the information content of equivalence scales*...richard blundell university of college london,...

Journal of Econometrics 50 (1991) 49-68. North-Holland

The information content of equivalence scales*

Richard Blundell University of College London, London WCIE 6BT, England

Arthur Lewbel Brandeis University, Waltham, MA 02254, USA

It is well known that equivalence scales, defined as ratios of cost functions between demographically different households, cannot be completely identified from demand data alone. This paper derives the exact components of equivalence scales that are identifiable. It is shown that demand equations alone provide no information about equivalence scales in any one price regime, but that if equivalence scales in any one price regime were known, then demand data can identify the unique true equivalence scales in all other price regimes. Sensible identifying assumptions and implications for the appropriate construction and interpretation of equivalence scales are discussed, and some empirical tests and estimates from UK micro-data are provided.

1. Introduction

The analysis of demands for goods by demographically different households identifies preferences conditional on household composition, i.e., conditional preference orderings. In contrast, welfare comparisons between households depend on the joint distribution of preferences over goods and household composition (unconditional preference orderings). Even when the demographic attributes are not subject to choice (e.g., age), it is still the case that welfare comparisons depend on all the effects of demographics on utility, not just the effects that appear in the demands for goods. Therefore,

*This research was supported in part by the ESRC under project BOO230027 and in part by the National Science Foundation through grant SES-8712787. Part of this work was done when Lewbel was visiting the MIT Sloan school. We thank the UK Department of Employment and the Institute for Fiscal Studies for providing data. Comments by Charles Blackorby, Martin Browning, Francois Laisney, Essie Maasoumi, Panos Pashardes, Dan Primont, David Ulph, and Guglielmo Weber are gratefully acknowledged.

0304-4076/91/$03.50 0 1991-Elsevier Science Publishers B.V. (North-Holland)

50 R. Blundell and A. Lewbel, The information content of equivalence scales

demand analysis can only identify a subset of the parameters required for welfare comparisons in general and equivalence scales in particular.’

Our aim in this paper is first to establish precisely what information on equivalence scales (defined as the ratio of cost functions on demographically different households) can be identified from data on demands for goods. We show that demand equations for goods alone provide no information about equivalence scales at any one point in time (i.e., at any given price regime), but demand equations completely determine the way equivalence scales change over time in response to price changes.

We next show that the level of information can be increased if one is willing to assume certain global, partly untestable properties of equivalence scales. For example, suppose that equivalence scales are IB, which stands for independent of the base or reference utility level [see Lewbel (1989a, 1989b) and Blackorby and Donaldson (198811. In this case complete identification of equivalence scales is, in general, possible. While the IB assumption is commonly made in empirical demand work [see, for example, Jorgenson and Slesnick (1983, 1987) and Ray (198311, it is a very strong restriction. Barten scales also imply a unique ‘natural’ specification for equivalence scale structure.

Although complete identification of equivalence scales requires prior assumptions on utility function structure, these assumptions (like IB or Barten scales) typically place testable restrictions on the way demographic variables enter empirical demand systems. For example, it is possible to empirically reject IB; however, if the demand restrictions implied by IB are not empirically rejected, it remains impossible to infer that IB actually holds.

These results are starkly illustrated by the case of demographic variables entering demands solely by scaling income (or total expenditure). In this case price changes have no effect on equivalence scales, so demand equations alone provide no information at all about the values of equivalence scales. However, the demand data implications of IB are not violated by this specification, and if IB is therefore assumed to hold, then the true equivalence scales are just completely identified Engel scales.

The issue of equivalence scale identification discussed above does not depend on whether demographic or composition variables are exogenous in the demand equations for goods. Equivalence scales are not completely identified from demand data because demands only reveal conditional preference orderings, regardless of whether households choose demands and demographic attributes simultaneously, sequentially, or independently (it is also true that simultaneous choice does not rule out statistical exogeneity). The determination of household composition is irrelevant to these identifica-

‘Pollak and Wales (1979) were among the first to recognise this point, but discussions are also to be found in Deaton and Muellbauer (1980) and more recently in Fisher (1987) and Blackorby and Donaldson (1988).

R. Blundell and A. Lewbel, The information content of equivalence scales 51

tion findings. Indeed, the identification results apply to the measurement of constant utility cost differences associated with any household-specific char- acteristic, including, for example, the valuation of health disabilities or of public goods like environmental amenities.

We begin in section 2 by proving the basic result that only changes in equivalence scales are identified from demand data. In section 3 we consider a variety of options for estimating equivalence scales given this fundamental identification problem. In section 4 we draw on some empirical results using repeated cross-section data from the UK Family Expenditure Survey for the years 1970-84. Section 5 concludes.

2. The identification of equivalence scales

Let q = D(p, x, z) be the vector of quantities of goods demanded by a household having total expenditures (income for short) X, facing price vector p, and having a vector of demographic characteristics z. Assume Marshallian demands D arise from maximization of a utility function U(q, z>. Define a cost function c(p, U, z) = min,[ p’qlU(q, z) > u]. Given standard assumptions on the form of U, the demands D can be uniquely recovered from the cost function c.~

For any given price vector p, demographic characteristics z, and reference

characteristics z’, we define a schedule of equivalence scales Z(p, z, z”> by

I( P, z, z”> = {c( P, u,z)/c(p,u,zO)lallrealu}. (2.1)

Each element of the set I(p, z, 2“) equals the minimum cost for a household having characteristics z necessary to attain a given utility level u divided by the corresponding cost of attaining utility level u for a reference household

with characteristics z’. The calculation of equivalence scales defined by (2.1) requires recovery of

the cost function c from demands D. However, there is a serious identification problem in doing so [see Pollak and Wales (1979)1, since for any function F([, z) that is increasing monotonic in its first (scalar-valued) element 5, maximization of the utility function F(U(q, z), z) over q will, by ordinality, yield the same demands D(p, x, z) that arise from U(q, z). Hence the cost function E defined as c’(p, U, z> = min,[p’qlF(U(q, z), z) > ul will yield the

*It will be assumed throughout that each household’s attained utility or welfare level is given by the same (unknown) utility function U(q, z). Some researchers object to this assumption [see, e.g., Fisher (1987)], but without interpersonal comparability of some sort the prospect of estimating an equivalence scale is hopeless almost by definition, unless it were possible to observe the demands of a single household over goods as its demographic composition were changed.


same demands D(p, X, z> that arise from c. Let f((p, z, 2“) be the schedule of equivalence scales defined using c’. If F is independent of z, then r’= Z, so no identification problem would arise (this corresponds to a simple renum- bering of the indifference curves in q-z space). However, when F depends on z, i# I, so the true schedule of equivalence scales is econometrically underidentified from demand equations D. This is because demands D represent preferences conditional on z and can only identify the indifference curves in q space, while equivalence scales depend on the indifference curves in q-z space. Changes in F correspond to changes in the shape of indifference curves in q-z space that do not change the q space slices of the curves.

As a result of this underidentification, Pollak and Wales (1979) argue that equivalence scales that are calculated from demand data cannot be used for welfare comparisons. This seems to be an overly negative assessment. De- mand data can be used to recover the cost of attaining each indifference curve in q space, i.e., conditional preferences, and hence contains some information on relative costs. In addition, this argument has no direct relationship to whether or not composition z is jointly determined with q. It simply requires preferences to be defined over q and z. Similarly, the estimation of conditional demands does not require that z be exogenous in the demand system.

The usual approach to equivalence scale estimation is to assume a particular (conditional) cost function c, and to calculate Z. As the above analysis shows, the resulting equivalence scale estimates depend on a subjective value judgment embodied in the implicit arbitrary choice of the function F. Some empirical applications make the selection of F more or less explicit (e.g., Jorgenson and Slesnick in their construction of inequality measures), but the results are no less subjective for that.

The question that needs to be addressed is, what information about equivalence scales can be identified from demand data alone? The following lemma answers this question.

Lemma. Let q = D(p, x, z) be the demands arising from maximization of a utility function U(q, z) under a budget constraint p’q = x. For any cost function ~(p, U, z), let O(p, z”) be the collection of values taken on by I(P, z,z”) for all real z. Let W’ be any collection of positive constants having the same dimension as fl(p, z”). For any demands D(p, x, z>, any positive constants P, and any positive price regime p’, there exists a unique cost function c(p, u, z) such that:

1. the Marshallian demands arising from c( P, U, z) are HP, X, Z>, 2. for all real u, c(p’, u, z”) = u, and 3. O(p”, z”) = no.


Proof. Let Oz, equal the constant that the equivalence scale C(PO, U, z)/c(pO, U, 2”) will be set equal to, so R” = 10~Ul all real z and ~1. Note that fi’$, = c(p”, U, z”)/c(p”, u, z”) = 1 for all U. Let E(p, U, z) be any cost function that yields demands q = D(P, x, z). Let u = h(q, X, zl equal the indirect utility function associated with x = E(p, U, z), so h is the inverse function of E over the element U. Note that cost functions are monotonically increasing in U, so indirect utility functions are uniquely defined and are monotonically increasing in x. For all p, U, and z, define c(p, u, Z) by c(p,u, z) = c’[h(p”, OzUu, z),p, z]. Since i is the inverse function of 5, it follows that c(p“,u, z) =@,u and c(P’,u, z”) = u, SO c(P“,u, z)/ c(p”, u, z”) = O;, for all real z and U.

Corollary. For any demands D( p, x, z 1 and any positive price regime PO, there exists a unique cost function c(p, u, z) such that:

1. the Marhsallian demands arising from c(p, u, z> are D(p, x, z>, and 2. for all real u and all real z, c(pO, u, z) = u.

Proof. The proof of the lemma shows c(p, u, z) can be constructed from any cost function c’ that yields demands D. The corollary follows from the lemma by taking all the elements of 0“ equal to one.

The main implication of this lemma is that demands for goods provide no information about equivalence scales in a single price regime, i.e., for any observed demand system, a cost function can be found that rationalizes the demand system and yields any possible values for equivalence scales in any one given price regime p”. However, the lemma also shows that this cost function is unique3 so that given the true values of equivalence scales in one price regime, Marshallian demands can be used to uniquely recover the true values of all equivalence scales in all other price regimes.

The (arbitrary) normalization c(p”, u, z’) = u used in the lemma was chosen because it is the unique money metric, i.e., for households z” in price regime p”, utility equals total expenditure. In the corollary 0“ is chosen to give all demographic groups a money metric utility in the base price regime.

The lemma shows that the standard practice of assuming a particular cardinalization E to calculate equivalence scales is identical to assuming an exact value of the equivalence scales 0“ (equal to R[p”, z“]) in one price regime, then estimating the implied values of the equivalence scales in all others. Another implication is that if equivalence scales do not vary with

3Up to the normalization c(p, u, z”) = u, that is, which is harmless because it only corresponds to the numbering of indifference curves.

54 R Blundell and A. Lewbel, The information content of equivalence scales

prices (i.e., Engels scales), then observed demands provide no information at all about true equivalence scales.

For any price regime p, any reference price regime p”, and any demographic characteristics z, define a schedule of (Konus) cost of living indices

L(P, PO, z) by

L(P, PO, 2) = (4 P, u,z)/c(p”,u,z)lallreal u}; (2.2)

see Afriat (1977). Each element of the set L(p, pa, z) equals the cost for a household having characteristics z and facing prices p to attain a given base utility level u, divided by the cost of the same household to attain the same utility level u when facing prices p”. Unlike equivalence scales, the schedule of cost of living indices L(p, p”, z) are identical for all choices of the cost function c that yield observed demand equations D, and so are uniquely determined by the demand data D alone.

Consider a specific equivalence scale given by

C(PYU,Z) C(P?~>Z)/C(PO,U,Z) C(PO,U,Z)

c( P, u, z”) = c(P,~,z”)/c(Po,u,zo) c(pO,u,zO)

L(P,PO,Z) C(PO,U,Z)

= L(P,PO,ZO) C(PO,U,ZO). (2.3)

It follows from this equation that the equivalence scale in price regime p equals the product of a ratio of household-specific cost of living indices [elements of L(p, p”, z) and L(p, pa, z”)], which are identified from demand data alone times the corresponding equivalence scale in the base price regime p”, which by the lemma is unidentified. This simple decomposition shows that any equivalence scale that researchers report based on demand estimation is the product of relative cost of living indices uncovered by the data, multiplied by an arbitrary constant that the researcher has implicitly selected by his choice of F.

3. Options for estimating equivalence scales

For any U(q, z) consistent with demands q = D(p, n, z), and any corre- spondingly chosen transformation F(q, z), let U*(q, z) = F(U(q, z), z), SO U*(q, z) is the chosen form of utility, consistent with demands q = D(p, x, z). Let C*(I), u, z) be the form of the cost function associated with U*(q, z).

Given the results of the previous section, there appear to be only three options for estimating equivalence scales. These are 6) report only what can be unambiguously calculated from demand data, i.e., results independent of the choice of F given U, or (ii) make some reasonable, though untestable,

R. B&dell and A. Lewbel, The information content of equivalence scales 55

assumptions to arrive at a unique U*(q, z), i.e., choose a particular F given U, and report the equivalence scales that result, or (iii) augment demand equations q = D(p, x, z) with additional data about preferences over z or psychometric data about attained utility levels to get more information about choice of F given U that corresponds to agents actual welfare. We briefly discuss each of these options in turn.

(i) Testable result only

If we wish to have results that depend only on the observable demand equations q = D(p, x, z) [and the assumption that the welfare of all agents be defined by the same unknown utility function U(q, z>l, then the only component of equivalence scales that can be estimated is ‘relative’ equivalence scales, which are ratios of cost of living indices for different demographic groups [the first term on the right side of eq. (2.3)1. By the lemma, nothing can be inferred about the rest of eq. (2.3) using demand data alone.

A simple technique for constructing relative equivalence scales is to select a base price regime p”, then construct the unique cost function c that rationalizes the observed demands q = D(p, x, z) and satisfies the corollary. This representation of the cost function makes all base period equivalence scales equal one, so for this representation only, equivalence scales calculated in the usual way for any z, p, and u will equal the ratio of cost of living indices for the different households, which by definition are relative equivalence scales. To illustrate, a relative equivalence scale of two with prices p means that if it cost the household s times the reference household’s expenditures to attain the same utility level as the reference household in price regime p”, then in price regime p the household must spend 2s times as much as the reference household. This is admittedly a much weaker result than knowing s, but relative equivalence scales are all that can be identified from the demand equations q = D(p, x, 2).

(ii) Analysis incorporating reasonable utility assumptions

If actual equivalence scale esimtates are required using only demand data q = D(p, x, z), then an untestable choice of U* (more specifically, an arbitrary choice of F given any U consistent with D> must be made. This is not necessarily more offensive than the standard assumptions required for welfare analysis, such as interpersonal comparability, as long as one is explicit about the dependence of the resulting estimates on the untestable choice of F.

If this approach is to be taken, then special structure in the demand equations may suggest a natural choice or range of choices of F given U. Suppose D(p, x, z) = D*(p, x, r(z)) for some vector-valued function r(z)


that has lower dimension than z itself, so observed demands depend on demographics z only through T(Z). For example, T(Z) could equal Barten (1964) style good-specific equivalence scales. In this case it may be sensible to only consider candidate U* functions that depend on z through T(Z). Note that the choice of U* that satisfies the corollary also satisfies this criterion.

In some cases, more specific structure in the demand equations may suggest a unique ‘natural’ choice for U *. For example, suppose the demand equations have the form

q=R(z)D*(R(+‘Iv), (3.1)

for some (nonsingular) matrix-valued function R(z). Demands of this form were proposed by Gorman (1976) and are a special case of the demand transformations considered by Lewbel (1985). Barten commodity-specific equivalence scales are the special case of R(z) being square and diagonal. The class of all utility functions yielding demands of this form is

U(q,z) =G(ff[R(+dz), (3.2)

for some functions G and H, which suggests that the ‘natural’ or ‘simplest’ choice for U* is U*(q, z) = H[R(z)q]. The corresponding choice of cost function is the unique form c(p, U, z) = C(R(z)-‘p, u>. Note that one can test whether demands q have the form of eq. (3.1), but assuming they do, the choice of utility representation U* versus any other choice U consistent with eq. (3.2) is an assumption that, while natural, is still untestable using only demand data for goods.

Another example of demand structures that lead to a single natural choice for U * is the IB property, defined as the situation in which equivalence scales are independent of the base level of utility u at which the cost comparison is being made. Formally, IB means that the set I(.) in eq. (2.1) has only one element. IB can be viewed as a functional form property of the cost function, since equivalence scales are IB if and only if the cost function c equals m(p, z)G( p, u) for some functions m( ) and G( ), yielding equivalence scales of the form m(p, z)/m(p, z”). The IB property places testable restrictions on the (conditional) demand equations D.4 In general, IB also implies a unique choice of equivalence scales (i.e., a unique U*), because any other form of c consistent with D would be of the form c(p, u, z) = m( p, z)G(p, H[u, z]) for some function H, which does not yield IB scales when H[u, zl depends on z, and always yields the same scales m(p, z)/m(p, z”) when H[u, zl does not depend on z.

4As described earlier these are enough to reject IB but, following the lemma, are not sufficient to ‘accept’.


Blackorby and Donaldson (1988) and Lewbel (1989a) independently ana- lyzed the restrictions on demand and cost functions implied by the IB property (Blackorby and Donaldson call the IB property ‘Equivalence Scale Exactness’). Many empirical studies in equivalence scale estimation posit cost functions that possess the IB property [examples are Engel scales, homo- thetic demands, and the models of Jorgenson and Slesnick (1983) and of Ray (1983)]. However, the demographically translated Linear and Quadratic Expenditure Systems of Pollak and Wales (1981) do not necessarily satisfy the IB restriction. Similarly, the extensions of Deaton and Muellbauer’s Almost Ideal model estimated in Blundell, Pashardes, and Weber (1989) do not in general support the IB restriction. We return to an assessment of whether demand data is likely to rule out the IB restriction in our empirical application presented in section 4 below.

An example of choosing a natural U * is Jorgenson and Slesnick (1983). They claim that their Translog specification implies unique ‘Translog’ equivalence scales, which they incorporate into social welfare functions to analyze the distribution of welfare (i.e., utility) in the economy. The results of the lemma in section 2 disprove this ‘uniqueness’ claim in terms of welfare measurement. However, the Translog equivalence scale is unique both in the sense that it corresponds to the only form of U* that gives IB scales in the Translog context, and in the sense of being the ‘natural’ form of U* for Barten scales. The former is also true of the model by Ray (1983).

In all these cases, the choice of U* is ‘natural’ only in the sense of having the least complicated expression of the dependence of utility on z. When nothing else is known about the dependence of the true measure of welfare on z, this simplest form is attractive form a purely descriptive point of view. However, there is no a priori nor any possible demand-data-based reason to believe that this natural choice of U *, when one exists, will more accurately measure comparative welfare than any other choice consistent with observed demands. Nevertheless, if one has prior beliefs about (or alternative estimates of) equivalence scales, the estimates that result from any natural choice of U * can be checked for reasonableness against these priors.

(iii) Use of additional data

It may be possible that additional data in the form of revealed preference for characteristics z (e.g., treating geographic location or household size as a choice variable) can be used to identify equivalence scales in a single price regime, or one may have opinions about equivalence scales arrived at by introspection. The extensive work by Van Praag and his colleagues [see Van Praag (1989), Van Praag and Kapteyn (1973), and Van Praag and Van der Sar (1988), for example] provides an important attempt to use psychometric data in the form of an Income Evaluation Question to overcome the

58 R. Blundeli and A. Lewbel, The information content of equivalence scales

identification problem. In either case, the lemma in section 2 suggests how such information could be combined with estimates of relative equivalence scales to yield estimates of true equivalence scales. Of course, revealed preference over characteristics cannot be used for attributes that are not subject to choice, such as race or age.

4. Empirical results

4.1. A specification for preferences

To provide an empirical setting for the issues raised in the previous two sections we turn to a microeconometric study of consumer behavior in the UK. For this we use a long time series of repeated cross-sections to estimate the impact of demographics, prices, and income on disaggreggated consumer decisions. Naturally there are many other factors which enter the determination of preferences at the household level but for the purposes of this application we will concentrate on these three factors. The estimated models (that include many other factors) are discussed fully in Blundell, Pashardes, and Weber (1989), where the procedure for the selection of the models presented here is discussed in detail. Here we shall pay particular attention to the way demographic variables enter the demand equations.

We begin by considering the class of conditional preferences due to Muellbauer (1976) for which the cost function has the form

lnc(p,u,z) =lna(p,z) +b(p,z)u, (4-l)

where a( p, z) and b( p, z) are demographic or composition-dependent price indices. This form turns out to nest a number of popular demand systems and was found by Blundell, Pashardes, and Weber (1989) to be particularly well suited for describing the behaviour of households in our sample. For example, a demographic generalisation of the Almost Ideal (AI) model of Deaton and Muellbauer (1980b) results from assuming a Translog form for In a(p, z) and a Cobb-Douglas form for b(p, z). In this case the Marshallian share

equations for good i take the form

Wi=ai(Z) + CY,jlnPj+Pi(z)ln(x/a(P,z)), (4.2)

for i = 1,. . . , n. To arrive at (4.2), the translog aggregator In a(p, z) is written as

lna(p,z)=cf,(Z) + ~CXj(Z)lnpj+ 2 iYjklnPjlnPk, (4-V j j k


and b(p, z) has the Cobb-Douglas form

lnb(p,Z) = ~~j(Z)lnp,. (4.4)

If each pi in (4.2) were independent of z, then demographics would enter cost function (4.1) through the index a(p, z) alone. For conditional preferences described by (4.1) this is equivalent to the IB property. The schedule of equivalence scales corresponding to (4.1) has the form

In I( p, 2, z”) = In u( p, z) - In u( p, z”)

+b(PJ) -b(P,Z”)l% (4.5)

and can be seen to be independent of base u when demographics enter through the a(p, z) index alone, or, more precisely, when the price derivatives of b(p, z) are independent of z. However, as the lemma in section 2 showed, using (4.5) as the equivalence scale schedule is equivalent to assuming that the form for u in (4.1) represents the indirect measure of household utility in q-z space. If each pj in (4.4) is independent of z, then without further information (in addition to that available from demand data alone), we cannot reject the IB property. For this reason we pay particular attention to the dependence of pi on z in (4.2) in our empirical application since dependence of pi in z will rule out the IB property in q-z space.

If we restrict (4.2) to the IB class, we can rewrite our share equations as

Wj=CXi(Z) -/3,Ci?o(Z) + f:(ni-f?iCX,(Z))lllpj i

and the log equivalence scale as

(4.6)

lnZ(p,z,z”) =(Y~(z) -(~c(zO) + i(n,(z) --j(zo))lnpj. (4.7) i

At base level prices, that is where pj = 1 for all j, this reduces to a,(z) -

(Y&z?.

60 R. Blundell and A. Len&l, The information content of equivalence scales

In IB restricted models with cost functions of the form (4.1) it is impossible to identify equivalence scales uniquely [see Blackorby and Donaldson (198911. In particular, for the IB restricted model (4.6) it is not possible to identify the parameters of a,(z) [see Pashardes (198911. If we normalize a,(z) to zero, then (4.5) can only measure the deviation of the overall cost ratio from its value in the base period. However, if there exists a commodity for which one can safely assume ai to be zero (alcohol, for example), then the intercept in the share eq. (4.6) can be used to identify (Y,(Z) directly. For such a good, z only influences the share through the ln(x/a(p, z>) term in (4.2). This is the so-called ‘composition separability’ case which Deaton et al. (1989) use to define adult goods.5

As we have shown in the lemma, (4.5) only represents a true equivalence scale measure if u is the correct measure of household utility in q-z space. Our estimation of conditional preference cannot reveal anything about that issue. However, if the price derivatives of the b( 1 index [and therefore p,(z) in (4.2)] are independent of z, the data cannot reject the IB property. If, in addition, IB can be assumed for the utility measure in q-z space, then the estimated equivalence scale is given by In a(p, z) - In a(p, z”>.

This latter approach is equivalent to the second option for estimating equivalence scales discussed in section 3 above. As noted there, demand data does provide sufficient information to reject IB (and a number of other plausible properties including Barten scaling) but does not provide sufficient information to check such a restriction in q-z space. In general, by imposing IB in q-z space a unique equivalence scale is generated. ‘However, we shall usually want to report only what is precisely recoverable from demand data. For this we shall present the elements of the relative equivalence scale described by the first ratio on the right-hand side of (2.3).

4.2. The estimated preference parameters and the IB restriction

In table 1 we present the parameter estimates that relate to our preferred specification of the Almost Ideal model in which the IB property is not imposed. The summary statistics for the repeated cross-section data used in this study are presented in the appendix. A seven-equation nondurables and services model is specified for food (fo), alcohol (aE), energy (en>, clothing (cl), transport (tr), and services (su). Table 1 reports results in which homogeneity and symmetry are imposed. As a result parameters for a

%uch a condition is necessary to identify a(~,) in the AI model, but for more general preferences other identifying restrictions are available [see Blackorby and Donaldson (198911. For example, if higher-order terms in In(x/a(p, z)) are required in the share equation, then all these terms in a(p, z) can be identified through the restriction that a(p, z) is the same deflator in all higher-order terms.

R. Blundell and A. Lewbel, The information content of equicalence scales 61

Table 1

Price. income, and composition effects in an Almost Ideal model.”

Food Alcohol Energy Clothing Transport

In p( fo)

In p(a0

In p(en)

In p(cl)

In p(W)

In P(W)

0.0936 0.0257 - 0.0245 - 0.0062 - 0.0426 - 0.0012 (0.011) (0.007) ( - 0.006) (0.009) (0.01) (0.009)

- 0.0766 0.0604 - 0.0054 0.0256 0.0028 (0.009) (0.006) (0.007) (0.01) (0.009)

0.0175 0.0021 - 0.0584 - 0.0154 (0.007) (0.006) (0.011) (0.0069)

0.0114 0.0040 - 0.025 1 (0.011) (0.01) (0.008)

0.061 0.0097 (0.03) (0.015)

0.0288 (0.0139)

In x -0.1221 0.0507 - 0.0790 0.0439 0.0546 0.0590 (0.007) (0.005) (0.004) (0.0076) (0.12) (0.007)

K In x 0.0093 - 0.0076 - 0.0016 0.0118 - 0.0239 0.0024 (0.004) (0.003) (0.002) (0.004) (0.006) (0.004)

K, 0.0177 - 0.0042 0.0081 - 0.0012 - 0.0108 - 0.0086 (0.001) (0.001) (0.0001) (0.001) (0.002) (0.001)

K, 0.0204 - 0.0068 0.0045 - 0.0033 - 0.0055 - 0.0069 (0.001) (0.001) (0.001) (0.001) (0.002) to.001)

K, 0.0265 - 0.0075 0.0032 0.0004 - 0.0096 - 0.0090 (0.001) (0.001) (0.001) (0.001) (0.002) (0.00 1)

K, 0.0289 - 0.0079 0.0034 0.0042 - 0.0116 - 0.01 I9 (0.001) (0.001) (0.001) (0.001) (0.002) (0.00 I)

Homo- geneity 0.764 - 1.727 0.656 - 1.617 0.264 0.968

“GMM estimates. In x and K In x terms are instrumented [see Blundell, Pashardes, and Weber (1989)]. Asymptotic standard errors in parentheses. Homogeneity is N(O, 1) under the null.

Services

six-equation system are given, the omitted demand being the ‘other goods’ category.

The upper triangular portion of table 1 provides the (symmetric) estimates of the yij price parameters on the In p, terms in each share eq. (4.2). The In x and K In x terms refer to log real expenditure, and log real expenditure interacted with the total number of children K. The K,, K,, K,, and K, terms refer to the way children of increasing age affect the level of each expenditure share (precise data definitions are provided in the appendix). As noted in the notes to table 2 all In x terms are treated as endogenous, and


Table 2

The IB restricted Almost Ideal model.”

Food Alcohol Energy Clothing

In PC fo)

In .&al)

In Aen)

In p(cl)

0.0943 0.0249 - 0.0249 (0.0111) (0.0076) (0.0061)

-0.0717 0.0617 (0.0119) (0.0077)

0.0102 (0.0079)

Inp(tr)

In p(su)

In x -0.1235 0.0528 - 0.0784 (0.0072) (0.0055) (1.0041)

Kl 0.0182 - 0.0046 0.0080 (0.0012) (0.0009) (0.0006)

K2 0.0206 - 0.0070 0.0044 (0.0012) (0.0009) (0.0006)

K, 0.0268 - 0.0078 0.0031 (0.0010) (0.0007) (0.0006)

K4 0.0291 - 0.0081 0.0033 (0.0013) (0.0009) (0.0007I

IB testb 5.263 6.160 0.506

“See footnote to table 1. ‘Wald test of IB property: x:, under IB.

- 0.0042 (0.0089)

- 0.0063 (0.0076)

0.0019 (0.0060)

0.0134 (0.109)

- 0.0441 (0.0142)

0.0314 (0.0140)

- 0.0571 (0.0111)

0.0002 (0.0142)

0.0599 (0.0337)

0.0421 0.0584 (0.0077) (0.117)

- 0.006 -0.0121 (0.0012) (0.0019)

- 0.0031 - 0.0058 (0.0012) (0.0018)

0.0003 - 0.0103 (0.0011) (0.0011)

0.0046 - 0.0122 (0.0013) (0.0021)

7.524 13.742

Transport Services

- 0.0016 (0.0093)

- 0.0023 (0.0093)

- 0.0168 (0.0077)

- 0.0251 (0.0092)

0.0055 (0.0176)

0.0349 (0.157)

0.0571 (0.008)

- 0.0084 (0.0012)

- 0.0067 (0.0010)

- 0.0088 (0.0011)

- 0.0012 (0.0013)

0.326

household income, other prices, and interest rates are used as instruments in a full-system Generalised Methods of Moments regression. As can be seen from the parameters in the K In x row, the IB property for conditional preferences is rejected. To be independent of base would require the /?, parameters to be independent of K.6

The intercept terms in each share equation are allowed to depend on demographics (as well as seasonals, regional dummies, etc.) and the parameters on the four ‘kids’ variables, K,, . . . , K,, are presented in table 1. These show a strong and varying impact as expected and although it was found that the different Ki variables could be grouped for the purposes of the K In x interaction terms this is not so for the intercept effects in the system of share

61n the estimated model seasonal dummies were included in the p,( 1 function but these are not reported here for presentational reasons. See Blundell, Pashardes, and Weber (1989).


equations. In table 2 we present results under the IB restriction for this Almost Ideal specification. As we note above the IB restriction is strongly rejected; the Wald test of this restriction is given in the row at the foot of table 2. It is useful to assess the impact of IB on the parameters of interest. For example, the change in the Ki parameters themselves is quite dramatic. However, it is as important to evaluate the impact of imposing IB on the estimated relative equivalence scales. We turn our attention to these issues in the next section.

4.3. The estimated relative equivalence scales

The component of equivalence scales that can be identified from demand data alone are relative equivalence scales, defined as the ratio of L(p, p”, z) to L(p,p’, z”) [see eq. (2.311. The denominator in this ratio is the cost of living index for our reference household evaluated at (indirect) utility level u. The relative equivalence scale therefore measures the deviation in this reference cost of living index for a particular household type. This ratio is unity at base period prices. In table 3a we present estimates of relative equivalence scales for four household types in periods 1975 (January) and 1980 (January). The base period is January 1970. The reference household is a married couple with no children and of average age and income (expenditure). The characteristics K, to K, once again refer to children of increasing age (see the appendix). Each scale relates to the addition of one child in this

Table 3a

Relative equivalence scales.a

1975

K, 1.0019 [0.001941 K2 1.0033 [0.00329] K3 1.0040 [0.003931 K, 1.0042 [0.00418]

“In L(p, p”, K,) in brackets.

1980

1.0025 [0.00250] 1.0038 [0.00380] 1.0039 [0.00389] 1.0035 [0.003511

Table 3b

Relative equivalence scales under IB.

1975 1980

KI 1.0013 1.0022 K2 1.0031 1.0034 K3 1.0036 1.0033 K, 1.0039 1.0037


Table 4

Component price indices (January 1970 = l).a

Food Alcohol Fuel Clothing Transport Services Other

LCP, P”, zO)

“In P, in brackets.

1975

1.90 [0.642] 3.94 Il.3711 1.39 [0.329] 2.88 [1.058] 1.62 [0.482] 3.60 [1.281] 1.64 [0.495] 2.92 [I.0721 1.82 [0.599] 3.75 [1.322] 1.67 [OS131 3.56 [1.270] 1.67 [0.513] 3.46 [1.241]

1.698 3.617

1980

Table 5

A base period equivalence scale.

AI

K, 2.181 K2 2.287

K, 2.327

K4 2.360

“Married couple with no children = 2. Evah- ated at base period (1970) prices.

age group. Table 3a shows that over the period relative prices have changed so as to increase the costs for families with children and generally more so for older children. The component prices and the cost of living index for the reference household is provided in table 4. Table 3b presents the same scales evaluated using the IB-restricted parameter estimates of table 2. Even though these are evaluated for the reference household on average income there is a noticeable difference.

In table 5 an attempt is made to estimate a plausible base period value for the second term in the equivalence scale decomposition described in eq. (2.3). This is calculated using the IB-restricted Almost Ideal model and assuming alcohol to be an ‘adult good’. Following the discussion of section 4.1, this latter assumption is sufficient to identify the base period equivalence scale from (4.5). This assumes that the (indirect) preferences represented by u in (4.1) describe preferences over 4-z space with b(.) independent of z.

Basing the scale at 2 for a married couple with no children, we find that costs increase with the age of a child and that, for example, the cost of an older child is 0.36 of an adult.


Throughout this empirical analysis we have assumed demographic variables to be exogenous to the determination of commodity demands. If we treat decisions over compositions as resulting from the same underlying utility-maximising model that generates consumption decisions, then the possibility of simultaneity is obvious. However, decisions over composition appear to have two characteristics by which they stand apart from day-to-day consumer decisions. Firstly, these decisions are irreversible. Secondly, they are likely to be made at a sequentially earlier stage in the decision-making process. Together these suggest a rather complex structural model, but they also suggest that demographic variables could be treated as exogenous for the estimation of current-period decisions. For most purposes therefore exogeneity can be assumed and where in question can be tested by standard residual addition techniques.7

5. Summary and conclusions

This paper derived the exact components of equivalence scales that can be completely identified from demand data alone. We have shown that conditional demand equations alone provide no information about the equivalence scales in any one price regime, but that if equivalence scales in any price regime were known, then empirical demand analysis can be used to estimate the true equivalence scales in all other price regimes. An alternative way to express this result is that demand equations can be used to construct distinct cost of living indices for households of any given composition, but demand equations alone provide no information about the relative cost of living of changing household composition in any selected reference price regime.

There are three possible responses to this fundamental underidentification. One is to combine demand data with other types of data (e.g., psychometric data) to estimate equivalence scales. An alternative is to only report the component of equivalence scales that is identified from demand data, which is cost of living indices for each household type. The third possibility is to make plausible identifying assumptions concerning the properties of equivalence scales, such as independence of base utility (IB), or the assumption that unconditional preference orderings depend on demographics only through Barten scales. These identifying assumptions have testable implications for demand data. Rejection of these testable implications means rejection of the proposed equivalence scale assumption, but failure to reject can never guarantee that the proposed assumptions hold.

‘See, for example, Smith and Blundell (1986). Difficulties only arise when the demand in question is censored as in the corner solution model of zeros in demand or the selectivity model of labour supply. In these cases an explicit reduced form cannot necessarily be derived and a coherency condition is required for identification [see Gourieroux et al. (1980) and Heckman (1978)].

66 R Blundell and A. Lewbel, The information content of equivalence scales

Using a time series of repeated cross-sections we estimate an empirical demand model which contains a reasonably general specification of demographic effects. The specification is a demographic generalisation of the Almost Ideal model of Deaton and Muellbauer in which both the intercept and slope parameters of the share equations are allowed to be demographically dependent. We consider the restriction that would be consistent with equivalence scales that are IB. This IB restriction is rejected. We then go on to estimate the various components of equivalence scales that are identified from demand data alone and assess the sensitivity of the estimated scales to the IB restriction.

Given our theoretical results, we argue that an appropriate procedure is to first estimate demands, then calculate and report cost of living indices for each demographic group, or equivalently report relative equivalence scales, defined as the ratio of cost of living indices for different groups (see table 3). This is the only component of equivalence scales that can be derived from demand data alone. If more information is desired, the researcher should then explicitly state what identifying assumption about utility structure is being made to construct equivalence scales, remembering that such assumptions can never be completely tested with demand data. Moreover, the implications of such assumptions that are testable should be tested. In our illustration we report equivalence scales based on imposing IB (table 5), and test the implications of IB that are testable in table 2. The testable implications of IB compare demands arising from cost functions of the form c(p, u, z) = m(p, z)G(p, H[u, z]) against more general alternatives, while the untestable implication of IB is the assumption that H[u, zl = u.

Instead of following the above procedure, it is standard practice in demand analysis to propose a functional form for utility, estimate parameters from demand data, and report the resulting equivalence scales implied by the model. Our results show that this standard practice is inherently dishonest or at least uninformative, since in a given price regime, any value of equivalence scales can be rationalized by any demand system.

Appendix

The family expenditure survey data and definitions

The data used refer to a pooled cross-section of 15 annual surveys drawn from the UK FES for the years 1970-1984. The demand system refers to seven broad commodity groups: food, alcohol, energy, clothing, transport, services, and other.

A sample is chosen such that the head of household has an age in the range 18-59 and is not in self-employment. The total sample size is 64,271. All prices are log price indices drawn from the component elements of the


Retail Price Index. In estimation the price indices are based at log(P1956) = 1 for each good. The composition dummies are defined as follows.8

Mean SE Min. Max.

Children aged O-2 KI 0.1838 0.4408 0 Children aged 3-5 K2 0.1898 0.4497 0 Children aged 6-10 K, 0.3305 0.6445 0 Children of age ll+ K, 0.3837 0.695 1 0

The expenditure shares are gicen by:

Share of food 0.3479 0.1258 Share of alcohol 0.0667 0.0778 Share of energy

(domestic fuel) 0.0852 0.0621 Share of clothing 0.1015 0.1029 Share of other goods 0.1035 0.0711 Share of services 0.1174 0.1049

Total number of children K=K,+K,+K,+K,

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the information content of equivalence scales*...richard blundell university of college london,...

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