UNIQUE EQUIVALENCE SCALES

ESTIMATION AND IMPLICATIONS FOR

DISTRIBUTIONAL ANALYSIS

David S. Johnson

US Bureau of Labor Statistics

and Thesia I. Garner

US Bureau of Labor Statistics and National Academy of

Sciences, Czech Republic

Equivalence scales are used to adjust income by family size to obtain income 

distribution measures. Recently, the concept of equivalence scale elasticity has been 

introduced to characterise the effect that scales have on distribution measures. We 

produce utility-based equivalence scales that have the property of constant elasticity. 

By assuming a particular functional form for the scales and that the scales are 

independent of the base level of utility, we obtain unique equivalence scales. In 

contrast to previous estimates of utility-based scales, we do not restrict our sample to 

particular family types. We determine price-dependent scales by estimating a 

characteristic-dependent almost ideal demand system using quarterly expenditure data 

from the US Consumer Expenditure Survey and price indices from the US Consumer 

Price Index. We use our scales and those implicit in the US official poverty thresholds 

to adjust expenditures and show that these scales have similar effects on inequality 

measures.

1. INTRODUCTION

In the public policy literature there has been an increased interest in the

estimation of equivalence scales. By equating measures of economic

welfare across households with different characteristics (such as family

Correspondence to: David S. Johnson, Division of Price and Index Number Research, US Bureau 

of Labor Statistics, Postal Square Building, Room 3105, 2 Massachusetts Avenue, NE, 

Washington, DC 20212.

Journal of Income Distribution, 4(2), 215234 Copyright ®1995 JAI Press Ltd

ISSN: 0926-6437 All rights of reproduction in any form reserved
size), equivalence scales are used to adjust poverty thresholds by family size. 

Equivalence scales are also used to adjust household income figures to obtain 

income distribution measures.1 Since different equivalence scales can yield 

different measures of poverty or inequality, different equivalence scales should 

be used depending on the policy being analysed. The key issues involved in 

selecting an equivalence scale include selecting an appropriate measure of 

economic welfare and choosing the household characteristics, the functional 

form of the scales and the data set to use to estimate the scales.

In this paper, we calculate a utility-based equivalence scale, which provides 

the adjustment that must be made to equate households’ actual welfare as 

determined by a utility function (see van der Gaag and Smolensky 1982, 

Jorgenson and Slesnick 1987, Betson 1990 and Nelson 1993). We determine the 

equivalence scales by using the cost functions obtained from estimating an 

almost ideal demand system. Using the results of Blackorby and Donaldson 

(1991) and Lewbel (1989), we restrict the class of equivalence scales to those 

that are independent of the reference level of utility. We obtain utility-based 

scales such that there are no other equivalence scales, which satisfy our 

assumptions, that are consistent with the demand data. Hence, our assumptions 

imply that our scales are unique in our particular class of equivalence scales.

For simplicity, we assume that our equivalence scale is given by the product of 

family size raised to some power and a function of prices and family size 

(number of adults and children). For this reason, our scale has the property of 

constant elasticity, i.e. the percentage change in the equivalence scale due to a 

percentage change in the family size remains constant for all family sizes.

This concept of elasticity with respect to family size suggests that equivalence 

scales can be viewed as measures of the economies of scale (in the production of 

economic welfare) within the household.2 To determine the effect of these 

economies of scale on income distribution measures, we use our scales to adjust 

total expenditures (which we use as a proxy for income) by family size and 

evaluate the distribution of expenditures. For comparison, we also adjust 

expenditures by equivalence scales which have different elasticities and find that 

the effect on various inequality measures is minimal. This is due to the U-shape 

relationship between an inequality measure and elasticity (see Coulter, Cowell 

and Jenkins 1992).

The characteristics and expenditures data used are from the US Consumer 

Expenditure Survey for 1987-1991. The price data are price indices from the US 

Consumer Price Index series for the same time period. The sample upon which

we base our analysis is not restricted by household characteristics. (Our sample 

is only restricted to households with positive total expenditures and income, 

non-negative health expenditures, and
Unique Equivalence Scales 217

which are complete income reporters).3 This is in contrast to the approach taken 

by previous researchers who, for example, restricted their samples to households 

with persons less than a certain age or to households with children (see, for 

example, Nelson 1993).

We base our scales on quarterly expenditure data, not annual (or four quarters 

of) data, again in contrast to the approach followed by previous researchers (see, 

for example, van der Gaag and Smolensky 1982; Jorgenson and Slesnick 1987 

and Nelson 1993). Characteristics of the sample from quarterly data are more like 

those for the total population than are the characteristics of samples from annual 

data. For this reason, equivalence scales based on our quarterly data should be 

more useful than other equivalence scales when adjusting incomes and 

expenditures for distributional analysis. While our scales are similar to the scales 

implicit in the US official poverty thresholds, our scales are steeper than other 

scales obtained using similar utility-based methods (see van der Gaag and 

Smolensky 1982, Betson 1990 and Nelson 1993).

The second section of this paper describes utility-based equivalence scales and 

presents the almost ideal demand system models that we use to estimate our 

equivalence scales. The third and fourth sections present the data used in and the 

results of our estimation, respectively. The fifth section presents the effect these 

scales have on inequality measures. The final section concludes the paper and 

provides directions for future research.
2.1 Equivalence Scale Exactness

A problem with utility-based equivalence scales is that they cannot be 

identified by examining demand data alone because different reference 

levels of utility will yield different equivalence scales. Blundell and 

Lewbel (1991) show that demand data can only identify a relative 

equivalence scale, which represents the ratio of cost-of-living indices for 

households with different characteristics. They show that Equation (1) can 

be represented as the product of a relative scale and an absolute scale, 

which determines the characteristic changes at base prices, p0. That, (1) 

can be rewritten as:
Recalling that equivalence scales are given by the ratio of the cost 

functions, (3) implies that the equivalence scale is given by:

Since the utility level does not enter into Equation (4), the use of 

different reference utility levels will not change the equivalence scales. In 

addition, arbitrary scaling of the households' income levels will not 

change the value of the equivalence scales. As a result, estimates of the 

equivalence scales from characteristic-specific demand data will be the 

same for all levels of income and utility.

Condition (4) also shows that prices can affect the scales. In this paper, 

we estimate price-dependent scales and scales that are independent of 

prices, which are called Full Engel scales (see Blackorby and Donaldson 

1991). By allowing price dependence, the scales in Equation (4) can be 

viewed as a generalisation of Engel scales.

2.2 THE ALMOST IDEAL DEMAND SYSTEM

To obtain equivalence scales, we use demand data to estimate 

characteristic-dependent cost functions. We modify the almost ideal 

demand system presented by Deaton and Muellbauer (1980) by allowing 

the cost function to depend on characteristics. This characteristicdependent almost ideal demand system is described by the following cost 

function:
These equivalence scales are similar to those in Blackorby and Donaldson 

(1991), Nelson (1993) and Phipps (1991).

Blackorby and Donaldson (1991) show that with the almost ideal 

demand system different scales that satisfy equivalence scale exactness 

can be consistent with the same demand behavior. They show that there 

are functions of characteristics and prices which when multiplied by the 

scale yield new scales while leaving the demand behavior unchanged. 

Since the first component in Equation (11) is independent of prices and 

the second component is Cobb-Douglas in prices, equivalence scale 

exactness and demand data yield unique equivalence scales.
To check the constant elasticity assumption, we estimate another Full 

Engel scale that allows different family sizes to have different effects on 

the size of the scale. The Full Engel scale without constant elasticity is 

given by:
differently to different family sizes. By excluding these terms, the Full 

Engel scales require family size to affect each share equation in the same 

manner.

The parameter bi in Equation (13) is independent of characteristics, that 

is, family size and income enter the share equation separately and there 

are no family size-income interaction terms. This shows equivalence 

scale exactness, which means that all households will have the same 

coefficients on income in their demand equations.7

3. THE DATA

The expenditure and consumer unit characteristics data are from the US 

Consumer Expenditure Survey Interview portion (CEX) for 1987 quarter 

two through 1991 quarter one, in which time 84187 CEX interviews were 

conducted (USDL 1991b). We analyse quarterly expenditure data from 

the internal Bureau of Labor Statistics (BLS) files (data are not topcoded).

We reduced the sample to 71593 by including only consumer units 

designated by the BLS as complete income reporters and having positive 

values for total expenditures and income, and non-negative health 

expenditures. We further reduced the sample by selecting a five per cent 

random sample to limit the costs of running the models. Our random 

sample includes 3550 consumer units; 5.8 per cent of these are households 

for which expenditures are observed for more than one quarter. 

Descriptive statistics of the sample are presented in Table 1.8

We use expenditures for consumption only; expenditures for gifts and 

capital improvements, and allocations to savings are excluded. We 

grouped expenditures into seven commodity groups: food, shelter, apparel 

and upkeep, transportation, fuels and utilities, health care, and ‘other’, 

which includes household furnishings and operations, entertainment, and 

other goods and services.

Price data are area-specific price indices from the Consumer Price Index 

(CPI) listed on the BLS LABSTAT data system for the same time period 

as the expenditure data (USDL 1992). The base period for the indices was 

1982 to 1984. We used the seasonally unadjusted consumer price indices 

in thirty areas for the first six commodity groups, while a weighted 

average price was computed for the `other' expenditures commodity 

group; the weights were the shares of total expenditures allocated to the 

three commodity groups included in the `other' category.

We used price indices from fifteen urban areas, while regional and 

citysize average price indices were used for the other areas represented in 

the

sample. To obtain quarterly prices, we averaged monthly prices (or bimonthly, if 

monthly prices were not available) over the three months of the interview 

quarter.

While the area-specific prices used in this paper do not provide a measure of 

price differences across areas, they do provide an accurate measure of price 

differences over time within areas and we show that they do not bias the 

parameter estimates in the equivalence scales. We tested whether these 

area-specific prices bias the results by estimating a fixed effects model which 

included a dummy variable for each area in each equation. We found that, while 

the price coefficients are different, the presence of fixed effects do not 

significantly improve the explanatory power of the model and hence, we cannot 

reject the hypothesis that the fixed effects were insignificant.9 To further check 

the possible bias caused by area-specific prices, we estimate the model using 

average national price indices (for each area) and obtain similar parameter 

estimates of the characteristic variables. While our results suggest that using 

area-specific prices do not bias the parameter estimates, in future research we 

plan to use new BLS research in which interarea prices are produced (see 

Kokoski, Cardiff and Moulton 1992).

4. THE RESULTS

We estimated Equation (13) for six share equations; the coefficients for the 

seventh equation (other) are calculated using the homogeneity and symmetry 

restrictions. Tests of the symmetry restriction (6b) allowed us to reject this 

restriction, while tests of the homogeneity restrictions (6a) did not allow us to 

reject this restriction. The estimates were obtained using the non-linear 

seemingly unrelated regression procedure in SAS.

The parameter estimates, standard errors and R2 values for the almost ideal 

demand systems using three different functional forms of the scales are given in 

Table 2. As the R2 values suggest, the model with pricedependent scales fits the 

data better than the models without these terms. This suggests that price changes 

affect different size households differently.

Equivalence scales are calculated by substituting the parameter estimates 

(from Table 2) and the average prices for the data set (over all four years) into 

Equations (12a,b and c). The equivalence scales and their approximate standard 

errors are given in Table 3. We normalised the scales so that a single-member 

household has an equivalence scale of one. Except for two-person households, 

the price-dependent scales are larger (although not significantly) than both of the 

Full Engel scales.

(Table 2 continued)

Parameter Variable Parameter Estimate by Model

(Standard Error)

Full Engel Full Engel

Price- Scale with Scale with

dependent Constant Family Size

Scale Elasticity Dummies

Shelter-Shelter 0.280' 0.272' 027I'

(0.032) (0.032) (0.032)

Shelter Apparel 0.038' 0.039' 0.039'

(0.014) 10.014) (0.014)

Shelter Transportation -0.125' -O.I19' -0.121'

(0.034) (0.034) (0.034)

Shelter-Fuel -0.115' -0.118' -0.1172

(0.014) (0.014) (0.014)

Shelter-Health -0.121' -0.122' -0.122'

(0.019) (0.019) (0.019)

Shelter-Other 0.031 0.033 0.033

Apparel-Apparel -0.016 -0.017 -0.017

(0.017) (0.017) (0.017)

Apparel Transportation 0.002 0.003 0.003

(0.027) (0.028) t0.028)

Apparel-Fuel 0.001 0.002 0.002

(0.011) (0.011) (0.011)

Apparel-Health -0.033 -0.031 -0.031

(0.017) (0.017) (0.017)

Apparel-Other 0.037 0.036 0.036

Transportation- - - -

Transportation 0.150 0.144 0.148

(0.087) (0.087) (0.087)

Transportation-Fuel 0.080' 0.082' 0.08(?'

(0.027) (0.027) (0.027)

Transportation-Health 0.007 0.006 0.007

(0.039) (0.039) (0.039)

Transportation-Other -0.096 -0.0% -0.096

Fuel Fuel -0.021 -0.023 -0.022

(0.015) (0.015) (0.015)

Fuel-Health 0.020 0.016 0.016

(0.016) (0.016) (0.016)

Fuel-Other 0.042 0.043 0.042

Health-Health 0.087' 0.091' 0.091'

(0.035) (0.035) (0.036)

Health-Other 0.005 -0.001 0.000

Other-Other 0.039 0.048 0.045

do Family Size 0.667' 0.567

(0.086) (0.025)

Family Size 2 -- - 0.475'

(0.038)

4.1 Constant Elasticity

The parameter d0 is a measure of the overall economies of scale due to family 

size and accounts for most of the differences in the scales across family size. The 

estimates of d0 are .67 in the price-dependent scales, .57 in the Full Engel scales 

with constant elasticity, and an approximated value of .58 in the Full Engel 

scales with dummy variables. These results suggest that there are economies of 

scale in the production of economic welfare.

We tested the assumption of constant elasticity by including dummy variables 

for family sizes three through six and seven or more in addition to the log of 

family size. We could not reject the hypothesis that all of the coefficients on the 

dummy variables were zero. This implies that different family sizes do not 

change the constant elasticity parameter.

Table 3 shows that the Full Engel scales with and without constant elasticity 

are almost identical, which suggests that the constant elasticity assumption is not 

overly restrictive. Constant elasticity and the fact that most of the families of 

size three or more consist of not more than two adults suggests that adults and 

children do not have different effects on the overall economies of scale, that is, it 

is the overall family size that is important in determining the economies of scale.
4.2 Price-characteristic Interaction Terms

We also tested whether the price-characteristic interaction terms significantly 

contribute to the explanatory power of the model and we were able to reject the 

hypothesis that these coefficients were zero. While the price-characteristic 

variables are significant, the homogeneity restrictions imply that the equivalence 

scales will not change significantly as prices change (see Phipps 1991).

Not only do the price-characteristic interaction terms improve the fit of the 

model, their inclusion causes the constant elasticity parameter to increase. This 

could be due to the fact that the interaction terms allow the share equations to 

depend on the characteristics in different ways. We would expect increases in 

family size to cause a greater increase in the food share than in all of the other 

shares since there are less economies of scale in food expenditures. As Table 2 

shows, the d0 parameter is larger in the price-dependent model and the 

interaction terms in the food share are close to zero and they are negative in 

other share equations.

While the constant elasticity parameter represents the absolute scale, 

price-characteristic interaction terms are used to determine the relative scales. 

Taken together, the price-characteristic terms yield relative scales that are less 

than one for households with children and greater than one for all-adult 

households. A negative coefficient on an interaction term implies that as the 

relative price of the commodity increases, the equivalence scale for larger 

households decreases, because it becomes less costly to live in a large 

household.

As expected, the coefficients in the shelter share equation are negative for 

both adults and children. The coefficients in the transportation equation are 

positive suggesting that increases in the relative price of transportation represent 

a higher cost for large households than for small households, perhaps reflecting 

the fact that the average number of cars per household increases with family 

size.

The positive coefficient for the interaction term between children and apparel 

supports the results presented in Nelson (1989) which suggest that children are 

‘clothing-purchase intensive.’ The negative coefficient for the interaction term 

between children and health may reflect the pricing structure of health plans in 

the US, with family health insurance policies usually covering children no 

matter how many are present.

4.3 Equivalence Scale Exactness

Finally, we tested the assumption of equivalence scale exactness. Recent 

attempts at estimating equivalence scales that satisfy equivalence scale
exactness have found this assumption to be overly restrictive; Blundell and 

Lewbel (1991) and Pashardes (1992) both reject the hypotheses of equivalence 

scale exactness. We obtained similar results.

We test for equivalence scale exactness by estimating a model is which we 

allowed the coefficients on income to depend on family size. The resulting share 

equations depended on a new term, the product of family size and income. We 

found these terms to be significant in each share equation and were able to reject 

the hypothesis that the parameters were all zero. This suggests that total 

expenditures and family size are correlated and that interaction terms should be 

included in the share equations. Future work will involve estimating a Rank-3 

demand system, which allows for this interaction while still allowing 

equivalence scale exactness (see Pashardes 1994).

5. EQUIVALENCE SCALES AND DISTRIBUT10N

MEASURES

In this section we examine the impact of different equivalence scales on the 

distribution of expenditures, which are used as a proxy for income. We compare 

three different measures of inequality over the entire sample. For these 

comparisons, we inflate the expenditures by the CPI so that all expenditures are 

in 1990 dollars. We adjust the expenditures by family size using the equivalence 

scales implicit in the US official poverty thresholds and by those produced from 

the estimations of the price-dependent model and the two Full Engel models. For 

comparison, we also present results in which expenditures are not adjusted for 

family size (household weighted) and in which expenditures are simply divided 

by family size (per capita).

The relationship between equivalence scale elasticities and inequality indices 

are shown in Table 4. Results are presented for three distributional measures; 

two are from the Generalised Entropy family of indices, the Theil and the Mean 

Logarithmic Deviation, and the other is the Gini coefficient.

Coulter, Cowell and Jenkins (1992) show the effect of changing the elasticity 

of the equivalence scale on various inequality indices. They show that the Theil 

and Mean Logarithmic Deviation inequality indices plotted against the 

equivalence scale elasticity yields a U-shaped curve. In other words, for small 

(close to zero) and large (close to one) elasticities, the inequality indices will be 

higher than for elasticities in the one-half range. This U-shape is also exhibited 

for the Gini coefficient for some distributions.

Our results using the US CEX confirm the U-shape relationship for the
three inequality indices. Table 4 shows that the indices are greatest when the 

expenditures are not adjusted by family size and when the per capita adjustment 

is used. The elasticities obtained in our models are in a fairly fiat segment of the 

U-shaped relationship (see Cowell, Coulter and Jerkins 1992) and hence, the 

differences in the inequality indices are minimal. Our equivalence scale estimates 

yield inequality indices that are also similar to those obtained by adjusting 

expenditures with the US poverty scales.

6. CONCLUDING REMARKS

Using expenditure data and price indices to estimate an almost ideal demand 

system, we produced three different equivalence scales. By assuming a particular 

functional form for the scales and that the scales are independent of the base level 

of utility, we obtained unique equivalence scales. We showed that price effects 

should be included in the equivalence scales and that the scales have the property 

of constant elasticity. In general, our scales are larger than earlier estimates using 

similar utilitybased methods (see, Nelson 1993 and Betson 1990).

Even though the elasticity of the scales differ, their impact on three measures 

of inequality is minimal. Since our scales are similar to the scales used in 

adjusting the US official poverty thresholds by family size, the difference in the 

inequality measures was also small. In our scales (as in the poverty scales), adults 

and children are given the same weight. Since there may be different economies 

of scale for adults and children as suggested by
the estimates of the price-characteristic terms, we plan to investigate the separate 

effects of children and adults and to compare the equivalence scales. Another 

area for future research is to compare single-parent and two-parent households.

In future research, we will continue our exploration of the methodological 

issues related to the estimation of equivalence scales. Specifically, we plan to test 

whether the use of interarea prices or a longer sample period will produce more 

price variation. Since other researchers have used restricted samples, we plan to 

use our model on similarly restricted samples to determine the sensitivity of 

sample selectivity. We also plan to further examine the implications of requiring 

equivalence scale exactness and particular functional forms. A complete analysis 

would examine different methods of obtaining equivalence scale exactness in an 

almost ideal demand system, including the translog form of the scales as 

presented in Jorgenson and Slesnick (1987), the form in Pashardes (1991) and a 

Rank-3 demand system as in Pashardes (1994).

ACKNOWLEDGMENT We would like to thank Fred Rowland for LABSTAT assistance and 

Richard Dietz for computer programming assistance. We would also like to thank Carsten 

Folkertsma, Eva Jacobs, Arthur Lewbel, Brent Moulton, J.G. Odink, Panos Pashardes and Marshall 

Reinsdorf for helpful comments as well as the participants at the 22nd General Conference of the 

International Association for Research in Income and Wealth and the Amsterdam Workshop on 

Household Equivalence Scales for helpful comments. The views expressed in this paper are those 

of the authors and do not reflect the policies of the BLS or views of other BLS staff members.
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