Estimating Consumption Economies of
Scale, Adult Equivalence Scales, and
Household Bargaining Power
PRELIMINARY - NOT FOR CIRCULATION
Arthur Lewbel, Pierre-André Chiappori, and Martin Browning
Boston College, University of Chicago, and University of Copenhagen
Revised January 2003
Abstract
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 are controlled by the
husband? How much money does a couple save on consumption goods
by living together versus living apart? We propose a collective model of
household behavior that permits identification and estimation of concepts
like these. We model households in terms of the utility functions of its members, a bargaining or social welfare function, and a consumption technology
function. We demonstrate generic identification of the model, and hence
of equivalence scales, consumption economies of scale, household members’
bargaining power and other related concepts. Our empirical application of
the model uses data from the UK Family Expenditure Surveys.
JEL codes: D11, D12, C30, I31, D63, J12.
Keywords: Consumer Demand, Collective Model, Adult Equivalence Scales, Household
Bargaining, Economies of Scale, Demand Systems, Bargaining Power, Barton Scales.
This research was supported in part by the NSF, grant SES-9905010.
Corresponding Author: Arthur Lewbel, Department of Economics, Boston College, 140
Commonwealth Ave., Chestnut Hill, MA, 02467, USA. (617)-552-3678, [email protected],
http://www2.bc.edu/~lewbel/.
1
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 benefit to the husband? How much money does
a couple save on consumption 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 like these.1
The above questions are standard in the literature on equivalence scales. Our
approach, however, is crucially different. Equivalence scales seek to answer the
question, ”how much money does a household need to spend to be as well off as
a single male living alone?” The equivalence scale itself is then the expenditures
of the household divided by the expenditures of the single male that enjoys the
same ”standard of living” as the household.2 The early equivalence scale literature
attempted to define this ratio of costs of living directly in terms of measurable
quantities such as the costs of acquiring a required number of calories, but this
was soon replaced by the concept of defining the household and the single male
as equally well off to mean that they attain an equal level of utility (see, e.g.,
Jorgenson and Slesnick 1987, and for a survey, Lewbel 1997). 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 were supposed to measure
the ratio of costs of attaining the same utility level under different household
compositions.
Unfortunately, unlike true cost of living indexes, equivalence scales defined in
1
A large literature exists on specification and estimation of ordinary demand systems, collective and household bargaining models, and on identification (and lack thereof) of equivalence
scales, bargaining power measures, 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).
2
Equivalence scales have many practical applications. For example, given a poverty line for
single males, that line may be multiplied by the household equivalence scale of a couple to
obtain a comparable poverty line for couples. Income inequality measures should be applied to
equivalence scaled income rather than observed income to adjust for household composition.(see,
e.g., Jorgenson 1997). Calculation of appropriate levels for alimony or life insurance require
comparing costs of living for couples versus those of singles. Lewbel (2002) shows that legal
statutes in the United States regarding appropriate compensation in cases of wrongful death
can be interpreted as requiring an equivalence scale (or a private goods equivalent) calculation.
this way can never be identified from revealed preference data (that is, from the
observed expenditures of households under different price and income regimes).
The reason is that defining a household to have the same utility level as a single
male requires that the utility functions of the household and of the single male be
comparable. We cannot avoid this problem by defining 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
single have different preferences and hence do not possess the same indifference
curves. Pollak and Wales (1979, 1992) describe these identification problems in
detail, while Blundell and Lewbel (1991) prove that only changes in traditional
equivalence scales, but not their levels, can be identified by revealed preference.3
We argue that the source of these identification problems is that the equivalence scale question is badly posed. The appropriate question to ask is, ”how
much income would an individual living alone need to attain the same indifference curve over goods that the individual attained as a member of the household?”
This latter question avoids issues of interpersonal comparability and differences in
indifference curves (it only depends on ordinal preferences), and hence is at least
in principle answerable from revealed preference data. Consequently, in sharp
contrast with the 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 groups (such as the household). On the contrary, following the basic ideas of the collective approach to household behavior,
we assume that each individual is characterized by his/her own utility function;
the only comparisons we make is between the same person’s welfare before and
after marriage.
In this context, the main practical obstacle to the definition of equivalence
scales is the inobservability of individual consumptions and the intrahousehold
allocation process. If individual consumptions within the household were directly
observable, one could simply add up the cost of buying what he or she consumed
within the household to determine the money that person would need to spend to
as a single to attain the same standard of living. In general, however, only house3
Many previous attempts to identify equivalence scales are based on assumptions such as
Independence of Base (IB) or Equivalence Scale Exactness (see, e.g., Lewbel 1989). Jorgenson
and Slesnick’s (1987) model is a special case of an IB assumption. IB imposes testable restrictions on demands, but it also requires untestable (that is, cardinal or comparable utility type)
restrictions. IB is an example of the paradigm of comparing an individuals’s utility to that
of a household, which is intrinsically impossible without some cardinalization or interpersonal
comparability of utility.
hold’s total purchases are observed. This raises two problems. First, since the
distribution of resources within the household is not recorded, it has to be identified from the aggregate household demand. Secondly, household consumption
entails shared consumptions and economies of scale. Typically, in a multiperson
household individual consumptions add up to ’more’ than total purchases. Again,
such economies of scales are not directly observed, and must also be identified.
In our framework, couples are characterized by two specific features aimed at
addressing these problems. One is the sharing rule, that drives the allocation of
resources within the household. The concept of a sharing rule is directly borrowed from the ’collective’ approach to household behavior, and specifically on
the assumption that household decisions are Pareto efficient (see Chiappori 1992
and Browning and Chiappori 1998). Considering the household as a small, open
economy, we know from the second welfare theorem that any efficient outcome
can be decentralized as an equilibrium, possibly after lump sum transfers between
members. The sharing rule summarizes those transfers, and can thus be seen
as a reduced form of the decision process taking place within the household.4
In particular, intrahousehold based equivalence scales does not impose particular
assumptions on intrahousehold distribution of resources (e.g., that all household
members are equally well off); on the contrary, they are fully compatible with bargaining models, and generally any sophisticated representation of the household
allocation process (provided it does not violate efficiency).
The second feature characterizing couples, that is original to the present paper
and plays a key role in our approach, is a consumption technology function. This
function captures the main issues (joint consumptions, economies of scale, etc.)
that are traditionally associated with consumption in households having more than
one member. Specifically, we propose a collective model of household consumption
behavior in which z, the n vector of purchased quantities of goods, is transformed
into an n vector of private good equivalents x. These private good equivalents
are then divided up between the household members via a household welfare or
bargaining function, with each member deriving utility from consuming their share
of x. The transformation from z to x (the household’s consumption technology)
embodies all of the technological economies of scale and scope that result from
living together. The distinction between public and private goods in a household
in the existing collective models literature (see, e.g., Vermeulen 2000 for a survey)
is a special case of a consumption technology function.
4
Note that neither the sharing rule nor our new definition of equivalence scales depend on a
particular cardinalization of individual utility functions.
In this collective framework, one can define an intrahousehold based equivalent
income (or expenditure) to be the income or total expenditure level y i∗ required
by household member i, when living alone, to attain the same standard of living
(i.e., the same indifference curve over goods) that he or she attains while living in
the household. Member i’s intrahousehold based equivalence scale is then y/y i∗ ,
the ratio of the household’s total expenditures to member i’s intrahousehold based
equivalent total expenditures. To see the appropriateness of intrahousehold based
equivalent income, consider the question of determining an appropriate 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 insurance policy that pays enough to permit spending y f ∗ dollars per year.
Similarly, in cases of wrongful death, juries are instructed to assess damages both
to compensate for the loss in ”standard of living,” (i.e., y i∗ ) and, separately for
”pain and suffering,” which would presumably be noneconomic effects (see Lewbel
2002).5
Throughout the paper, we assume that the household consumption technology
function is linear. Our main result is that, given observable data on the demands
of couples and singles, the proposed model is generically identified. In particular, the household’s sharing and consumption technology functions are identified.
Identification and estimation of the relative bargaining power of the members and
household economies of scale in consumption directly follow. This result is important per se, but also because it shows that, unlike traditional equivalence scales,
our new definition of the concept of a household equivalence scale is nonparametrically identifiable from observed behavior. It also shows that our household
economies of scale and bargaining power measures are similarly nonparametrically
identified.
We also provide duality results that yield a convenient way to parameterize
the household’s behavior for empirical analysis, analogous to the use of indirect
utility functions to conveniently parameterize ordinary empirical demand functions. These duality results include specification of an income sharing function in
5
In addition to its role in equivalence scales identification, the household consumption technology function can also be used to construct measures of the economies of scale that results
from joint consumption within the household. When a couple buys the bundle z at a cost pe0 z,
it is equivalent in terms of consumption to two singles living apart who in total buy the larger
bundle x, which would cost pe0 x, where x solves z = F (x). Distinct from individual member
equivalent incomes used for intrahousehold based equivalence scales, we can define the household equivalent income as p0 x, and the economies of scale from joint consumption as pe0 z/e
p0 x,
which equals the cost savings, in percentage terms, that results from shared consumption.
place of the direct bargaining or social welfare function, and specification of the
consumption technology in a form that is analogous to Barten (1964) scales or
Gorman’s (1976) general linear technology.
Finally, we provide an empirical application of our model using the Canadian
FAMEX survey. The results are used to calculate estimates of equivalence scales
and economies of scale of married couples versus their single counterparts.
The next section describes the basic framework and summarizes the main theoretical results of the paper, for the most part using the special case of linear
technologies for clarity. Later sections then provide more formal derivations, describe our empirical implementation of the model, and discuss possible extensions
and variants.
2. The Model and Applications
This section describes the basic form of the proposed household model, along with
its identification, specification, and applications. The purpose of this section is
expository, so results will be stated here without proofs. Formal derivations and
proofs are provided later.
2.1. The basic framework
To keep notation simple, let superscripts refer to household members and subscripts refer to goods. Let U i (xi ) be the direct utility function for a consumer i,
consuming the vector of goods xi = (xi1 , ..., xin ). We consider households consisting
of two members, which we will for convenience refer to as the husband (i = m)
and the wife (i = 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.
Singles When living alone, household member i maximizes a monotonically increasing, continuously twice differentiable and strictly quasiconcave utility function U i (z i ) when facing the n vector of prices pe with income level y i . Hence i
solves the optimization program
U i (z i )
max
i
z
pe0 z i = y i
(P i )
Let z i = hi (e
p/y i ) denote the solution to this individual optimization program, so
the vector valued function hi is the set of Marshallian demands of member i when
living alone. Define V i by
p/y i ) = U i [hi (e
p/y i )]
V i (e
(1)
so V i is the indirect utility function of member i when living alone. The functional
form of individual demand functions hi can be obtained from a specification of
V i using Roy’s identity. The Marshallian demand functions hm and hf may be
estimated using ordinary demand data on observed prices, total expenditures, and
quantities purchased for individuals living alone.
Households: decision process Now consider a household consisting of a couple living together, and facing the budget constraint y ≥ pe0 z, where pe is a vector of market prices. Following the standard collective approach, our key assumption regarding decision making within the household is that outcomes are
Pareto efficient. This is equivalent to the existence of some increasing function
e f (xf ), U m (xm ), p] that is maximized under budget and technology constraints.
U[U
e is twice differentiable. It is
In what follows, we assume for convenience that U
e may in general depend on prices. Thus, U
e can be interimportant to note that U
preted as a social welfare function for the household, in which individual weights
e may stem from some specific bargaining
may vary with prices. Alternatively, U
model (Nash bargaining for instance). Also, all of the above defined functions
may depend on other variables that we have suppressed for notational simplicity.
e and F could deFor example, the bargaining and sharing technology functions U
pend on relative wages of the household members, and both these and the utility
functions U f and U m might depend on demographic characteristics.
A standard result of welfare theory is that one can, without loss of generality,
e as µ(p)U f (xf ) + U m (xm ) for some function µ(p). The Pareto
write the function U
weight µ(p) can be seen as a measure of f ’s influence in the decision process: the
larger the value of µ is, the greater is the weight that member f ’s preferences
receive in the resulting household program, and the greater will be the resulting
private equivalent quantities xf versus xm . However, a difficulty with using µ as
a measure of the weight given to (or the bargaining power of) member f is that
the magnitude of µ will depend on the arbitrary cardinalizations of the functions
U f and U m .
The Consumption Technology Function As stated in introduction, the
household purchases some bundle z, but individual consumptions add up to some
other bundle x. In the technological relation z = F (x), we can interpret z as the
inputs and x as the outputs in the household’s consumption technology, described
by the production function F −1 , though what is being ’produced’ is consumption.
This framework is similar to a Becker (1965) type household production model,
except that instead of using market goods to produce commodities that contribute
to utility, the household essentially produces the equivalent of a greater quantity
of market goods via sharing.
We assume throughout the paper that the consumption technology is linear:
F (x) = Ax + a
(2)
where A is a nonsingular n by n matrix and a is a n vector. Then the unobserved
n vectors xf , xm , and x are private good equivalent vectors, that is, they are
respectively the quantities of transformed goods that are consumed by the female,
the male, and in total.
Consider some simple examples. Let good j be food. Suppose that if an
individual or a couple buy a quantity of food zj , the then total amount of food
that the individual or couple can actually consume (that is, get utility from) is
zj − a, where a is waste in food preparation, spoilage, etc.,. If the individuals
lived apart, each would waste an amount a, so the total amount wasted would
be 2a, while living together results in only a waste of a. In this simple example
the economies of scale to food consumption from living together is a reduction in
waste from 2a to a, implying that xj = zj + a, so the consumption technology
function for food takes the simple form zj = Fj (x) = xj − a.
Economies of scale arise directly from sharing. For example, let good j be automobile use. If xfj and xm
j are the distances traveled by each household member
in the car (or some consumed good that is proportional distance, perhaps gasoline), then the total distance the car travels is z = (xfj + xm
j )/(1 + r), where r is
the fraction of distance that the couple ride together, which yields a consumption
technology function for automobile use of zj = Fj (x) = xj /(1 + r). This example
is similar to the usual motivation for Barten (1964) scales, but it is operationally
more complicated, because Barten scales fail to distinguish the separate utility
functions, and hence the separate consumptions, of the household members.
More complicated consumption technologies can arise in a variety of ways. The
fraction of time r that the couple share the car could depend on the total usage,
resulting in Fj being a nonlinear function of xj . There could also be economies
(or diseconomies) of scope as well as scale in the consumption technology, e.g.,
the shared travel time percentage r could be related to expenditures on vacations,
resulting in Fj (x) being a function of other elements of x in addition to xj .
Consumption technologies can also embody some differences in preferences
between singles and households. For example, married couples may tend eat at
home, and to prepare meals from scratch, more often than singles do. This could
be because couples desire such meals more than singles, or it could be due to
economies in food preparation. Either effect would be included in the consumption
technology.
A limitation of this framework is that it assumes that the total private good
equivalents bundle x that results from a purchased bundle z is not a function of
how x is divided amongst the household members. This assumption greatly simplifies model identification, specification, and interpretation by separating bargaining
from consumption economies of scale. This assumption is consistent with the food
example, where economies arise from reduced waste relative to total consumption.
The assumption is less plausible in the car example, but it should still be at least
a reasonable approximation if, for goods with large scale economies (i.e., when
f
xj /zj is large), the relative division of the good, xm
j /xj , does not differ greatly
from one. The model is certainly more general than most existing collective models, which assume that every good is either completely private, implying z = x,
f
or completely public, implying z = xm
j = xj (see, e.g., Chiuri and Simmons 1997
and references therein). The model also permits the technology function F to
depend on observable variables other than p that affect sharing or bargaining, for
example, F can be a function of characteristics of the family members, such as
the amount of time that the husband is out of town for work or the relative wages
of the husband and wife.
The basic program Our proposed model of this household’s consumption behavior is thus the optimization program
e [U f (xf ), U m (xm ), p]
max U
(P∗ )
xf ,xm ,z
x = xf + xm ,
z = F (x),
p0 z = 1
where z is the vector of quantities of the n goods the household purchases, p = pe/y
where pe is prices and y is the household’s total expenditures.
The solution of the program (P∗ ) yields the household’s demand functions,
which we denote as z = h(e
p/y) = h(p), and also private good equivalent demand
functions xi = xi (p). The vector valued demand function z = h (p) can be estimated using ordinary demand data on observed prices, incomes, and quantities
purchased by the couple.
One way to interpret the program P∗ is to consider two extreme cases. If
all goods were private and there were no shared or joint consumption, then
F (x) = x, so z would equal xf + xm and the program P ∗ would reduce to
e f (xf ), U m (z − xf ), p] such that p0 z = 1, which is exactly equivalent
maxxf ,z U[U
to a standard specification of the collective model (see, e.g., Bourguignon and
Chiappori 1994 or Vermeulen 2000). At the other extreme, imagine a household
that has a consumption technology function F , but assume that the household’s
utility function for transformed goods just equaled U m (x), as might happen if the
male were a dictator that forced the other household member to consume goods
in the same proportion that he does. In that case, the model would reduce to
maxz U m [F −1 (z)] such that p0 z = 1, which for linear F is equivalent to Gorman’s
(1976) general linear technology model, a special case of which is Barten (1964)
scales (corresponding to A diagonal and a zero). Our general model, program P ∗ ,
combines the consumption technology logic of the Gorman or Barten framework
with the collective model of a household as either a bargaining or a social welfare
maximizing group.
Assuming that U m and U f are the utility functions of singles, the model
permits changes in utility to result from living together. Specifically, arbitrary
changes in utility that are separable from goods consumption are allowed, e.g.,
one could be generally happier as a result of living with a spouse, and a wife’s
utility could depend on both her own attained utility level over goods (the value
e In addition,
of U f ) and on U m . These effects are absorbed into the function U.
as will be discussed later, the consumption technology function F can incorporate
some kinds of taste changes that result from living together.
For our identification and empirical results, we use data on single men and
women to estimate the Marshallian demand functions hf and hm arising from
the utility functions U m and U f , and the married couple’s demand functions
z = h(p) derived from the program P∗ . These estimates are combined to identify
the consumption technology function F and relevant features of the bargaining
e which in turn yields estimates of the private good
or social welfare function U,
f
m
equivalents x and x , the sharing rule, economies of scale from consumption, and
the intrahousehold based equivalence scales associated with any price and income
regime.
The Form of Household Demands An equivalent formulation of the problem,
that is standardly used in collective models, relies on the notion of a sharing
rule. Consider the household as an open economy. From the second welfare
theorem, any Pareto efficient allocation can be implemented as an equilibrium
of this economy, possibly after lump sum transfers between members. Transfers
can be summarized by the sharing rule η(p), describing the fraction of household
resources consumed by member f . The household’s behavior is equivalent to
allocating the fraction η f = η(p) to member f , and the fraction η m = 1 − η(p) to
member m. Each member i then maximizes their own utility function U i given the
shadow price vector π and their own shadow income η i to calculate their desired
private good equivalent consumption vectors xf and xm .
Formally:
Proposition 1. Define π(p) by
π(p) =
A0 p
1 − a0 p
Then there exists a scalar valued sharing rule η(p), with 0 < η(p) < 1, such that
the solutions to program P ∗ satisfy
µ
¶
π(p)
f
f
x (p) = h
η(p)
¶
µ
π(p)
m
m
x (p) = h
(3)
1 − η(p)
h(p) = F [xf (p) + xm (p)]
µ
µ
¶
¶
A0 p
A0 p
1
1
f
m
= Ah
+ Ah
+a
1 − a0 p η(p)
1 − a0 p 1 − η(p)
Conversely, given any regular Marshallian demand functions hm (e
p/y m ) and hf (e
p/y m ),
and any function η(p) that always lies between zero and one, there exists a utility
e (or, equivalently, a Pareto weight µ (p)) such that the solutions to the
functin U
corresponding program P ∗ are given by (3).
Proof. See Appendix
Here, π (p) denotes the equilibrium (shadow) prices within the household economy; note that the above definitions make shadow income π 0 (xf + xm ) = 1. Given
estimates of scaled shadow prices π and private good equivalents xf and xm , the
sharing rule η is given by η = π0 xf .
Clearly, η(p) can be seen as a direct measure of the weight given to member
f is the household decision process. There is a one to one, strictly monotonic
relationship between µ(p) and η(p), that can be written:
µ(p) =
∂V f π
( )
∂y η
− ∂V m π
( )
∂y 1−η
(see Appendix 2 for a proof)
An advantage of the measure η is that (unlike µ) it does not depend on any
chosen cardinalization of utility. Also, unlike Browning and Chiappori (1998), who
find using household data alone that relative bargaining power measures can only
be identified up to an arbitrary location, we show in our model that by combining
data from households and from singles living alone, η is completely identified.
2.2. Applications
A Measure of Economies of Scale in Consumption Previous attempts to
measure economies of scale in consumption have required very restrictive assumptions regarding the preferences of household members (see, e.g., Nelson 1988). In
contrast, given estimates of the private good equivalents vector x = xf + xm from
our model, we may calculate 1/p0 x = y/e
p0 x, which is a measure of the overall
economies of scale from living together, since y is what the household spends to
buy z and pe0 x would be the cost of buying the private good equivalents of z.
Technically, this economy of scale measure takes the form
y
1
= 0 f
0
pe x p (h [π/η] + hm [π/(1 − η)])
where π = A0 p/(1 − a0 p).
The measure relates the household’s total expenditures to the total cost of
independently purchasing the same individual consumptions. It is important to
note that this measure does not directly provide an estimation of adult equivalent
scales. The reason is that, in general, the shadow prices used within the household
are not proportional to market prices (this is exactly Barten’s intuition). It follows
that individuals, on their own, would not buy the bundles xf and xm : they could
in general reach a higher level of utility by purchasing and consuming a different
combination.
Adult Equivalence Scales Given each household member i’s private good
equivalents xi , we define member i’s intrahousehold based equivalent income, y i∗ ,
as the minimum expenditures required to buy a vector of goods that is on the same
member i indifference curve as xi . The ratio of y/y i∗ is then our intrahousehold,
collective model based adult equivalence scale, since y is what the household
spends to get member i onto the same indifference curve over goods that he or
she could attain while living alone with an income level of y i∗ . Like the above
described economies of scale and bargaining power measures, these equivalence
scales do not depend on arbitrary utility cardinalizations.
Recall that V i (e
p/y i ) is the indirect utility function of member i when living
alone, and the private good equivalent vector consumed by member i while in the
household is xi (p) = hi [π(p)/η i (p)], where η f (p) = η(p) and η m (p) = 1 − η(p).
Then y i∗ (e
p, y) is defined as the solution to the equation
µ
µ
¶
¶
pe
π(e
p/y)
i
i
V
=V
(4)
y i∗ (e
p, y)
η i (e
p/y)
Note that this definition is ordinal, i.e., it does not depend on the chosen
cardinalization for member i’s utility function. In an application, functional forms
for V i and η i would be directly specified, and in the example of a linear technology
we have π(e
p/y) = A0 pe/(y − a0 pe).
p, y), we may calculate equivalence scales y/y i∗ (e
p, y) or the difference
Given y i∗ (e
in income required, y−y i∗ (e
p, y). For example, at prices pe, y would equal the amount
of money that the household must spend so that the man in that household has
the same standard of living (i.e., attains the same indifference curve over goods)
that he would attain if he were to live alone with total expenditures y m∗ (e
p, y).
This could be applied to the calculation of poverty lines for different households.
Similarly, in the case of a wrongful death or insurance calculation, a woman (or
a mother and her children, if V f is defined as a joint utility function of a mother
and her children) would need income y f ∗ (e
p, y) to attain the same standard of
living without her husband that she (or they) attained while in the household
with the husband present and total expenditures y. This would be be sufficient to
compensate for the loss of economies of scale and scope from shared consumption
(and for any changes in preferences over goods that result from living in the
household), but would not account for grieving, loss of companionship, or other
components of utility that are assumed to be separable from consumption of goods.
Another interesting equivalence scale to construct might be y f ∗ /y m∗ , the ratio
of how much income a woman needs when living alone to the income a man
needs when living alone to make each as well off as they would be in a household.
Interestingly, even if men and women had identical preferences, this ratio need
not equal one in general, because for example if the sharing rule η is bigger than
one half, then the wife receives more than half of the household’s resources, and
hence would need more income when living alone to attain the same standard of
living.
To separate bargaining effects from other considerations, other measures of
intrahousehold based equivalent income could be calculated. For example, the
ratio y f ∗ /y m∗ might better match the intuition of an equivalence scale if each y i∗
equation were calculated as the solution to
µ
µ
¶
¶
pe
π(e
p/y)
i
i
V
=V
(5)
y i∗ (e
p, y)
1/2
which gives equivalent incomes assuming equal sharing of resources.
In other applications, one might want to consider the roles of equivalent income
and sharing rule jointly. For example, given poverty lines for singles, one might
define the corresponding poverty line for the couple as the minimum y such that,
by choosing η optimally, each member of the couple would have an equivalent
income y i∗ equal to his or her poverty line as a single.
2.3. Identification
In our context, the demand functions of singles and the aggregate demands of
household are observable. The identification question is thus, given the observable
demand functions hm , hf , and h, can the consumption technology function F (i.e.,
matrix A and vector a) and the private good equivalent demand functions xf (p)
and xm (p) be estimated? We show that these functions are ’generically’ identified,
meaning that identification will only fail if the utility and technology functions are
too simple (for example, F is not identified if demands are of the linear expenditure
system form). This identification in turn implies (with some caveats) identification
of features of the model including the sharing rule, economies of scale, and adult
equivalence scales.
A question closely related to identification is, given functional forms for the
technology, the sharing rule, and the demands of singles, what is the implied
functional form for household demands? This also is generically identified, and
has a simple, explicit solution in the case of linear consumption technologies.
Uniqueness of the consumption technology and sharing rule We first
investigate whether, given the observable Marshallian demand functions of singles
and households, one can in general uniquely recover the private good equivalent
functions xm and xf , the linear consumption technology F, and the sharing rule
η. In general, the answer is positive, provided that n ≥ 3.
To see why, take some given (finite) set of price vectors p1 , ..., pT where T
can be chosen conveniently large. For any pt , let z t denote the corresponding
household demand; from Proposition 1, we know that
µ
µ
¶
¶
1
1
A0 pt
A0 pt
t
f
m
+ Ah
+ a, t = 1, ..., T ((Id))
z = Ah
1 − a0 pt η(pt )
1 − a0 pt 1 − η(pt )
This provides, for each t, (n − 1) independent equations, hence a total of
(n − 1) .T equations. The unknowns, on the other hand, are the matrix A, the
vector a and the scalars η t = η(pt ), t = 1, ..., T ; hence a total of n2 + n + T
unknowns. If n ≥ 3, then whenever T ≥ n + 10, the number of equations strictly
exceeds the number of unknowns, and the system is identified. This allows recover
the consumption technology. Finally, once A and a are known, the sharing rule
η (p) is identified for all p from the equation
µ
µ
¶
¶
A0 p
A0 p
1
1
f
m
z (p) = Ah
+ Ah
+a
1 − a0 p η(p)
1 − a0 p 1 − η(p)
The private good equivalent functions xm and xf follow, since
µ
¶
A0 p
1
f
f
x (p) = h
1 − a0 p η(p)
and
µ
¶
1
A0 p
x (p) = h
1 − a0 p 1 − η(p)
Clearly, the result is only generic, because for particular functional forms equations (??) may fail to be linearly independent. The reader may check, for instance,
that this is the case when the individual demands hf and hm belong to the LES
class. However, such problems do not appear for for flexible enough functional
forms. This claim is illustrated in the next subsection for the QUAIDS form,
which will be used for our empirical estimation.
m
f
The case of QUAIDS individual demands For our empirical application,
we use a parametric approach. Instead of specifying the structural model as in
program (P∗ ), the following method can be used to construct functional forms for
estimation. First, choose ordinary indirect utility functions for members m and f ,
and let hm and hf be the corresponding ordinary Marshallian demand functions.
Next choose a functional form for the sharing rule η, which could simply be a
constant, or a function of measures of bargaining power such as relative wages
of the household members. This sharing rule function must lie between zero and
one. Proposition 1 then provides the resulting functional form for the household
demand function h(p), and ensures that a corresponding household program exists
that rationalizes the choice of functions hm , hf , and η.
In practice, we will assume singles have preferences given by the Integrable
QUAIDS demand system of Banks, Blundell and Lewbel (1997). For i = f or m,
let wi = ω i (e
p/y i ) denote the n-vector of member i’s budget shares wki (k = 1, ..., n)
when living as a single, facing prices pe and having total expenditure level y i . The
QUAIDS demand system we estimate takes the vector form
¤
¤2
£ ¡ ¢
λi £ ¡ i ¢
ω i (e
ln y − ci (e
p/y i ) = αi + Γi ln pe + β i ln y i − ci (e
p) + i
p)
b (e
p)
(6)
where ci (e
p) and bi (e
p) are price indices defined as
1
ci (e
p) = δ i + (ln pe)0 αi + (ln pe)0 Γi ln pe
2
ln[bi (e
p)] = (ln pe)0 β i .
(7)
(8)
Here αi , β i and λi are n-vectors of parameters, Γi is an n×n matrix of parameters
and δ i is a scalar parameter which we take to equal zero, based on the insensitivity
reported in Banks, Blundell, and Lewbel (1997). Adding up implies that e0 αi = 1
and e0 β i = e0 λi = Γi e = 0 where e is an n-vector of ones. Homogeneity implies
that Γi0 e = 0 and Slutsky symmetry is equivalent to Γi being symmetric. The
above restrictions yield the integrable QUAIDS demand system, which has the
indirect utility function
#−1
µ ¶ "µ
¶−1
i
i
)
−
c
(e
p
)
p
e
ln
(y
=
Vi
+ λi0 ln (e
p)
(9)
yi
bi (e
p)
for i = f and i = m. The singles demand functions ω i (e
p/y i ) in equation (6) are
obtained by applying Roy’s identity to equation (9). Deaton and Muellbauer’s
(1980) Almost Ideal Demand System (AIDS) is the special case of the integrable
QUAIDS in which λi = 0.
With these restrictions, these singles demand functions ωi (e
p/y i ) can be obtained by applying Roy’s identity to the Integrable QUAIDS indirect utility function
Proposition 2. Assume singles have preferences given by the AIDS or integrable
QUAIDS, and that couples have a linear household technology. Assume β f 6= β m
and each element of β f , β m , and the diagonal of A is nonzero. Then the household
technology and the sharing rule are identified.
Proposition 2 confirms that the AIDS and QUAIDS models for single’s preferences, with a linear household technology, are sufficiently nonlinear to permit
identification of all the components of the couple’s model, as discussed in Theorem
1, and moreover, that identification here does not require a monotonic or exclusive
good. The assumption regarding nonzero and unequal elements in Theorem 3 can
be relaxed. See the proof of Theorem 3 in the Appendix for details.
3. Additional Results
3.1. Barten and Gorman Scales
Gorman’s (1976) general linear technology model assumes that household demands are given by
µ
¶
A0 p
m
z = Ah
+a
(10)
1 − a0 p
Barten (1964) scaling (also known as demographic scaling) is the special case of
Gorman’s model in which a = 0 and A is a diagonal matrix. See also Muellbauer
(1977). Demographic translation is Gorman’s linear technology with A equal
to the identity matrix and a non zero, and what Pollak and Wales (1992) call
the Gorman and reverse Gorman forms have both A diagonal and a nonzero.
These are all standard models for incorporating demographic variation (such as the
difference between couples and individuals) into demand systems. The motivation
for these models is identical to the motivation for our linear technology F , but
they fail to account for the structure of the household’s program. Even if the
household members have identical preferences (z f = z m ) and identical private
equivalent incomes (η = 1/2), comparison of equations (??) and (10) shows that
household demand functions will still not actually be given by the Gorman or
Barten model. In fact, comparison of these models shows that household demands
will take the form of Gorman’s linear technology, or some special case of Gorman
such as Barten, only if either one household member consumes all the goods (η is
zero or one), or if demands are linear in prices (i.e., the linear expenditure system).
3.2. Technology and Preference Changes
The consumption technology function F may incorporate some changes in preferences resulting from living in a household versus living alone, as well as purely
technological economies of scale or scope in consumption (i.e., sharing). This
means that some calculations, in particular economies of scale and adult equivalence scales, must be interpreted with caution. For example, a change in taste
for dancing as a result of marriage could show up as economies or diseconomies
of scale in the consumption of dancing shoes.
One way to address this issue could be to explicitly incorporate parameters to
measure taste changes in addition to the household technology parameters. To
illustrate this point, let the mapping from market prices to shadow prices be given
by:
π k = Lk Ak pk
(11)
where Ak ∈ [0.5, 1] is a technological transformation parameter with value 0.5 for
a purely public good and value 1 for a purely private good, and the parameter Lk
captures common taste changes over the ‘private good equivalent’ xk = (zk /Ak ).
If Lk = 1 then tastes for good k are unchanged. If Lk < 1 then the individuals in
the partnership like good k more than when they are single.
In this example, the household consumption technology is of the Barten type
z = Ax where A is diagonal, and the utility function of an individual i living alone
is, as before, U i (z i ), but now in addition the individual’s tastes change in a purely
Barten way, from U i (z i ) before marriage to U i (Lz i ) aftern marriage, where L is
diagonal. The household’s optimization function is then
e f (Lxf ), U m (Lxm ), pe/y]
max U[U
xf ,xm ,z
z = A(xf + xm )
pe0 z = y
and the resulting household demands are exactly as before, except that everywhere
we had Ak we will now have Lk Ak . Without additional information, this model is
observationally equivalent to our original model with technology z = LA(xf +xm ).
In estimation we cannot separately identify the Ak ’s and the Lk ’s but only their
product. Nonetheless, we can interpret an estimated value for the product Lk Ak
of less than 0.5 or greater than 1 as partially reflecting taste changes. If we are
prepared to assert a priori that some good is purely private, (e.g., clothing) then
this serves to identify the change in tastes for that good.
Identifying information could also come from measurement of some sharing
within the household, e.g., direct observation of the percentage of time that couples use their car jointly. Additional economic restrictions or functional form
assumptions could provide information for separating taste changes from true
economies of sharing. For example, a sensible constraint on L would be that, if
Ak were set equal to one for all goods k, then the estimated economies of scale
(which will be a function of L) should equal to one.
More generally, if we do not try to separate taste changes from the technology function F , then to interpret the model’s estimates of economies of scale as
being due to the technology of sharing and not on taste changes, it must be assumed that the dollar effect of any change in tastes for goods is small, so that F
largely represents true technological changes in consumption. For example, joint
consumption of heating is likely to have a much larger effect on measured cost
savings of living together than on any change in tastes for heat.
The same caveat applies to our intrahousehold based adult equivalence scales,
if they are to be interpreted as purely based on consumption technology. However, many applications of equivalence scales deal with appropriate compensation
for a change in status, for example, calculation of the appropriate level of life
insurance on a spouse, or compensation in legal cases of wrongful death. In these
circumstances, it may be completely appropriate to compensate for both changes
in tastes and the loss of purely technological economies of scale in consumption,
and there would be no need to separately identify each. This would still exclude
noneconomic effects such as grief and loss of companionship.
The portion of F that corresponds to taste changes versus consumption technology is also largely irrelevant for calculations such as demand elasticites, or the
interpretation of the sharing rule η as a measure of bargaining power.
Future work may lead to additional data or alternative assumptions that allow
for more complete separation of taste changes and purely technological economies
from sharing. While our particular identifying assumptions are open to debate
and (we hope) improvement, we believe that in any case the model we provide
here is an appropriate framework for analyzing these issues.
4. Empirical Application
From an empirical perspective, one may directly specify a functional form for η
e which greatly simplifies empirical implementation of the model
instead of for U,
and estimation of relative bargaining power. As a result, in our empirical implementation the sharing rule η is directly parameterized and estimated as part of
the household’s demand functions. The simplest specification simply lets η equal
a constant to be estimated. More generally, η is specified as a function of variables
that affect bargaining power, such as the relative incomes and other demographic
characteristics of the household members.
Given estimates of ordinary demand functions hf and hm from singles data,
equation (??) may then be estimated from household demand data, where the
parameters to be estimated are the technology parameters A and a, and any
parameters in the sharing rule η(p).
4.1. Data
We use Canadian FAMEX data from 1974, 1978, 1982, 1984, 1986, 1990 and 1992.
The Canadian FAMEX is a multistaged stratified clustered survey that collects
information on annual expenditures, incomes, labour supply and demographics for
individual (‘economic’) households. The survey is not nationally representative.
In particular, in most years only information from respondents living in large cities
is collected (hence the hgih proportion of city dwellers in our samples below). The
survey is run in the Spring after the survey year (that is, the information for 1978
was collected in Spring 1979). All of the information is collected by interview so
that the expenditure and income data are subject to recall bias. Although this
may give rise to problems, the FAMEX surveying method has the great advantage
that information on annual expenditures is collected. Thus the FAMEX has much
less problem with infrequency bias than do surveys based on short diaries. It is
also the case that since the survey year coincides with the tax year (January to
December) the income information is thought to be unusually reliable since it is
collected at about the time that Canadians are filing their (individual) tax returns.
These are often explicitly referenced by the enumerators.
Prices are taken from Statistics Canada. When composite commodities are
created, the new composite commodity price is the weighted geometric mean of
the component prices (a Stone price index) with budget shares averaged across the
strata (couples, single males and single females) for weights. Thus, the weights
are not the individual household budget shares.
We have three strata: single females, single males and couples with no on else
present in the household. We sample only younger agents: the single females and
wives in couples are aged less than or equal to 42; single males and husbands are all
aged less than or equal to 45. All agents are in full year full time employment and
we have excluded any household with non—negative net income or non-negative
individual gross incomes. This results in sample sizes of 1393, 1574 and 1692 for
single females, single males and couples respectively. The (non-durable) goods we
model are: food at home, restaurant expenditures, clothing, recreation, transport,
services and vices (alcohol and tobacco).
Single Single
Couples
females males Husband Wife
Net income*
26, 137 30, 890
56, 481
Total expenditure*
11, 855 13, 645
23, 259
Wife’s share of gross income
0.43
bs(food at home)
0.18
0.16
0.19
bs(restaurant)
0.11
0.15
0.11
bs(clothing)
0.16
0.09
0.14
bs(recreation)
0.11
0.13
0.11
bs(transport)
0.21
0.25
0.25
bs(services)
0.17
0.10
0.12
bs(vices)
0.07
0.12
0.08
car
0.64
0.78
0.95
home owner
0.13
0.23
0.57
city dweller
0.85
0.81
0.83
age
29.9
31.1
30.1
28.1
higher education
0.20
0.25
0.21
0.19
Francophone
0.19
0.17
0.21
0.18
Allophone
0.08
0.08
0.09
0.07
White collar
0.43
0.40
0.39
0.37
Notes. * mean for 1992 (when prices are unity).
These data possess some significant features. Income and total expenditures
are lowest for single females, highest for couples. Car ownership and home ownership are significantly higher for couples than for singles. Total expenditure on
non-durables is between 41% and 45% of net income. Single males have significantly higher budget shares for restaurant, transport and vices than single
females. Single females have higher budget share for food at home, clothing and
services. Of course, this does not necessarily mean that single men and women
have different preferences since these are conditional on demographics and real total expenditure which differ across the two strata. Below we shall present tests for
whether single men and women have the same preferences. Apart from transport
and services, couples’ budget shares resemble single female’s shares much more
closely than those of single males.
4.2. Barten Technology, Overheads, and Budget Shares
Our empirical application will use a Barten type technology function with overheads, and express demands in budget share form. This technology function is
defined as
zk = Ak xk + ak
(12)
for each good k, and so is equivalent to the general linear technology z = Ax + a
when the matrix A is diagonal. The shadow prices for this technology are
πk =
Ak pek
y − a0 pe
(13)
where the couple faces prices pe and has total expenditure level y. This technolp/y)
ogy function is particularly convenient for budget share models. Let wk = ωk (e
be the budget share of good k for the household, so ω k (p) = pek hk (e
p/y)/y. Similarly, let wki = ω ik (e
p/y i ) be member i’s budget share of consumption of good k
when living as a single, for i = f or m. With the technology function (12) and
corresponding shadow prices (13), based on Theorem 2 we obtain the following
simple expression for the couple’s budget shares
µ ¶
µ
¶·
¶¸
µ
π
π
ak pek
a0 pe
f
m
ηωk
+ (1 − η)ωk
+
(14)
ω k (p) = 1 −
y
η
1−η
y
Apart from adjusting for overheads, equation (14) shows that the budget shares
of the couple equal a weighted average of the budget shares of the members, with
weights given by the income sharing rule η and 1 − η.
Couple’s budget shares may be estimated by substituting into equation (14)
the estimated single’s budget share functions ωik , the shadow price equation (13),
and a specification for the income sharing rule η. The singles data identify the
parameters of ω ik , so the only additional parameters that require the household
data are the parameters in A, a, and η. For efficiency, the singles and couples
models will be estimated simultaneously rather than sequentially.
Given estimates of the budget share systems for singles, wki = ω ik (e
p/y i ), and for
households (equation 14), the private good equivalent quantities for each household member may be calculated as
µ ¶
η f π
f
xk = ωk
(15)
πk
η
µ
¶
π
1−η m
m
,
(16)
ω
xk =
πk k 1 − η
the economies of scale to consumption are
y
pe0 z
=
,
pe0 x pe0 (xfk + xfk )
(17)
and adult equivalence scales are obtained as follows. Let V i (e
p/y i ) be the indirect
utility function (up to any arbitrary monotonic transformation) that gives rise to
member i’s budget share demand functions wk = ω k (e
p/y i ). Then the minimum
expenditure required by member i, while living alone, to attain the same indifference curve over goods that he or she attains as a member of the household is the
number y i∗ that solves the equation
µ
¶
µ ¶
pe
π
f
f
V
(18)
=V
f
∗
y
η
for i = f , and
µ
¶
µ
¶
π
V
=V
(19)
y m∗
1−η
for i = m. We may then calculate the male adult equivalence scale for the household as y/y m∗ and for a female as y f ∗ /y m∗ . Alternatively, to separate issues of
bargaining power from other considerations, we could evaluate these equations
and the resulting equivalence scales substituting η = .5 into equations (18) and
(19).
m∗
Note in all of the above that π, η, xfk , xm
, and y f ∗ are implicitly functions
k , y
of p = pe/y, and of demographic and other characteristics that affect individual
preferences and bargaining.
m
pe
m
4.3. Functional Form
For our application, we assume singles demands have the integrable QUAIDS
functional form described in equations (6) and (9). The vectors of parameters αi
and β i in this model are specified to depend on demographics. Specifically, we
take:
Mα
X
i
i
αk = αk0 +
αikm dm
(20)
m=1
β ik
=
β ik0
+
Mβ
X
β ikm dm
(21)
m=1
where Mβ = 2, with the demographics dm being dummies for owning a car and
being a home owner. For the intercept demographics we have Mα = 13. These
demographics dm include four regional dummies; a dummy for owning a car, a
house ownership dummy and a dummy for living in a city. As well we include
individual specific demographics: age and age squared, a dummy for having more
than high school education, dummies for being French speaking or neither English
nor French speaking and an occupation dummy for being in a white collar job.
The model results in 24 parameters per good/equation for singles
We will also estimate a separate QUAIDS model for couples, to use for comparison to our formal household model. This comparison QUAIDS model is identical
to the above specifications for singles, except that the individual specific demographics for both members of the household are included, resulting in Mα = 19
and hence in 30 parameters per good/equation for couples.
Given the demand functions for singles, our household model for couples is
given by equations (13), (14), and specifications for the income sharing rule η and
the technology parameters Ak and ak , k = 1, ..., n. The Barton type scales Ak we
specify as
!
Ã
MA
X
(22)
Akm dem
Ak = exp Ak0 +
m=1
where MA = 3, with the demographics characteristics dem being dummies for couples car and home ownership (which differ greatly between singles and couples and
have a strong impact on transport and services expenditures), and on a dummy
for the couple having different education levels (where there are only two education levels: ‘high school or less’ and ‘more than high school’). The exponent form
ensures that the scales are positive.
We model overheads ak as constants. To ensure that a0 p remains less than
y, so that we can take logs of y − a0 p, we introduce a penalty function into the
optimization criterion. The optimal estimates are not close to the threshold.
Based on Browning, Bourguignon, Chiappori and Lechene (1994) and Browning and Chiappori (1998), our model for the sharing rule is
η = η0 + η1
³y´
Yf
+
η
ln
2
Yf +Ym
P
(23)
where Y i is the gross income of household member i and P is a Stone price
index for the couple. This sharing rule depends on the wife’s share of total gross
income (a potential measure of bargaining power) and on the couple’s total real
expenditures.
4.4. Estimation
Our estimation procedure proceeds in two stages. First we estimate the parameters of the system for single males and single females separately. We drop the last
equation to accomodate adding up and we impose homogeneity by including only
relative prices in each equation. To take acccount of the endogeneity of total expenditure we estimate by GMM. For instruments we take all of the demographics
plus log real net income, (defined as the log of nominal net income divided by
a Stone price index computed for our seven non-durables goods) and its square,
the product of real net income with the car and home ownership dummies and
absolute log prices. Thus there is one degree of over-identification for each equation. The χ2 (6) statistics for the over-identifying restrictions are 5.8 (probability
= 44%) and 12.4 (5.4%) for single females and single males respectively. We then
impose symmetry on both sets of estimates; the respective χ2 (15) statistics were
25.3 (4.6%) and 25.3 (4.6%). We denote the symmetry constrained sets of estimates by θ̂f and θ̂m for females and males respectively. Since so much of our
analysis is based on single men and women having different preferences, we present
tests for the sets of parameters being the same. For the six equations modelled we
have the following χ2 (24) statistics for equality: 52 (food), 118 (restaurant), 686
(clothing), 95 (recreation), 117 (transport) and 622 (services). Thus we decisively
reject that single men and women have the same preferences.
In the second step we take the singles’ estimates, θ̂f and θ̂m , the data on
couples and trial values for the parameters in equation (14). The latter are the
parameters of the sharing function η, the Barten terms A1 , ...An and the fixed
costs elements a1 ...an . Let ζ denote this set of parameters. From these we can
form predicted budget shares for each couple in our sample, based on their prices,
total expenditure and demographics, denote these ω̂ (ζ). We then minimise, with
respect to the trial parameter values, a weighted sum of squares of the difference
between the predicted budget shares and the actual budget shares, ω:
min (ω̂ (ζ) − ω)0 W (ω̂ (ζ) − ω)
ζ
(24)
For the weighting matrix W we use the weighting matrix from GMM estimates
of the QUAIDS system for couples:
W = (In−1 ⊗ Z)V̂ (In−1 ⊗ Z)0
(25)
where Z is an (H × g) matrix of instruments and V̂ is derived from the first round
of GMM estimates in the usual way. Testing is based on the minimised criterion
having a χ2 distribution with degrees of freedom equal to ((n − 1)g − dim (ζ)).
To derive the weighting matrix we estimate the QUAIDS for couples. For instruments we use the instruments used for singles (with individual specific values for
husbands and wives, where appropriate). We also include logs of the individual
gross incomes and the ratio of the wife’s gross income to total gross income (we
shall see why below). In all we have g = 34.
Before presenting estimates with our restrictions imposed, we graph the fits
for the different goods for the three strata. To do this we first estimate QUAIDS
system for each strata (with homogeneity and symmetry imposed) and then caculate predictions for agents who live in Ontario, living in city, owning a car, renting
their accmodation, aged 30, with high school education, Anglophone and all having a blue collar job. We predict for 1992, the year in whcih prices are close to
unity. For total expenditure we take a range from the first decile to the ninth
decile. For couples we divide total expenditure by 1.7, the scale recommended
by the OECD, to make the utility levels roughly constant (later we present.our
own estimate of the required scale). Figures 1 to 7 present the plots for our seven
goods.
4.5. Estimates of Couples Model
* THIS SECTION IS PRELIMINARY AND INCOMPLETE. THE REPORTED
RESULTS ARE NOT YET ITERATED TO CONVERGENCE*
We start with an extremely parsimonious specification and add extra parameters until we find a satisfactory fit. As a benchmark, we first estimate with
all scales Ak equal to one, all overheads ak equal to zero, and the sharing rule
η constant, so η = η 0 Our criterion is minimised at a value of η = 0.62 with a
criterion value of 1270. The criterion value for setting η = 0.5 (equal sharing) is
1583, so that we reject equal sharing for this simplest model (formally we have
a χ2 (1) statistic of 1583 − 1270 = 313). The value of η implies that the budget
shares of couples are somewhat closer to those of single females than to those of
single males.
The first model we estimate (model 1) has adds constant Barten terms to the
benchmark model; thus we have only the parameters η 0 , A01 , ...A0n . The results
are given in the first column of the Table below. As can be seen, adding the
Barten terms improves the fit dramatically: the χ2 (7) statistic for them all being
zero is 733. However we still reject the overall model.
Model 2 adds some characteristics to the sharing function, with η given by
equation (23). In the Table we present the value of η evaluated at the means of
its arguments. The fit improves a small amount; the χ2 (2) for excluding the three
new parameters is 6.5 which has a probability value of 3.9%. Consistent with this,
the mean value of η is unchanged and the other parameter estimates are not much
changed.
In model 3 we introduce fixed costs into model 2. The extra parameters
a1 , ..., an are highly significant (a χ2 (7) of 84) but induce only small changes in
the other parameters.
Model 4 introduces demographics into the Barten scales, with each Ak given
by equation (22). The new parameters are highly significant (a χ2 (21) of 109).
The Barten scale for ‘vices’ is rather high, but it is probably not significantly
different from a more reasonable value.
1
2
3
4
5
η̂
0.55 0.55 0.52 0.54
Â1 1.12 1.21 0.99 1.35
Â2 0.52 0.52 0.33 0.33
Â3 2.47 2.54 3.21 4.75
Â4 0.75 0.76 0.74 0.82
Â5 0.39 0.39 0.42 0.48
Â6 1.00 0.99 0.79 0.94
Â7 1.69 1.84 6.93 15.2
χ2 536.9 530.4 446.9 337.9
df
196
194
187
166
Additional models to try: restricting Barton scales to less than two, or restricting overheads to be nonnegative, allowing δ i (the intercept in the c functions) to
be nonzero and possibly contain demographics, estimating couples using an ordinary GMM weighting matrix, doing a full optimization to simultaneously estimate
the parameters for singles and couples, and trying some additional demographic
variables .
4.6. Implications of parameter estimates
* THIS SECTION IS PRELIMINARY AND INCOMPLETE *
We present some implications of our parameter estimates. We evaluate everything at the mean total expenditures in 1992 (see table 1) using unit prices. This
includes mean values of the Barton scales, overheads, and the income sharing rule.
We also use equations (15), (16), (18),and (19) to construct private good equivalents xik and equivalent incomes y i∗ for i = f and i = m. We may then construct
the household economies of scale by equation (17), and household equivalence
scales for the household as y/y m∗ , and for a female as y f ∗ /y m∗ .We also report
the what the economies of scale and adult equivalence scales would equal using a
sharing parameter of η = 1/2, to remove the role of bargaining power from these
calculations.
4.7. Figures
5. Conclusions
We model households in terms of the utility functions of its members, a bargaining
or social welfare function, and a consumption technology function. By employing a collective model and a separate consumption technology, we combine data
from singles and couples to identify equivalence scales, consumption economies of
scale, household members’ bargaining power and other related concepts, without
assumptions regarding cardinalizations or interpersonal comparability of preferences. We also provide duality results that facilitate the empirical application of
the model. These include the use of an income sharing rule to implicitly model the
outcome of household bargaining or social welfare calculations, and shadow prices
that embody economies of scale and scope arising from joint consumption. Our
empirical application of the model uses data from the UK Family Expenditure
Surveys. We demonstrate generic identification of our model.
Given our framework, one useful area for further work would be the development of alternative identifying assumptions. For example, with linear technologies
the model is very much over identified, in fact, some examples in the Appendix
suggest that the entire consumption technology can be identified from the couple’s
demands for a single good. This overidentification might be exploited by relaxing the model to permit greater differences in preferences between individuals in
households and those living alone. Direct observation of some elements of the
equivalent consumption vectors x, xf , or xm would similarly aide identification,
by distinguishing components of F that are due to changes in preferences versus
pure economies of scale or scope in consumption. Such data might include direct
measures of the quantity of food wasted (discarded) by singles and couples, or
measures of the fraction of time couples drive together versus alone. Alternative
identification conditions that minimize the data required on singles would be valuable not only to minimize assumptions regarding changes in preferences, but also
to extend the model to directly include the utility functions of children, which for
now must be assumed to be joint with the utility of an adult household member.
Other useful extensions would be explicit parameterizations of household demands with nonlinear technologies, and extension to more general technology
functions, such as z = F (xf , xm , p). Technology functions that depend on xf and
xm other than through the sum x = xf + xm will result in shadow prices that
differ for different household members. A particularly useful extension in terms
of realism would a model that adds the constraint zj ≥ max(xfj , xm
j ). A special
f
case would be the inclusion of pure public goods j, where zj = xj = xm
j . Our proposed model can easily accomodate these extensions in theory, but identification,
specification, and duality results like our Theorems 1 and 2 would be required to
make these models practical in empirical applications.
While our particular identifying assumptions are open to debate and (we hope)
improvement, we believe that the model we have provided is an appropriate
framework for estimating and analyzing adult equivalence scales, consumption
economies of scale, household members’ bargaining power, and other concepts
relating to household preferences, consumption, and demand behavior.
Appendix
Here we provide proofs of lemmas and theorems in the paper, and some additional related results.
A. Proof of Proposition 1
Consider the household as a two person economy facing the linear technological
constraint
p (Ax + a) = 1
In this economy, the ’production’ side pins down equilibrium relative prices: any
equilibrium price vector π must satisfy the condition
π(p) =
A0 p
1 − a0 p
>From the welfare theorems, any equilibrium is Pareto efficient, and any
Pareto efficient allocation can be decentralized as an equilibrium for some lumpsum transfers. In particular,
the Pareto efficient allocation characterized
¡ mconsider
¢
∗
f
by program (P ), and let x , x be the corresponding individual demands. Normalize the price vector π = π(p) such that π0 x = 1, and define the intrahousehold
income sharing rule η(p) by
η(p) = η f (p) = π.xf
Then
η m (p) = 1 − η f (p)
and the second welfare theorem implies that agent i, facing prices π and income
η i , will choose the bundle xi .
The sharing rule η = η f is the fraction of the household’s shadow income that
is allocated to member f , and so can be interpreted as a measure of her weight
or bargaining power in the household’s program. She then uses this income to
’buy’ her vector of private good equivalents xf , paying shadow prices π. Similarly,
member m allocates income 1 − η = η m to obtain private good equivalents xm ,
paying shadow prices π.
B. Relationship between η and µ
A first order condition of the household’s program is
e ∂U i (xi )
∂U
= λ0 π k
∂U i ∂xik
for each member i and good k, where λ0 is a Lagrange multiplier. It follows that
so
e ∂V i (π/η i )
e xi0 DU i (xi )
∂U
∂U
=
= λ0
∂U i
∂η i
∂U i
ηi
f
e
∂ U/∂U
∂V f (π/η f )/∂η f
=
e /∂U m
∂V m (π/η m )/∂η m
∂U
(26)
It can then be directly verified that all of the first order conditions from the
e is replaced with µ(p)U f (xf ) + U m (xm ).
program P∗ are unchanged when U
C. Proof of Proposition ??
The AIDS quantity demand for good j consumed by single i is
p/y i ) =
hij (e
where
¤¢
£ ¡ ¢
yi ¡ i
p)
αj + Γij ln pe + β ij ln y i − ci (e
pej
1
ci (e
p) = (ln pe)0 αi + (ln pe)0 Γi ln pe
2
i
ln[b (e
p)] = (ln pe)0 β i
αi and β i are n-vectors of parameters, Γi is an n×n matrix of parameters, e0 αi = 1,
e0 β i = Γi e = 0 where e is an n-vector of ones, and Γi is symmetric.
Linear technology imples that the shadow price for good j is
P
Akj pek
A·j pe
πj =
= k 0
0
y − a pe
y − a pe
where A·j is the transpose of the j’th column of A and Akj is the k’th element of
A·j .
With AIDS singles and linear technology, the couple’s quantity demand for
good k is
hk (p) =
X
Akj
j
hk (p) = (y − a0 pe)
X
j
"
³
η
αfj + Γfj ln π
πj
¡
1−η
m
αm
j + Γj ln π
πj



Akj 

η
³
¤´
£
+ β fj ln (η) − cf (π) +
¢
m
+ βm
j [ln (1 − η) − c (π)]
#
+ ak
³
´³
´
h
³
´i´
f
Ae
p
Ae
p
f
αfj + Γfj ln y−a
+
β
ln
(η)
−
c
0p
0
j
e
´ y−a pe 
³

Ae
p
m
m
³
´
α + Γj ln y−a0 pe
hj
³
´i 
+ (1 − η) A1·j pe 
Ae
p
m
m
+β j ln (1 − η) − c
y−a0 pe
1
A·j pe
Given hk (p) for each good k, the constants ak are identified as the intercept
terms, and we may identify the summation by [hk (p) − ak ]/ (y − a0 pe). Now use
µ
¶
Ae
p
1
i
i
p)
c (π) = c
p))0 Γi ln (Ae
= (ln Ae
p)0 αi − (ln y − a0 pe) + (ln (Ae
0
y − a pe
2



+ak

to write this summation term as
µ
¶

αfj + Γfj ln (Ae
p) +
¢¤
£
¡
η
f
X Ajk 
1
0 f
0
0 f

β
α
−
(ln
y
−
a
p
e
)
+
(ln
(Ae
p
))
Γ
ln
(Ae
p
)
ln
(η)
−
(ln
Ae
p
)
j

2
¶ 
µ
m
m


A·j pe
αj + Γj ln (Ae
p) +
¢¤
£
¡
j
+ (1 − η)
βm
p))0 Γm ln (Ae
p)
p)0 αm − (ln y − a0 pe) + 12 (ln (Ae
j ln (1 − η) − (ln Ae

For each good k, the coefficient of ln y in this expression is
X ηβ fj + (1 − η) β m
j
P
−
(A
/A
)e
p
`j
kj `
`
j
where the summation is now over all goods j for which Akj is not equal to zero.
Variation in y therefore provides identification of the above expression for each
good k. Variation of pe in the above expression for each k identifies
ηβ fj + (1 − η) β m
j
A`j /Akj
for triplets of goods j, k, ` having Akj 6= 0 and A`j 6= 0. The above expression
f
with ` = k identifies η, since β m
j and β j are identified from singles, and therefore
A`j /Akj is also identified. Define A`j = A`j /Ajj and dj = Ajj . Identification of
A`j /Akj implies identification of A`j for all j, `. What remains is to identify each
dj .
P
The term Akj /A·j pe = 1/ ` (A`j /Akj )e
p` is identified for each j, which implies
based on the large summation term above that
³
¢¤´
£
¡
f
f
f
1
0 f
0 f
p) + β j ln (η) − (ln Ae
p)) Γ ln (Ae
p)
p) α + 2 (ln (Ae
η αj + Γj ln (Ae
£
¡
¢¤¢
¡ m
m
1
m
0 m
0 m
p) α + 2 (ln (Ae
p) + β j ln (1 − η) − (ln Ae
p)) Γ ln (Ae
p)
+ (1 − η) αj + Γj ln (Ae
is identified for each good j. The component of this expression that is quadratic
in ln (Ae
p) is
µ
¶
0
f1 f
m1 m
ln (Ae
p) −ηβ j Γ − (1 − η) β j Γ
ln (Ae
p)
2
2
! Ã
!
Ã
X
X
XX
ln
dj Akj pej ln
dj A`j pej ϕk`
=
k
`
j
j
where ϕk` and Akj are already identified for all j, k, `. Variation in pe in this
expression then identifies dj , which completes the identification.
Note that the technology and sharing rule are substantially overidentified.
Given the demands of singles, the couple’s demands for each good k separately
identify η, a (from the y − a0 pe term) and A`j /Akj for all j, k, ` having Akj 6=
0. Therefore, observing couple’s demands for all goods, not just one, greatly
overidentifies these terms. In particular, if the k0 th row of A has all nonzero
elements, then (given the demands of singles) the technology and the sharing rule
can be completely identified just by observing couple’s demands for the k0 th good.
In addition to the overidentification resulting from observing the couple’s demands
for multiple goods, only the terms involving the intercept, the coefficient of ln y,
and the quadratic in ln (Ae
p) were used for identification, so all the other terms
in the couple’s demands, such as the linear in ln (Ae
p) term, provide additional
overidentifying restrictions. These various overidentifying assumptions may be
used to relax the assumptions regarding unequal and nonzero parameters in the
statement of Theorem 3.
Similarly, the above proof applies when single’s have QUAIDS demands, since
in that case the only difference in the model is the addition of more terms (in
particular, a quadratic in ln y) which provide more overidentifying restrictions.
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