Estimating Consumption Economies of Scale,
Adult Equivalence Scales, and Household
Bargaining Power
Martin Browning, Pierre-André Chiappori and Arthur Lewbel
Oxford University, Columbia University, and Boston College
Revised September 2010
Abstract
How much income would a woman living alone require to attain the same standard
of living that she would have if she were married? What percentage of a married cou-
ple's expenditures are controlled by the husband? How much money does a couple save
on consumption goods by living together versus living apart? We propose and estimate
a collective model of household behavior that permits identication and estimation of
concepts such as these. We model households in terms of the utility functions of its mem-
bers, a bargaining or social welfare function, and a consumption technology function.
We demonstrate generic nonparametric identication of the model, and hence of a ver-
sion of adult equivalence scales that we call "indifference scales," as well as consumption
economies of scale, the household's resource sharing rule or members' bargaining power,
and other related concepts.
JEL codes: D11, D12, C30, I31, D63, J12. Keywords: Consumer Demand, Collective Model, Adult
Equivalence Scales, Indifference Scales, Household Bargaining, Economies of Scale, Demand Sys-
tems, Bargaining Power, Barten Scales, Sharing rule, Nonparametric Identication. We would like to
thank the editor, anonymous referees, and Krishna Pendakur for numerous helpful comments, and the
Danish National Research Foundation for support through its grant to CAM. Corresponding Author:
Arthur Lewbel, Department of Economics, Boston College, 140 Commonwealth Ave., Chestnut Hill,
MA, 02467, USA. (617)-552-3678, lewbel@bc.edu, http://www2.bc.edu/~lewbel/
1 Introduction
On average, how much income would a woman living alone require to attain the same standard
of living that she would have if she were married? What percentage of a married couple's
expenditures benet the husband? How much money does a couple save on consumption
1
goods by living together versus living apart? The goal of this paper is to propose a collective
model of household behavior aimed at answering questions such as these.
A large literature exists on specication and estimation of ordinary demand systems, col-
lective and household bargaining models, and on identication (and lack thereof) of equiva-
lence scales, bargaining power measures, resource shares, consumption economies scale, and
other related household welfare measures. Some surveys of this literature include Deaton and
Muellbauer (1980), Blundell (1988), Browning (1992), Pollak and Wales (1992), Blundell,
Preston, and Walker (1994), Blackorby and Donaldson (1994), Bourguignon and Chiappori
(1994), Lewbel (1997), Jorgenson (1997), Slesnick (1998), and Vermeulen (2000).
We propose a new model of household consumption behavior that has three components,
which are separate utility functions over goods for each household member, a consumption
technology function that characterizes the jointness or publicness of goods and hence the
economies of scale and scope in consumption, and a sharing rule that denes the relative
allocation of household resources among the household members. The basic structure of
this model is that households purchase a bundle (an n vector of quantities) of goods z. By
economies of scale and scope in consumption, that is, by sharing, this z for the household
is equivalent to purchasing bundle of privately consumed goods x where each element of x
is typically greater than or equal to the corresponding element of z. The vector of quanti-
ties x is then divided up among the household members, and each member i derives utility
from consuming their bundle of goods x
i
, so the sum of these x
i
vectors across household
members i is x. The conversion of z to x, which is embodied by the consumption technol-
ogy function, is essentially an application of the models of Barten (1964) and Gorman (1976)
that characterize how demands differ across households of different sizes. The allocation of
shares of x to different household members, characterized by a sharing rule, is essentially the
collective household model of Chiappori (1988, 1992), Bourguignon and Chiappori (1994)
and Vermeulen (2000). Our model combines the features of both approaches, which is what
allows us to identify consumption economies of scale, adult equivalence scales, and household
bargaining power.
We provide a dual representation of our collective model that facilitates empirical applica-
tion, and show that the model is generically nonparametrically identied. We then show how
the model overcomes traditional problems regarding nonidentication of equivalence scales,
and can be used to address the questions listed above. Our results only require ordinal rep-
resentations of preferences, and do not depend on any utility cardinalization or interpersonal
comparability assumptions. We apply the model to Canadian consumption data on couples
and singles.
2 Equivalence Scales and Indifference Scales
Equivalence scales seek to answer the question, “how much money does a household need to
spend to be as well off as a single person living alone?” Equivalence scales have many practical
applications. They are commonly used for generating poverty lines for households of various
compositions given a poverty line for single males. Income inequality measures have been
applied to equivalence scaled income rather than observed income to adjust for household
2
composition (see, e.g., Jorgenson 1997). Calculation of appropriate levels of alimony or life
insurance also entail comparisons of costs of living for couples versus those of singles.
An equivalence scale is traditionally dened as the expenditures of the household divided
by the expenditures of a single person that enjoys the same “standard of living” as the house-
hold. Just as a true cost of living price index measures the ratio of costs of attaining the same
utility level under different price regimes, equivalence scales are supposed to measure the ratio
of costs of attaining the same utility level under different household compositions. Unfortu-
nately, unlike true cost of living indexes, equivalence scales dened in this way can never be
identied from revealed preference data (that is, from the observed expenditures of households
under different price and income regimes). The reason is that dening a household to have
the same utility level as a single individual requires that the utility functions of the household
and of the single individual be comparable. We cannot avoid this problem by dening the
household and the single to be equally well off when they attain the same indifference curve,
analogous to the construction of true cost of living indices, because the household and the sin-
gle have different preferences and hence do not possess the same indifference curves. Pollak
and Wales (1979, 1992) describe these identication problems in detail, while Blundell and
Lewbel (1991) prove that only changes in traditional equivalence scales, but not their levels,
can be identied by revealed preference. See Lewbel (1997) for a survey of equivalence scale
identication issues.
We argue that the source of these identication problems is that the standard equivalence
scale question is badly posed, for two reasons. First, by denition any comparison between
the preferences of two distinct decision units entails interpersonal utility comparisons. Second,
and perhaps more fundamental, the notion of a household utility is awed. Individuals have
utility, not households. What is relevant is not the 'preferences' of a given household, but
rather the preferences of the individuals that compose it.
We propose therefore that meaningful comparisons must be undertaken at the individual
level, and that the appropriate question to ask is, “how much income would an individual liv-
ing alone need to attain the same indifference curve over goods that the individual attains as a
member of the household?” This latter question avoids issues of interpersonal comparability
and does not require us to compare the utility levels of different indifference curves. This
question also does not depend on the utility level that is assigned to an indifference curve,
i.e., it is unaffected by the fact that a person's utility associated with a particular indifference
curve over goods might change as a result of living with a partner. The question only depends
on ordinal preferences, and hence is at least in principle answerable from revealed prefer-
ence data. Consequently, in sharp contrast with the existing equivalence scale literature, our
framework does not assume the existence of a unique household utility function, nor does it
require comparability of utility between individuals and collectives (such as the household).
Instead, following the basic ideas of the collective approach to household behavior, we as-
sume that each individual is characterized by his/her own ordinal utility function, so the only
comparisons we make is between the same person's welfare (dened by indifference curves)
in different living arrangements.
Dene a equivalent income (or expenditure) to be the income or total expenditure level y
i
required by an individual household member i purchasing goods privately, to be as well off
3
materially as he or she is while living with others in a household that has joint income y. In
our model, this means that when the household spends y to buy a bundle z, household member
i consumes a bundle x
i
(determined by the consumption technology and sharing rule). Then
y
i
is the least expenditure required to buy a bundle of goods that lies on member i's same
indifference curve as the bundle x
i
. Then, instead of a traditional adult equivalence scale, we
dene individual i 's "indifference scale" to be S
i
D y
i
=y.
If member i were given the fraction S
i
of the household's total expenditures, then by buy-
ing goods on the open market individual i could get herself to the same indifference curve
(dened in terms of her own utility function) that she attained as a member of the household,
taking into account whatever economies of scale in consumption she enjoyed by sharing and
joint consumption within the household. Indifference scales depend only on the indifference
curves of household members, the resources of the household, and on the degree to which
consumption is shared within the household, and so can be identied without any utility car-
dinalization or interpersonal comparability assumptions.
To see the usefulness of indifference scales, consider the question of determining an ap-
propriate level of life insurance for a spouse. If the couple spends y dollars per year then for
the wife to maintain the same standard of living after the husband dies, she will need an insur-
ance policy that pays enough to permit spending S
f
y dollars per year. Note that this amount
of money only compensates the wife enough to reach the same indifference curve over goods
that she attained while she was a household member. It does not compensate for any loss of
utility due to grief, or for any change in her preferences that might result from the death of her
husband. Similarly, in cases of wrongful death, juries are instructed to assess damages both to
compensate for the loss in “standard of living,” (i.e., S
i
) and, separately for “pain and suffer-
ing,” which would presumably be noneconomic effects (see Lewbel 2003). Another example
is poverty lines. If poverty lines for individuals have been established, then the poverty line
for a couple could be dened as the expenditures required for each member of the couple to
attain his or her own poverty line indifference curve.
Traditional equivalence scales do not properly answer these questions, because they at-
tempt to relate the utility of an individual to that of a household, instead of relating the utility
of the same individual in two different settings, e.g. living with a husband versus without. This
is similar to the distinction Pollak and Wales (1992) make between what they call a welfare
comparison versus a situation comparison.
3 The Model
This section describes the proposed household model. Let superscripts refer to household
members and subscripts refer to goods. Let U
i
.x
i
/ be the direct utility function for a consumer
i, consuming the vector of goods x
i
D .x
i
1
; :::; x
i
n
/. We consider households consisting of
two members, which we will for convenience refer to as the husband (i D m/ and the wife
(i D f ). For many applications, it may be useful to interpret one of these utility functions as
a joint utility function for all but one member of the household, e.g., U
f
could be the joint
utility function of a wife and her children.
4
3.1 Household Members
ASSUMPTION A1: Each household member i has a monotonically increasing, continuously
twice differentiable and strictly quasi-concave utility function U
i
.x
i
/ over a bundle of n goods
x
i
.
If member i were to face an n vector of prices p with income level y
i
and this utility
function, he or she would solve the optimization program
max
x
i
U
i
.x
i
/ subject to p
0
x
i
D y
i
. (1)
Let x
i
D h
i
.p=y
i
/ denote the solution to this individual optimization program, so the vector
valued function h
i
is the set of Marshallian demand functions corresponding to U
i
.x
i
/. Dene
V
i
by
V
i
p=y
i
D U
i
h
i
p=y
i
(2)
so V
i
is the indirect utility function corresponding to U
i
. The functional form of individual
demand functions h
i
could be obtained from a functional specication of V
i
using Roy's
identity.
3.2 The Household Decision Process
Now consider a household consisting of a couple living together, and facing the budget con-
straint p
0
z y. Following the standard collective approach, our key assumption regarding
decision making within the household is that outcomes are Pareto efcient. A standard result
of welfare theory (see, e.g., Bourguignon and Chiappori 1994) is that, given ordinality, one can
without loss of generality write Pareto efcient decisions as a constrained maximization of the
weighted sum U
f
.x
f
/ C U
m
.x
m
/. Here the constraints are the technology constraints that
dene feasible values of individual consumption vectors x
f
and x
m
given z, and the budget
constraint that denes feasible values of the vector of purchases z. It is important to note that
the Pareto weight may in general depend on prices, total expenditures, and on a vector s of
distribution factors, the latter being dened as variables with no direct impact on preferences,
technology or budget constraint, but that may inuence the decision process.
We assume the household does not suffer from money illusion, and so write the Pareto
weight function as . p=y; s/. Possible examples of distribution factors s include individual
wages (as in Browning et al., 1994) or non labor income (Thomas 1990), sex ratio on the rele-
vant marriage market and divorce legislation (Chiappori, Fortin and Lacroix 2002), generosity
of single parent benets (Rubalcava and Thomas 2000), spouses' wealth at marriage (Thomas,
Contreras and Frankenberg 1997), and the targeting of specic benets to particular members
(Duo 2000). See also Chiappori and Ekeland (2005) for a general discussion.
ASSUMPTION A2: Given budget and technology constraints and the absence of money
illusion, the household makes Pareto efcient decisions, that is, it's choice of x
f
and x
m
5
