Equivalence Scales and Intra-household
Distribution
Hilde Bojer
Department of Economics, University of Oslo
Pobox 1095 Blindern
0317 Oslo
Norway
E-mail: hilde.bojer@econ.uio.no
Abstract
Equivalence scales are dened as the ratio between the cost functions of households of dierent size and composition. It is shown that
the household cost function depends on the distribution of income or
utility within the household. Therefore, neither cost functions nor
equivalence scales are uniquely determined by the size and composition of households.
In analysis of personal income distribution, income is usually dened
as total household income per equivalent adult. The number of equivalent
adults assigned to a household depends on the size and composition of the
household.
Household income is used because members of a household are assumed to
pool their income. Pooling of income means that the consumption possibilities of each individual may be dierent from her individual income, larger or
smaller as the case may be. There are two reasons why household income is
not then simply divided by the number of persons. Firstly, economies of scale
are assumed to be present in household consumption, for instance because of
public goods in the household. Even if two cannot live as cheaply as one, they
certainly do not need two kitchens. Secondly, there may be systematically
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dierent preferences between dierent kinds of persons. Typically, children
are assumed to dier from adults.
Intuitively, equivalent adult scales or equivalence scales should give the
answer to the following question: How big an income does a household
consisting of several persons need in order to obtain the same (material)
standard of living as a single person with a given income? A widely used
equivalence scale, the OECD scale, assigns the number 1 to a single adult, 0.7
to the second and each following adult and 0.5 to each child. A household
consisting of two adults and one child will thus consist of 2.2 equivalent
adults, or be assigned the number 2.2 on the OECD equivalence scale.
Formally, equivalence scales are dened as the ratio between the cost of
living functions of households. The standard of living is represented by the
utility function of the household. Utility is assumed to be a function of
household composition as well as commodity quantities. The cost function
is derived from the indirect utility function. For a given price vector p and
a vector of demographic characteristics z it is written c(u; p; z). Let the
reference household have characteristics zr. A schedule of equivalence scales1
is given by (See e.g. Blundell, Preston and Walker 1994)
m(u; p; z; zr) = cc((u;u;p;p;zzr))
(1)
The utility level u represents the utility level of the household as a whole,
in accordance with the traditional approach in economics of treating the private household as a single entity without distinguishing between the dierent
individual members of the household. In the literature on equivalence scales
there is very little discussion of the relationship between the welfare of the
individual and the welfare of the household as a whole. One exception is Nelson 1993, another Blackorby and Donaldson 1994, where internal household
distribution is assumed to make all individual utility levels equal. Household
utility is then set equal to this common utility level.
There is a growing literature in economics analysing decision processes
within the household, with particular attention being paid to the possibility
that distribution within the household may well not secure equality. (Apps and Rees 1988, Bourgignon and Chiappori 1994, Haddad and Kanbur
The term equivalent adult scales seems to imply that the reference household is a
single adult. The term equivalence scales may be taken to be more general.
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2
1990, Wooley and Marshall 1995) The purpose of this paper is to explore
some consequences for the theory of equivalence scales of the individualistic
approach, and of intra-household inequalities.
Let us then take a closer look at the household as a collection of individuals. In the following, I shall simplify the analysis by assuming that all
individuals have the same preferences. This assumption, while acceptable in
a world of adults, means that children are not included in my analysis.2 I
shall study a household consisting of two identical adults, while the reference
household consists of one adult. Prices will be assumed to be constant, so
that I can ignore the price vector in (1). To start with, I shall also assume
that all consumption goods are individual: there are no public goods within
the household and no economies of scale. The concept of a household is still
meaningful: the two adults constitute a household because income is shared
between them.
One advantage of these strong assumptions is to create a quite clear
intuition of what the value of our household ought to be on an equivalence
scale: it should equal two. In the absence of public goods and economies of
scale, two adults need exactly twice as much as one adult to obtain the same
utility level.
In this simple world, all individuals have the same individual indirect
utility function, u = V (x), where x is total consumption expenditure (individual). The utility function is assumed to possess the usual properties, in
particular it is strictly increasing and strictly concave in x. Inverting V gives
the cost function c = C (u). From the properties of V (:), it follows that C (:)
is strictly increasing and strictly convex in u.
The cost of living function of the household is now a simple sum of the two
identical cost of living functions. The two household members may, however
have dierent levels of utility. Therefore, the cost function of the household
will have as arguments the two individual utility levels. Superscripts h, 1
and 2 denote the household and the two individuals respectively.
ch (u1; u2) = c(u1) + c(u2)
(2)
The question now is, what should we mean by the utility of the household
as a whole? One denition of household utility might be the total utility uh =
2 As I argue in e. g. Bojer 1996, it is not enough to allow for dierent preferences
in order to incorporate children in the analysis. There are great diculties in assigning
preferences to children.
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u1+u2. However, this denition is clearly not useful for comparing households
of dierent sizes. A better denition is average utility: u = 1=2(u1 + u2).
None of these denitions gives cost as a well dened function of household
utility. To obtain cost as a function of average utility, the internal distribution
needs to be known. A simple and attractive rule would be to postulate that
the household acts so as to maximise average utility for a given total income.
It is well known that, when preferences are equal, maximum average utility is
obtained when the two utility levels are equal. If u = u1 = u2, (2) becomes:
ch(u) = c(u) + c(u) = 2c(u)
(3)
The equivalence scales value of the two-person household then simply becomes m(u; 2; 1) = 2c(u)=c(u) = 2, as it should.
It follows from duality theory that the cost function (3) represents the
minimum expenditure required to obtain a specied average utility level.
The rule of equal utilities within the household postulated by Blackorby
and Donaldson (1994) is therefore in some ways the natural assumption to
make; it makes the denition of the household cost function similar to the
denition of individual cost functions, it results in the intuitively right value
of the equivalence scales, and it makes for a straightforward interpretation
of household income per equivalent adult.
But the assumption of equal utilities is a questionable one, and certainly
not the only possibility. If utility levels are not equal, the cost function (2)
no longer minimises the expenditure necessary to obtain the average utility
of the household. Hence, we may be certain that, if u1 = u2, then
c(u1) + c(u2) > ch(u)
(4)
Moreover, the cost-function depends on distribution of utility as well as
average utility in the household. Let the distribution of utility be characterised by a parameter such that = 1 when utilities are equal, while
0 < < 1 when they are not. In general, therefore, the equivalent scales
corresponding to a given average household utility must be written
1 + c(u2)
m(u; ; 2; 1) = c(u )c(
(5)
u)
It follows from (4) that m(u; 2; ; 1) > 2 whenever = 1. In other words,
if internal distribution within the household results in unequal utilities, a
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two person household needs more than twice the income of a single person
household to achieve a given level of (average) utility.
Now, household utility as such may well not be the important concern,
either in distributional analysis or in distributional policies. What concerns
us may well be the welfare of one particular household member. Let us make
the reasonable assumption that number one has the higher utility level, but
does not consume the whole household income. Since the preferences are
identical, this implies c(u1) > c(u2) > 0 and therefore c(u1)=c(u2) > 1.
If the utility of number one is what concerns us, the equivalence scales
should be dened by
1
2
2
m(u1; 2; 1) = c(u c)(+u1c)(u ) = 1 + cc((uu1))
(6)
In this case, 1 < m(u1; 2; 1) < 2:
If what concerns us is the utility of number two, the denition is:
1
2
1
m(u2; 2; 1) = c(u c)(+u2c)(u ) = 1 + cc((uu2))
(7)
In this case, we see that m(u2; 2; 1) > 2:
It costs more to ensure the well-being of number two than that of number
one. The unequal internal distribution implies that in order for one ECU to
reach number two, more than one ECU must reach number one, and more
than two ECUS accrue to the household as a whole. On the other hand, for
one ECU to reach number one, number two receives less than one ECU and
the household needs less than two ECUS. This is also the reason why equality
of utilities is necessary for the household to have an equivalence scale value
of exactly 2.
Now, let us relax the assumption that preferences are identical. If the
household distributes income to ensure maximum average utility, marginal
utilities but not utility levels will be equal. If, on the other hand, income is
distributed to secure equal utilities, the ensuing household cost function will
not minimise the expenditure corresponding to the average utility level in the
household. The researcher working with equivalence scales will either have to
choose between one of two equally attractive assumptions when interpreting
the scales, or fall back on the assumption of equal preferences. In the last
case, equivalence scales handle the eect of household size and composition
only in a limited sense: they only measure economies of scale.
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Actual behaviour of households will probably neither maximise average
utility nor secure equal utilities. Moreover, if internal distribution varies from
household to household, cost functions will vary from household to household.
It is, for instance, realistic to assume that internal distribution between adults
will depend on individual earning power, which may well vary even between
individuals who otherwise have the same preferences. Such variation raises
problems for empirical estimation of equivalence scales.
References
Blackorby, Charles and David Donaldson (1994) 'Measuring the cost
of children: a theoretical framework' in Richard Blundell, Ian Preston
and Ian Walker (Eds) The Measurement of Household Welfare 51{69
Cambridge University Press.
Blundell, Richard, Ian Preston and Ian Walker (1994) 'An introduction to applied welfare analysis' in Richard Blundell, Ian Preston and
Ian Walker (Eds) The Measurement of Household Welfare 1{50. Cambridge University Press.
Bourgignon, Francois and Pierre{Andre Chiappori (1994) 'The collective approach to household behaviour' in Richard Blundell, Ian Preston
and Ian Walker (Eds) The Measurement of Household Welfare Cambridge University Press. 70{85.
Haddad, Lawrence and Ravi Kanbur (1990) 'How Serious is the Neglect of Intra{household Inequality?', The Economic Journal, 100: 866{
881.
Nelson, Julie (1993) Household Equivalence Scales: Theory versus Policy?', Journal of Labor Economics, 11:471{492
Woolley, Frances R. and Judith Marshall (1995) 'Measuring Inequality within the Household', Review of Income and Wealth, 40, 415{432.
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