephen Jenkins (1992).61 The importance of accounting for charac-

teristics other than family size has been demon-strated by Banks and Paul Johnson (1994).

62 See Roberts (1980a) and Alan Slivinski(1983).


where Vk = V(p, Mk, Ak) is the indirectutility function and AR is the vector ofattributes of a reference household.These monetary measures of welfarewere first used as arguments of a socialwelfare function by Muellbauer (1974)who estimated the expenditure functioneconometrically using a linear expendi-ture demand system. Subsequent appli-cations of this general approach havebeen used in a variety of contextsby Apps and Elizabeth Savage (1989),Deaton and Muellbauer (1980a), King(1983), and Martin Ravallion and S.Subramanian (1996).

The price index Pk in (36) accountsfor the fact that price changes have dif-ferent effects on different householdsdepending on their expenditure pat-terns and is defined by:

Pk(p0, p, Vk, Ak) = M(p, Vk, Ak)M(p0, Vk, Ak)

.

If the prices of necessities increase rela-tive to luxuries, households that consumelarge proportions of these goods (e.g. thepoor) will experience larger increases inthe cost of living and, all other thingsequal, will attain lower welfare levels.

In contrast with the ad hoc formula-tions used in (33)–(35), equivalencescales in (36) are designed to addressthe welfare theoretic question of theadditional expenditure required tomaintain a given level of well-being ashousehold characteristics change. Re-call from (20) that household equiva-lence scales are defined as the expendi-ture, relative to a reference household,necessary to attain utility level VR:

m0(p, VR, Ak) = M(p, VR, Ak)M(p, VR, AR)

.

If the equivalence scale is evaluated atthe actual utility level Vk, per equivalentexpenditure provides a reasonable basisfor comparing the welfare of households

with different compositions. If a familyof four needs three times the expendi-ture of a single individual to attain thesame utility, then (ignoring the differen-tial effect of prices) we know that it isworse off with $40,000 compared to theindividual’s $20,000 and, moreover,would need another $20,000 to be aswell off.

Rather than account for heterogene-ity through household size alone,equivalence scales depend on additionalcharacteristics that influence demandpatterns such as the age, race and gen-der of the members. It is worth repeat-ing that using equivalence scales (de-rived from household expenditurepatterns) for welfare comparisons re-mains an issue of some controversy dat-ing back to Pollak and Wales (1979).What is lacking from the debate is a dis-cussion of how one might compare well-being and develop aggregate measuresof welfare without equivalence scales.Simply stating impossibility without analternative proposal precludes the ap-plication of welfare economics to keyelements of policy analysis includingpoverty and inequality studies, theanalysis of growth, and the assessmentof the impacts of taxes and transfers.63

The money-metric utility function(36) provides an exact representation ofhouseholds’ preferences. As long as theexpenditure function is estimated usingobserved demands, it is consumption-based, incorporates price effects as wellas preference heterogeneity. As an ar-gument of a social welfare function,however, it suffers from the same prob-lems as the aggregate surplus measuresidentified by Roberts (1980a) andBlackorby and Donaldson (1988a).Money-metric utility functions requirethe (arbitrary) choice of a reference

63 Notable exceptions are the papers by Atkin-son (1992) and Atkinson andFrancois Bourguignon(1987).


price vector and the ranking of socialstates need not be invariant to thischoice. Social welfare functions definedover their distribution need not bequasiconcave which yields an orderingof outcomes with adverse implicationsfor distributive ethics.

An alternative approach proposed byJorgenson and Slesnick (1984) is to usean indirect utility function to representhousehold welfare:64

Wk = ln V(p, Mk, Ak) = αp′ln p + 1 ⁄ 2ln p′Bppln p

− D(p)ln (Mk ⁄ m0(p, Ak)), (37)

where αp and Bpp are unknown parame-ters that are estimated econometrically.As with Muellbauer’s model, this measureof well-being is a function of prices, thelevel of total expenditure, and the demo-graphic characteristics of the household.Household welfare is not converted intomonetary units using the expenditurefunction so the associated problems areavoided. Note, though, that the measur-ability issue does not go away, and it isstill necessary to make assumptions con-cerning the cardinality and comparabilityof this indicator across households.

To this point, I have restricted the fo-cus to social orderings that depend onlyon welfare functions that are them-selves functions of the material well-be-ing of households. Sen (1977) has ar-gued that such a framework is undulyrestrictive because it precludes the useof “non-welfare” information in rankingsocial states. He believes that socialwelfare should depend not only on theutility derived from consumption butalso on characteristics that are not argu-ments of the welfare function. Nonwel-farist orderings of outcomes could bebased on societal attributes such as the

degree of personal liberty, the level ofsocial justice and the ability of individu-als to live in the absence of disease andmalnutrition.

To accommodate this broader defini-tion, Sen (1985, 1987) developed atheoretical framework in which the util-ity derived from consumption (“wel-farism”) is replaced by a function of in-dividuals’ “capabilities” such as anindividual’s life expectancy, healthstatus, personal liberty and the like.Commodities are consumed for thecharacteristics they provide which, inturn, are inputs to the production of ca-pabilities or “functionings” from whichwell-being is derived. Food is consumedto provide the characteristic related tonutrition which gives the capability ofliving in the absence of malnutrition.The medical care characteristic pro-vides the capability of living a long lifeand the transportation characteristicprovides the capability of physical mo-bility. In this context, consumption isbest viewed as a means to an end ratherthan an end itself.65

It is difficult to imagine how this ap-proach might be implemented empiri-cally to provide a comprehensive wel-fare measure. Individuals’ capabilitiesare not always the result of revealedpreference so we have little prospectfor measuring individuals’ valuations ofthese capabilities. For example, it mightbe important to know how they tradeoff material well-being for, say, life ex-pectancy. This can only be answered inan arbitrary way since data provide littleinformation on this issue. Most empiri-cal applications examine each capabilityseparately but make no attempt to ag-gregate the joint effects into an overallmeasure of welfare. Nevertheless, it isdifficult to deny the importance of

64 See also Deaton (1977) and, more recently,Banks, Blundell, and Lewbel (1996) and DavidNewbery (1995).

65 For further elaboration of this point, see Sud-hir Anand and Ravallion (1993).


these issues in the assessment of socialwelfare.

Incorporating Principles of Equity.The axiomatic social choice frameworkoften yields a canonical representationfor the social welfare function. Oncethe household welfare function is cho-sen, the specification is completed bychoosing an explicit functional form forthe social welfare function. As with theissue of the measurability and compara-bility of welfare, the final decision onthe functional form has important ethi-cal implications related to the principleof vertical equity.

The concept of vertical equity isoften identified with the notion that, allother things equal, more egalitarian dis-tributions are preferred to those thatare more dispersed. This condition istypically imposed by requiring the so-cial welfare function to be quasicon-cave, which implies that a convex com-bination of two distributions yields alevel of social welfare that is not lowerthan that attained at the worst of thetwo. The condition of quasiconcavitydoes not narrow the set of feasible so-cial welfare functions sufficiently, andincludes functions as ethically diverseas the utilitarian specification and theRawlsian maximin criterion. There isstill flexibility on the part of the analystto incorporate different levels of con-cern for equity by choosing social wel-fare functions with different degrees ofcurvature.

The weight one places on equity canbe adjusted systematically using a socialwelfare function of the form:

W(u) =

11 − ρ

∑ k = 1

K

Wk1 − ρ. (38)

This representation coincides with thatproposed by Atkinson (1970) whenhousehold welfare is represented by thelevel of income. In addition to the condi-

tions of unlimited domain, independenceof irrelevant alternatives and the weakPareto principle, the social welfare func-tion in (38) requires three additional restric-tions. The social ordering must be invari-ant to the scaling of household welfarefunctions, and it must satisfy the condi-tions of anonymity and separability.66

The sensitivity of the social welfarefunction to disparities in well-being isrelated to the parameter ρ. A positivevalue ensures concavity and higher val-ues imply greater concern for equity. Inthe limit, a value of ρ equal to zeroyields a utilitarian social welfare func-tion and the Rawlsian maximin welfarefunction is obtained as ρ approachesinfinity. The choice of this parameteris arbitrary and reflects the analyst’saversion to inequality in ranking out-comes. In most applications, the socialwelfare function is evaluated at a num-ber of different values to assess the sen-sitivity of policy conclusions to differentlevels of concern for vertical equity.

An alternative method of charac-terizing concern for vertical equity is todescribe the effect of frictionless trans-fers of income or expenditure on themeasured level of social welfare. Ham-mond (1977) described a distributionMk

to be “more equitable” than

Mk′

if:

(i) Mi + Mj = Mi′ + Mj′

(ii) Mk = Mk′ (for k ≠ i, j)

(iii) V(p, Mi′, Ai) > V(p, Mi, Ai) > V(p, Mj, Aj) > V(p, Mj′, Aj).

66 This measurability and comparability assump-tion is referred to as “cardinal ratio scale compara-bility”. Anonymity requires that permutations ofassignments of welfare to households leaves theordering unaffected. As in most contexts, the as-sumption of separability is quite strong. The nor-mative implications in social choice theory is thatindividuals who are indifferent between outcomeshave no influence on the social ordering. SeeRoberts (1980c) for further elaboration of the axi-omatic basis of this social welfare function.


In other words, a distribution is moreequitable than another if it results froma progressive transfer of expenditurefrom one household to another. He de-fined a social welfare function to be “eq-uity regarding” if it is always larger for amore equitable distribution.

Given differences in the compositionof households, it might be desirable toassign different weights to householdswith different characteristics. We may,for example, want to give a couple twicethe influence of a single individual inordering social states. The ultimatechoice, which reflects a personal valuejudgement on the part of the analyst,influences the transfer sensitivity of thesocial welfare function in a significantway. The requirement that the socialwelfare function be equity-regardingcan help determine the final choice ofthe weights and, simultaneously, makeexplicit the normative implications ofthis choice.

To illustrate, assume that the socialchoice axioms imply a general utilitar-ian social welfare function:

W(u) = ∑ak

k = 1

K

Wk.

If the welfare function is taken to be ofthe type (37) and the weights are nor-malized to sum to one, then the require-ment that the social welfare function beequity-regarding implies weights of theform:

ak = m0(p, Ak)

Σm0(p, Ak).

For these representations of individualand social welfare, the condition that thesocial welfare function be equity-regard-ing implies identical weights for house-holds with the same demographic char-acteristics. Moreover, these weightscorrespond to the number of adultequivalent members in each household

rather than the number of persons.Data Issues. Since household welfare

functions are critical inputs to socialwelfare measurement, one needs all ofthe data necessary to estimate thesefunctions. This includes information onspending patterns, the prices paid forgoods and services and the demographiccharacteristics that influence consump-tion patterns. The aggregation of thehousehold welfare functions adds sev-eral additional data requirements.

The most important requirement isthat of comprehensive coverage overthe set of households for which welfareis to be measured. While this seems ob-vious, it is often difficult to achieve be-cause many surveys restrict the samplecoverage. In the early 1980s, for exam-ple, the Consumer Expenditure Surveyslimited the sample to urban households,which precludes its use for an assess-ment of the standard of living and itsdistribution across the entire popula-tion. Surveys in developing countriesfrequently do not include the rural sec-tor, which is a particularly serious prob-lem because this sector often representsa large segment of the population.

The absence of price information inthe surveys creates special problems forthe measurement of social welfare, in-equality and poverty. Regional pricevariation in the U.S. is important formany goods (particularly housing), andignoring its effects could influence so-cial welfare judgements. Most empiricalwork links micro data with nationalprice series on different types of goodsso cross sectional price variation is ig-nored. Access to more disaggregated in-formation on prices will enhance ourability to measure social welfare, al-though it remains to be seen whetherfundamental conclusions concerningdistributional issues will be affected.

Since welfare comparisons are criticalto the calculation of social welfare,


future efforts should focus on the devel-opment of data sources that facilitatethe identification of the (unconditional)effect of demographic variables onhousehold welfare. These data could in-clude information on the intertemporalallocation of consumption, where pro-gress has been made using synthetic co-horts to identify preference parameters.It is not clear whether this approach ishelpful in the current context, sinceaggregation across age cohorts is re-quired, and results in a loss of informa-tion that would identify the welfare ef-fects across households distinguished byother characteristics. Subjective assess-ments of well-being and further infor-mation on the intrahousehold allocationof resources are also likely to provideimportant information in measuringhousehold welfare.

3.4 Applications of Social Welfare Analysis

The aggregation issue in welfare eco-nomics lies at the core of some of themost important problems in appliedeconomics. I present three examples asa means of illustrating the practicalproblems associated with the variousapplications of social welfare analysis.

Policy Evaluation. Following Harber-ger’s (1971) lead, the sum of theequivalent or compensating variations iscommonly used to represent the changein social welfare. While this has the ap-pearance of being a positive measure ofthe change in aggregate welfare, it is noless normative than methods based onexplicit social welfare functions. Thesum of Hicksian variations depends onthe distribution of well-being and theunderlying ethical assumptions areoften ambiguous. As pointed out severaltimes already, problems are also associ-ated with using individual expenditurefunctions as arguments of a social wel-fare function.

Once the social welfare function isspecified, policy evaluation is a rela-tively straightforward exercise. A poten-tial complication arises from the factthat social welfare need not be meas-ured in monetary units and could bedifficult to interpret. Money measuresoccur naturally if the arguments of thesocial welfare function are income orsome other money metric but will notoccur if utility functions such as (37)are used. In this case, social welfare canbe converted into equivalent monetaryunits using Pollak’s (1981) social expen-diture function:

M(p, W) = min M: W(u) = W, ΣMk = M

.

This function is exactly analogous to itsmicro level counterpart and is the mini-mum level of aggregate expenditure re-quired to attain a given social welfarecontour. The social expenditure functionprovides an exact monetary measure ofsocial welfare, so the change in welfarethat results from moving from policy 0 topolicy 1 is:67

∆W = M(p, W1) − M(p, W0).This general approach has been used

in a variety of contexts. Deaton (1977)recovered the indirect utility functionunderlying an estimated linear expendi-ture system and examined the socialwelfare impact of indirect taxes in theUnited Kingdom. King (1983) defined asocial welfare function over the distri-bution of indirect money-metric utilityfunctions and measured the effect ofchanging the tax treatment of owner-occupied housing in the U.K.68 Jorgen-son and Slesnick (1985a,b) developed a

67 Note that while the monetary measure is ordi-nally equivalent to the underlying social welfarefunction, the precise value of the money metricdepends on the choice of reference prices .

68 King recognized the problems with usingmoney metric utility functions as arguments of thesocial welfare function and appropriately re-stricted consideration to a model with homotheticpreferences.


model for analyzing the effects of en-ergy taxes using a social welfare func-tion that was defined over translog indi-rect utility functions and converted tomonetary equivalents using the socialexpenditure function.

In these latter papers, we developeda decomposition of the change in socialwelfare into efficiency and equity com-ponents. To describe it, define Wmax tobe the potential level of welfare that isattained when aggregate expenditure isreallocated so as to eliminate disparitiesbetween households. If the social wel-fare function is equity-regarding, socialwelfare is maximized at this distributionfor a fixed level of aggregate expendi-ture. The monetary equivalent of themaximum level of social welfare isM(p, Wmax) and the change in efficiencyis:

∆EFF = M(p, Wmax1 ) − M(p, Wmax

0 ).

Differences between the maximumlevel of social welfare and that attainedat the existing distribution, [M(p, W) −M(p, Wmax)], is a monetary measure ofthe loss in social welfare due to aninequitable distribution of householdwelfare. A measure of the change inequity can be represented as:

∆EQ = [M(p, W1) − M(p, Wmax1 )]

− [M(p, W0) − M(p,Wmax0 )],

and the change in social welfare is thesum of the change in efficiency and thechange in equity:

∆W = ∆EFF + ∆EQ.

Implementation of this approach re-quires observations on the effects ofpolicies on all segments of the popula-tion. Except in the unusual case inwhich we have a true natural experi-ment, it is not possible to discern theisolated (i.e. ceteris paribus) effects ofpolicies using the data sets that are typi-cally available. As a result, the effects

are simulated and the behavioral re-sponses (and associated welfare effects)are estimated using an econometricmodel. For this type of analysis it is notgenerally possible to use a non-econo-metric index number approach since,in this framework, the responses tochanges in policy cannot be predicted.

The method of simulating these poli-cies, therefore, plays an important rolein the calculations. For ease of imple-mentation this is most often done in apartial equilibrium framework wherethe broader consequences of a policychange are ignored. To examine theeffects of changes in indirect taxes,for example, complete pass-through isoften assumed and the effect is to in-crease the fixed producer prices by theamount of the tax. Over the last twodecades, computable general equilib-rium models have grown in popularityas a means of simulating alternativepolicies.69 While these models are use-ful in assessing how prices and levels ofincome and consumption change, mostare not sufficiently detailed to modelendogenous changes in the distribution.These are undoubtedly important inmeasuring the social welfare effects ofmany of the policies of interest.70

After all is said and done, our confi-dence in policy recommendations ulti-mately depends on the robustness ofour conclusions. Since equity concernsare based on subjective judgements, theeffects are often evaluated for socialwelfare functions that exhibit different

69 Early examples of computable general equi-librium models used for this purpose were pro-vided by John Shoven and John Whalley (1972)and Edward Hudson and Dale Jorgenson (1974).More recent applications include Auerbach andLaurence Kotlikoff (1987), Jorgenson and Kun-Young Yun (1990) and Fullerton and Rogers(1993) among many others.

70 Fullerton and Rogers (1993) provide the onlycomputable general equilibrium model that I amaware of that models endogenous changes to thepersonal distribution of income.


sensitivity to changes in the distri-bution. This is typically done by evalu-ating social welfare for different valuesof a curvature parameter such as ρ in(38). Less common is an assessment ofthe role of the econometric model inthe final conclusions. The level of socialwelfare is determined using statisticalestimates of the household welfarefunction which, of course, have sam-pling distributions. Given a sufficientlylarge sample of households, the “deltamethod” could be used to estimate as-ymptotic standard errors for the changein social welfare. What is often over-looked, though, is that the general equi-librium simulations are themselvesbased on econometric estimates and theprojections of outcomes of policies willlikewise have sampling distributions whichwill affect the precision of the results.

A number of recent efforts have fo-cused on reforms that are only smallperturbations of the policy instruments.The justification for this narrower focusis that reforms are restricted (due to,for example, political considerations) sothat large changes in, say, tax variablesare infeasible. This has become knownas the “tax reform” literature and hasbeen used to describe directions of taxchanges that improve social welfare.Ehtishan Ahmad and Nicholas Stern(1984) considered the reform of indi-rect taxes in India and measured thechange in social welfare that resultsfrom tax-induced changes in prices.71

More precisely, let social welfare bedefined by W = W(W1, W2, …, WK) whereWk = V(p, Mk, Ak). Using Roy’s Identity,the gradient of the change in social wel-fare resulting from price changes is:

∂W∂p

= − ∑ k = 1

K

λk xk , (39)

where λk is the social marginal utility ofincome for the kth household:

λk = ∂W∂Vk

∂Vk

∂Mk.

It appears that for small changes inprices, the welfare effects can be ap-proximated by (39) without estimatingdemand functions econometrically.Only observed quantities are requiredalong with a specification of the rele-vant distributive ethics embodied ineach household’s social marginal utilityof income. In fact, this initial impres-sion is illusory. The social marginal util-ity of income will generally be a func-tion of preference parameters whichmust be estimated. As Banks, Blundelland Lewbel (1996) point out, the as-sumption that the λk depend only onincome imposes the condition of priceindependence, which is empiricallyuntenable. They show, in addition, thatthe first order approximations providedby (39) can be quite inaccurate for theprice changes that are typically of inter-est and that inclusion of higher orderterms requires demand system estima-tion.

The Social Cost of Living. Estimatesof the cost of living are among the mostimportant statistics produced by thefederal government. They are used toadjust wages, Social Security benefits,pensions and government transfers inan effort to preserve recipients’ stan-dard of living. Tax brackets are indexedto avoid what is commonly known as“bracket creep”. Relatively minor biasesin the price index used for these ad-justments can have enormous impactson fiscal policy. The key question iswhich index should be used for thispurpose?

Over the years, the CPI has beenused extensively and virtually exclu-sively as the government’s estimate ofthe cost of living, despite increasing

71 See also the applications by Newbery (1995)and Joram Mayshar and Shlomo Yitzhaki (1995).


evidence of systematic biases.72 A panelof experts convened to study the CPIconcluded that the inflation bias is ap-proximately 1.1 percent per year.73 This is in addition to permanent errorsthat were previously introduced in theestimate of the price level that can beattributed to the treatment of owner-occupied housing. One source of errorarises from the fact that the CPI is aLaspeyres index that uses fixed weightsin averaging the inflation rates of theindividual goods, so substitution is nottaken into account as relative priceschange. Attention has also focused onthe biases that result from the treat-ment of quality changes, the introduc-tion of new goods, the treatment of ele-mentary goods and statistical problemsrelated to the sample rotation.74

These problems are of a mechanicalnature and are largely unrelated tomore serious conceptual issues relatedto the measurement of the cost of liv-ing. In most applications, the central is-sue is the amount nominal wages, bene-fits, or expenditures need to change tomaintain a constant level of well-being.This is a basic welfare economic ques-tion that the CPI is ill-equipped to an-swer. In order for the CPI to be inter-preted as a cost-of-living index, onemust make the assumption that aggre-gate expenditure patterns are consistentwith a representative consumer whosepreferences are of the Leontief fixedcoefficient form.75

A sound conceptual framework forcost-of-living measurement was pro-vided by Konus (1939) who defined aprice index at the micro level to be the

relative cost of attaining a utility level Vas prices change:

Pk(p1, p0, V, Ak) = M(p1, V, Ak)M(p0, V, Ak)

,

where p1 is the vector of prices in thecurrent period, and p0 is the vector ofprices in the base period. By construc-tion, this index answers the question ofthe ratio by which expenditure mustchange to maintain a household’s well-being.76

While micro level price indexes maybe of independent interest, of primaryconcern for policy analysis is the cost ofliving for a group of households. Themost common approach, and the oneimplicit in the use of national price in-dexes, is to assume that market de-mands are generated by a repre-sentative consumer and the social costof living is estimated using the samemethods that were designed for house-holds. I have already described themany problems associated with the ap-plication of the representative agentmodel to welfare economic problems.What, then, is the best way to measurethe cost of living for groups?77

Prais (1959) proposed a “plutocratic”index as a weighted sum of household-specific price indexes where the weightsare households’ shares of aggregate ex-penditure:

PP(p1, p0, u) = ΣM(p1, Vk, Ak)ΣM(p0, Vk, Ak)

= ΣskPk(p1, p0, Vk, Ak),

and sk = M(p0, Vk, Ak) ⁄ ΣM(p0, Vk, Ak). Thisindex can be interpreted as the relativecost of attaining a utility possibility

72 The BLS produces several consumer price in-dexes. Unless otherwise stated, I am referring tothe CPI-U.

73 See Michael Boskin et. al. (1997).74 An excellent description of the CPI and the

methods used to calculate it is provided by BrentMoulton (1996). See, also, Pollak (forthcoming).

75 See Pollak (1983) for further discussion.

76 Empirical implementation of household costof living indexes are provided by Bert Balk (1990),Jorgenson and Slesnick (1983) and Mary Kokoski(1987).

77 For a more exhaustive discussion of the ag-gregation problem in the context of the cost ofliving, see Pollak (1980, 1981).


frontier under the two sets of prices. Asa measure of the cost of living, it has ad-verse ethical implications because therich are more influential (i.e. have largerweight) than the poor in determining thecost of living.

To soften the normative basis, Praisalso proposed a “democratic” index asthe (unweighted) mean of the house-hold indexes:

PD(p1, p0, u) = (1 ⁄ K)ΣPk(p1, p0, Vk, Ak),

where K is the number of households.While Prais emphasized the normativeimplications of his approach, the welfaretheoretic foundation for his index is, atbest, ambiguous. Neither the plutocraticnor the democratic index answer thequestion of the amount of additional ex-penditure required to maintain a level ofgroup welfare as prices change.

A solution to the conceptual problemof aggregation was provided by Pollak’s(1981) social cost of living index. It isthe ratio of the minimum cost of attain-ing social welfare W at current prices tothat needed to attain the same welfareat base prices:

P(p1, p0, W) = M(p1, W)M(p0, W)

,

where M( ) is a social expenditure func-tion and W is the reference level of socialwelfare. The normative basis of this indexis made explicit through the choice of asocial welfare function. Most important,it can be used to compare well-being acrossgroups of households as prices change. Ifthe index doubles between 1980 and1990, twice the expenditure is requiredto maintain the level of social welfare.

This added feature comes at the costof adding a necessarily arbitrary ele-ment to the measurement of inflationbecause of the requirement of an ex-plicit social welfare function. This sug-gests that aggregate price indexes areinherently normative and they cannot

be expected to be invariant across dif-ferent specifications of the social wel-fare function. Will one obtain radicallydifferent inflation estimates using amaximin social welfare function com-pared to utilitarian specification? Forthe United States, I’ve found that theunderlying social welfare function haslittle influence on the price index (Sles-nick 1991b). Pollak (forthcoming) pointsout, however, that the role of aggrega-tion is likely to be more important (andis certainly more complex) once qualityissues are introduced. That is, differen-tial access to high quality goods will un-doubtedly influence estimates of the av-erage cost of living. Indeed, muchrecent discussion involves the role ofsubstitution biases, the “new goods”problem and the role of quality changeson estimates of the cost of living. Scantlittle attention has been given to the in-teraction of these issues with the aggre-gation question.

In the empirical implementation ofthe aggregate cost of living, debate hascentered on whether an econometric orindex number approach should be used.Government agencies are reticent touse econometric models out of concernthat widely-used inflation estimates willchange when the model is reestimatedor because the estimates are sensitive tomodel specification. Index number ap-proaches, such as Diewert’s superlativeindexes, do not use an econometricmodel (although a functional form as-sumption on preferences is often im-plicit) but still require repeated obser-vations on demands. Since these dataare nonexistent at the household level,this method can only be implemented ifdemands are aggregated over groups ofhouseholds and the averages are used tocalculate the price index. This requiresa representative agent assumption, withall the attendant problems, to provide awelfare theoretic interpretation.


Another practical concern is the levelof aggregation at which the cost-of-liv-ing indexes should be applied. If the in-dex is used to adjust benefits that aretargeted to a subgroup of the popula-tion, it would seem appropriate to use acost of living index that represents theconsumption patterns of that grouprather than a national price index. If so-cial security benefits are to be indexed,a cost of living index specific to the el-derly should be used. As Pollak (forth-coming) points out, the importance ofthis issue is ultimately an empirical is-sue and the evidence is scarce. Boskinand Michael Hurd (1985), Todd Idsonand Cynthia Miller (1997) and Jorgen-son and Slesnick (1998) find that infla-tion rates are similar across demographicgroups while Robert Michael (1979) andRobert Hagemann (1982) report some-what larger differences.

Given the fundamental importance ofcost of living measurement to publicpolicy, it is surprising that a number ofpotentially important issues have gonelargely unnoticed in the empirical lit-erature. The role of labor supply, publicgoods and intertemporal life cycle influ-ences have not been addressed to theextent that they deserve, and all repre-sent areas that would benefit from fu-ture research.

Inequality. Has everybody sharedequally in the rise in the standard of liv-ing or have some segments of the popu-lation been left behind? Should growthbe enhanced at the expense of a moreegalitarian distribution? Is such a trade-off inevitable or is it possible to have anincrease in living standards without arise in inequality? Fundamental ques-tions such as these suggest that distri-butional issues, along with growth, areparticularly influential in the design ofpublic policy.

In quantifying the degree of disper-sion in well-being, the policy analyst

must decide on the representation ofhousehold welfare and the appropriatemeans of comparing distributions. At-kinson (1970) showed that rankingdistributions is straightforward if theLorenz curves do not intersect. If theydo, then no unambiguous ordering ispossible, and we must rely on sum-mary statistics to quantify the levels ofinequality. In this case, researchersmust decide between a number of al-ternatives and we have no guaranteethat judgements concerning levels andtrends of inequality are invariant tothis decision.78 The choice of one indexover another requires subjective judge-ments related to its sensitivity to trans-fers between households. The Gini co-efficient is the ratio of the area betweenthe Lorenz curve and the diagonal tothe total area below the diagonal. It ismost sensitive to changes in the middleof the distribution. The coefficient ofvariation is the ratio of the standard de-viation to the mean, and mean-preserv-ing changes in the distribution have thesame effect no matter where they oc-cur. Theil’s entropy index is most sensi-tive to changes at the lower end of thedistribution.79

The link between inequality indexesand their ethical basis can be made ex-plicit by defining inequality to be theloss in social welfare attributable to dis-parities in welfare across households.This approach was pioneered by Atkin-son (1970) who demonstrated that, for afixed mean, Lorenz dominance of onedistribution over another implies ahigher level of social welfare for a utili-tarian social welfare function regardlessof the form of the household welfare

78 The sensitivity of inequality estimates to theindex used has been pointed out by Frank Levyand Richard Murnane (1992) in their study ofearnings inequality in the U.S.

79 For further discussion of the properties of awide variety of inequality indexes, see Blackorbyand Donaldson (1978) and Cowell (1977).


function.80 Failing such a relationship,a functional form of the household wel-fare function must be specified. As-sume, initially, that the welfare func-tions are identical across householdsand are functions only of income. Atkin-son defined “equally distributed equiva-lent income” (say y∗) as the amount ofincome which, if allocated to everyone,yields the same level of social welfare asthe observed distribution. It is implic-itly represented as:

W(y∗, y∗, …, y∗) = W(y1, y2, …, yK),and the inequality index is defined asone minus the ratio of equally distrib-uted equivalent income to mean in-come:

I(y1, y2, …, yK) = 1 − y∗

y_ .

If household welfare (in this case,represented by income) is the same foreach household, social welfare is maxi-mized (for a fixed level of aggregate in-come), there is no inequality, and theindex is equal to zero. If one householdhas all of the resources and the othersnothing, social welfare takes its lowestpossible value and the index is equal toone. Blackorby and Donaldson (1978)have shown that this general approachto inequality measurement has, as spe-cial cases, virtually all of the commonlyused indexes including the Gini coeffi-cient, Theil’s entropy coefficient andthe coefficient of variation.

While Atkinson’s measure of inequal-ity is not as commonly used as statisticalmeasures of dispersion, it has severaladvantages. The normative basis of theindex, which is part of any measure, ismade explicit. This facilitates an exami-nation of the sensitivity of conclusionsto different assumptions about the mag-

nitude by which inequality changes withmean-preserving transfers. Since the in-dex is based on a social welfare func-tion, the implied policy prescriptionsare built in. For example, the gains thatresult from poverty relief are readilydetermined. As important, the inequal-ity indexes are coherently related to themeasures of social welfare.

Atkinson’s framework can be ex-tended to the case in which the welfarefunction depends on prices, expendi-ture and the demographic characteris-tics of the household. In particular, letthe social welfare function W be de-fined as:

W = W(V(p, M1, Ak), .... , V(p, MK, AK)),

where V( ) is an indirect utility function.If the social welfare function is equity-regarding, it is maximized at the per-fectly egalitarian distribution of welfareand an inequality measure of the Atkin-son type is given by:81

I(p, W, Wmax) = 1 − M(p, W)

M(p, Wmax),

where M( ) is the social expenditurefunction, Wmax is the social welfare at-tained at the perfectly egalitarian distri-bution, and W is the actual level of socialwelfare. As with Atkinson’s original defi-nition, this index lies between zero andone and is equal to zero when welfare isequalized across all households.

In recent empirical studies there isincreasing concern with such aggrega-tion issues, but the choice of the house-hold welfare measure is almost an after-thought. The sheer volume of papersmeasuring income (or earnings) in-equality suggests that a tremendous in-tellectual investment is associated withthe use of annual income as a measure

80 See, also, Serge Kolm (1969) and Sen (1973).Partha Dasgupta, Sen and David Starrett (1973)generalized Atkinson’s (1970) result to include anysocial welfare function that is Schur-concave.

81 This generalization was proposed by Jorgen-son and Slesnick (1984) and subsequently used(Slesnick 1994) to examine trends in inequality inthe United States.


of economic welfare.82 The vast major-ity of these papers estimate inequalitywithin a single period and, as I’veargued repeatedly, income is unlikelyto provide an accurate representationof welfare. The burning question iswhether inequality measures that arefounded on welfare functions with a soundtheoretical foundation give noticeablydifferent depictions of inequality.83

The biases associated with family orhousehold income as a welfare measurearise from the use of income instead ofexpenditure, and the omission of priceand demographic effects in measuringwell-being. An implication of the per-manent income hypothesis is that thedistribution of total expenditure is quitedifferent from the distribution of in-come. Households with low incomes aredisproportionately represented by thosewith temporary reductions in currentincome and will have high ratios of con-sumption to income. Households withhigh income levels are over-representedby those with transitory increases in in-come and will exhibit low ratios of con-sumption to income. All other thingsequal, one would expect less dispersionin the distribution of total expenditurerelative to the income distribution.

In fact, this is typically what is ob-served when the household income dis-tribution is compared to the distri-bution of expenditures. Using data from

the Consumer Expenditure Surveys,I’ve found (Slesnick 1994) that the ex-penditure distribution is more equallydistributed than the income distributionin the U.S. and exhibits substantiallyless movement over time. Susan Mayerand Christopher Jencks (1992) andDavid Johnson and Stephanie Shipp(1997) also find the expenditure distri-bution to be more equally distributedcompared to income but report that thetrend varies with the sample definition,the consumption definition and themethod of quantifying inequality. Re-gardless, with little doubt, the expendi-ture distribution differs significantlyfrom the income distribution and its usebiases inequality estimates in an impor-tant way.

The symmetric treatment of hetero-geneous households, which is commonin income inequality studies, implicitlyassumes that households with the sameincome but different compositions areequally well-off. If household structureinfluences expenditure patterns, needsand welfare, such effects are likely tohave an important impact on both thelevel and trend of inequality. In particu-lar, the demographic changes that havetaken place in the postwar UnitedStates suggest that movements in thefamily income distribution are likely tobe quite different from those based oneven the simplest form of an attribute-specific welfare measure.

The importance of household compo-sition on inequality estimates can beseen by examining differences in house-hold size at different points of the dis-tribution. In the U.S., household expen-ditures and the average size of thehousehold have a strong positive rela-tionship. This suggests that adjustmentsfor differences in household composi-tion will have an important effect onthe estimated level of inequality. De-pending on the degree of economies of

82 Reviews of the voluminous empirical litera-ture are provided by Levy and Murnane (1992) forearnings inequality in the U.S. and PeterGottschalk and Timothy Smeeding (1997) cross-nationally.

83 Of greater interest are intertemporal or life-time inequality estimates. These are few and farbetween although see Kathryn Shaw (1989) andJoel Slemrod (1992) and Fullerton and Rogers(1993). Deaton and Christina Paxson (1994) takean important step in actually linking intertemporalconsumption behavior with observed patterns ofwithin cohort inequality. This is one of the fewpapers that attempt to explain patterns of inequal-ity rather than simply describe them.


scale in consumption, there will be a re-ordering of relative welfare levels, and achange in the estimated level of in-equality. In fact, Coulter, Cowell andJenkins (1992) find a systematic rela-tionship between the scale economiesand the level of inequality in the U.K.The relationship becomes more compli-cated once one accounts for other formsof heterogeneity (Banks and Preston1994). In general, though, ignoring dif-ferences between households will yieldbiased estimates of inequality.84

A measure of welfare based solely onincome also ignores the potential im-pact of prices on the distribution. In-creases in the prices of necessities rela-tive to luxuries will hurt the poorrelatively more than the rich, and willthus increase the dispersion in welfare.The large increases in the relativeprices of energy goods in the 1970scould have had a substantial impact onthe relative welfare levels. This effectwould obviously be missed by looking atthe distribution of household incomealone. Empirically, however, it appearsthat the distributional effects of pricechanges have been quite small.85

4. Conclusions

The influence of Harberger’s (1971)proposal that consumer’s surplus beused as a measure of individual and so-cial welfare is reflected by its wide-spread use in applied welfare econom-

ics. Despite years of criticism, it re-mains the method of choice among non-specialists. It is easy to use, the intui-tion underlying its interpretation as awelfare measure is transparent, and thedata requirements for implementationare minimal.

Since Harberger’s paper, the limita-tions and pitfalls of consumer’s surplushave been demonstrated systematicallyand definitively by Chipman and Mooreamong many others. To avoid these con-ceptual problems, methods have beendeveloped to estimate Hicksian surplusmeasures. These methods have thesame data requirements as consumer’ssurplus and are just as easy to compute.Recent focus has been on the develop-ment of innovative methods of imple-mentation that provide the greatestflexibility in modeling the welfare ef-fects subject to the limitations imposedby the data.

While the conceptual issues of wel-fare measurement have been resolvedat the micro level, there is more ambi-guity as to how to proceed to aggregatethe welfare effects of households. Mostagree that the procedure of assuming arepresentative agent and applying thetechniques designed for households toaggregate data has serious shortcom-ings. Moreover, the work of Blackorbyand Donaldson, Roberts, and othersshows that the use of income or othermoney-metric welfare indicators as ar-guments of a social welfare functionalso has many conceptual problems.

The rejection of these natural exten-sions of welfare economics to the aggre-gate level left a gap as to what to do inthe critical area of measuring welfarefor groups of households. Roberts andSen provided a social choice theoreticframework by shifting the emphasis inthe existing literature from the impossi-bility of the aggregation of welfare out-comes to conditions for possibility.

84 Accounting for differences across householdsaffects both the levels and trends of inequality inthe U.S. The precise magnitudes of the differ-ences depend on the characteristics used to distin-guish households and the equivalence scales. SeeSlesnick (1994).

85 The distributional effects of prices have beenexamined by Muellbauer (1974) and Newbery(1995) for the United Kingdom, Ravallion andSubramanian (1996) for India, and Stoker (1986)and Slesnick (1994) for the United States. Fuller-ton and Rogers (1997) find somewhat larger distri-butional effects when simulating tax policy in acomputable general equilibrium model.


Their descriptions of the conditionsnecessary for the existence of socialwelfare functions facilitates consistentaggregation and a sound foundationfor social policy evaluation, inequalityand poverty measurement, and themeasurement of the cost of living. Al-though the use of this framework isrelatively new to practitioners, continu-ation of the recent research effortswill yield empirical methods of socialwelfare measurement that, like the mi-cro level counter-parts, are based on asolid theoretical footing.

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