ESTIMATING MARGINAL UTILITIES WITHOUT
AGGREGATION
ETHAN LIGON
Contents
1.
Introduction
1
2.
A Wrong Turn
2
3.
An Alternative Approach
6
4.
Model of Household Behavior
8
5.
Choosing Subsets of Goods and Services with which to
6.
The Variable Elasticity System of Demand
7.
Estimation
14
8.
Conclusion
19
9.
References
20
Estimate Marginal Utilities
10
12
20
Appendix A.
Application in Uganda
22
1. Introduction
The models economists use to describe dynamic consumer behavior almost invariably boil down to a description of how consumers'
marginal utilities evolve over time.
A central example involves the
canonical Euler equation, which describes how consumers smooth consumption over time when they have access to credit markets (Bewley,
1977; Hall, 1978) or some more general set of asset markets (Hansen
and Singleton, 1982); this same Euler equation tells us how to price
assets using the mechanics of the Consumption Capital-Asset Pricing
Model (Lucas, Jr., 1978; Breeden, 1979).
So, taking dynamic models of consumption to the data means measuring marginal utilities.
servable.
But marginal utilities are not directly ob-
The usual approach to measuring these indirectly involves
Date : March 21, 2014.
This is an incomplete draft of a paper describing research in progress. Please do
not distribute without rst consulting the author.
1
2
ETHAN LIGON
constructing measures of consumers' total consumption expenditures,
and then plugging these total expenditures into a parametric utility
function, where marginal utilities may (Hansen and Singleton, 1982;
Ogaki and Atkeson, 1997) or may not Hall (1978) also depend on unknown parameters which have to be estimated.
There are problems with this approach, both in principle and in
practice. In principle this plugging of total expenditures into a direct
utility function only makes sense if Engel curves are all linear, and
we know that they are not (Engel, 1857). In practice the exercise of
collecting the necessary data on quantities and prices of all consumption
goods and services is extremely dicult, and even well-nanced eorts
by well-trained, ingenious economists and statisticians have yielded less
than satisfactory results.
In this paper we describe an alternative approach to measuring
consumers' marginal utilities which can completely avoid aggregation
across dierent commodities.
Avoiding aggregation allows us to also
avoid the most serious of the problems described above.
Our ap-
proach is principled, in the sense that our approach allows us to impose
much less structure on consumer preferences, and we can accommodate
desiderata such as Engel's Law. Indeed, the key to our approach is to
take advantage of the variation in the composition of dierent consumers' consumption bundles; this is the very variation the existence
of which is either denied or ignored by the usual approach. Our approach is also practical: we simply don't need to use data on goods
or services for which data or prices are suspect; and we entirely avoid
the diculties of constructing comprehensive aggregate; and it's simply
unnecessary to construct price indices to recover real expenditures;
nominal prices and expenditures are all that we need.
2. A Wrong Turn
In
the
standard
case
in
which
utility
takes
a
von
Neumann-
Morgenstern form and is thought to be separable across periods, the
Euler equation for a consumer
(1)
j
might be written
u0 (cjt ) = βj Et Rt+1 u0 (cjt+1 ),
u is a momentary utility function, βj is the discount factor for the
j th consumer, Rt are returns to some asset realized at time t, and where
cjt is a measure of total expenditures or consumption by consumer j
0
at time t, so that u (cjt ) is the marginal utility of consumption for the
j th household at time t.
where
ESTIMATING MARGINAL UTILITIES WITHOUT AGGREGATION
3
These same marginal utilities are often used to characterize not
only intertemporal behavior of a representative consumer often featured in the macroeconomic literature, but also tests of risk sharing
across households in the US (Mace, 1991; Cochrane, 1991), other high
income countries (Deaton and Paxson, 1996), and low income countries
(Townsend, 1994; Ligon, 1998; Ligon et al., 2002; Angelucci and Giorgi,
2009).
Estimating or testing models using the kinds of restrictions of which
0
(1) is an example requires one to take a stand on just what u (c) is. The
0
very notation seems to imply that u : R → R; that is that marginal
utility depends only on a scalar quantity. To construct measures of marginal utilities, empirical papers of the sorts mentioned above typically
begin by constructing a
consumption aggregate,
which is typically de-
signed to capture total expenditures on non-durable goods and services
over some period of time; and then plug that consumption aggregate
into some parametric direct momentary utility function. For example,
Hall (1978) substitutes annual per capita US consumption into a quadratic utility function; Runkle (1991) substitutes household-level nondurable expenditures into the Constant Relative Risk Aversion (CRRA)
power utility function; Townsend (1994) substitutes household-level
adult-equivalent consumption into an exponential utility function;
and Ogaki and Zhang (2001) use household-level measures of consumption expenditures into a power utility function, but with a translation
to allow for the possibility that relative risk aversion might vary with
wealth.
2.1. Expenditure Data. What data is collected to support the construction of a consumption aggregate? Expenditures (or consumptions)
are better than income, because they're a better measure of
permanent
income or wealth than is realized income in a particular year.
Careful surveys of consumer expenditures are conducted occasionally
in many countries, often with the aim of collecting the data necessary
to calculate consumer price indices of some sort (which typically rely
on estimates of the composition of consumption bundles). Such surveys
are, however, in particularly widespread use in low income countries. Of
particular note are the Living Standards Measurement Surveys (LSMS)
rst designed and introduced by researchers at the World Bank in 1979
(Deaton, 1997). These surveys typically feature quite comprehensive
modules designed to collect data on expenditures of nondurable consumption and services. The World Bank has had great success in using
expenditures over time using its LSMS family of surveys.
The main
complaints about these are simply that there aren't enough of them,
4
ETHAN LIGON
and that too seldom do they form a panel. Both of these complaints
presumably have a great deal to do with the associated costs; Lanjouw
and Lanjouw (2001) report that the cost of elding a single round of
an LSMS survey ranges from $300,000 to $1,500,000, or about $300 per
household.
The LSMS expenditure modules typically collect data on expenditures on dozens or even hundreds of dierent goods and services. However, it's unusual for these disaggregate data to feature directly in any
intertemporal analysis. Instead, the usual practice is to use these disaggregated data to construct a comprehensive expenditure aggregate
(Deaton and Zaidi, 2002).
This is intended to be a measure of all
current consumption expenditures (including all goods and services).
There is nothing wrong with these consumption aggregates in principle: to the contrary, theory suggests that period-by-period total expenditures on non-durables and services are exactly the object that we
ought to think intertemporally-maximizing households are making decisions about. However, in practice constructing such aggregates may
be rather like making sausage. It's not that the issues, both practical
and theoretical haven't been carefully considered: Deaton and Zaidi
(2002) provide what amounts to an instruction manual. The problem
instead is simply that the data demands of this exercise are extreme.
To indicate just a couple of the challenges: Even when the list of goods
and services is comprehensive, it may be extremely dicult to back
out the value of services from assets. The value of housing services is
a particular problem, particularly since in many low income countries
houses may be sold or exchanged very infrequently, but in general nding the right prices to go with dierent consumption items may be very
challenging.
2.2. From Direct to Indirect Utility. For the moment, let us set
aside the problem of constructing a consumption aggregate. In what
world does it even make sense to model consumer preferences in this
way?
The assumption that momentary utility depends only on the
quantity of total expenditures (perhaps adjusted for household size or
composition) is, on its face, an odd one.
Nobody really thinks con-
sumers are just consuming a single num\'eraire good, denominated in
some currency units. Instead, we should think of
u as an indirect utility
function.
Provided preferences time separable, we can think of
and of indirect utility:
v(x, p) = maxn u(c)
c∈R
such that
p 0 c ≤ x.
u : Rn → R,
ESTIMATING MARGINAL UTILITIES WITHOUT AGGREGATION
5
Now, if one knows (or can estimate) v , then it's possible to swap in
∂v/∂x for u0 in intertemporal restrictions such as (1), which gives something with greater logical consistency; e.g.,
∂v
∂v
(xjt , pt ) = β Et Rt+1 (xjt+1 , pt+1 ).
∂x
∂x
But now a problem with using the indirect utility function emerges:
one needs not only data on total expenditures, but also information
about (all) prices.
However, for many datasets (including most sur-
veys in the LSMS family) data on prices is collected, sometimes both
a the community and at the household level. Further, the same analysts who constructed the consumption aggregate are also likely to have
constructed a price index, say
π(p),
which we assume to be a contin-
uously dierentiable function of prices, and which is further assumed
not
to depend on an individual's expenditures
x.
Then to complete
our justication for using a consumption aggregate, we merely require
that
v(x, p) ≡ v(x/π(p), 1),
substituting a measure of
real
expenditures for nominal expenditures.
Now, when will using a simple price index like this be valid, so that
v(x, p) ≡ v(x/π(p), 1)
hold? And in particular, what restrictions does
this place on the underlying direct utility function? Roy's identity tells
that we can write the Marshallian demand for good
ci (x, p) =
where
πi
i
as
v 0 (x/π) xππ i
∂v/∂pi
πi
= 0
,
1 = x
∂v/∂x
π
v (x/π) π
is the partial derivative of the price index with respect to the
price of the ith good. This tells us that demands are all linear in total
expenditures
x,
and pass through the origin.
And this is the case if
and only if the utility function is homothetic.
2.3. The Dead End. The scenario we've described (homothetic, timeseparable preferences) is the
only
scenario in which it is correct to use
deated expenditure aggregates in dynamic consumer analysis. If utility is in fact homothetic, then Engel curves must be linear, and the
linear expenditure system is the correct way of describing consumer
demand. But the rst fact implies unitary income (expenditure) elasticities for all goods, and is thus at odds with Engel's Law, while the
second ies in the face of decades of empirical rejections of the linear
expenditure system.
Of course, though the assumptions an empirical researcher must
make to use the usual deated consumption aggregates in dynamic
analysis seem implausible, unrealistic assumptions on their own needn't
6
ETHAN LIGON
deter a dedicated economist (Friedman, 1953). And even if those assumptions seem to lead to predictions that are sharply at odds with
one set of stylized empirical facts (e.g., Engel's Law), they may nevertheless allow the researcher to explain
other
empirical facts (Kydland
and Prescott, 1996).
However, it's far from clear that homothetic utility and aggregating
consumption is important for explaining
any
of the important facts.
And for all the convenience they may oer the econometrician (only
a single random variable needs to measured), the
construction
of real-
ization of that random variable is extremely dicult, expensive, and
involves some intractable measurement problems. The household surveys and analysis necessary to collect comprehensive data on expenditures are very complicated and are hard to systematize across dierent
environments. Heroic assumptions are typically required to value ows
of services (Deaton and Zaidi, 2002), or deal with variation in quality
(Deaton and Kozel, 2005). Additional heroics are required to construct
price indices (Boskin et al., 1998).
Is there a better way? In this paper I'll argue that by starting with
aggregated consumption we've taken a serious
wrong turn,
by simply backing up and making
data the center of our
disaggregated
and that
analytical focus we can make important progress without complicating
our dynamic analysis.
3. An Alternative Approach
In the rest of this paper we'll describe an alternative approach
to measuring marginal utilities which which is theoretically consistent; which uses Engel-style facts about the composition of dierentlysituated consumers consumption bundles; which has comparatively
modest data requirements; which allows us to simply
ignore
expen-
ditures on goods and services which are too dicult or expensive to
measure well; and which completely avoids the price index problem
by simply avoiding the need to construct price indices. The approach
doesn't require (though it permits) (quasi-)homothetic utility; avoids
the usual sausage factory from which consumption aggregates are extruded; should allow for much less expensive data collection; and directly yields measures of the marginal utilities that theory tells us we
want.
The basic approach we take involves using disaggregated data from
one or more rounds of a household expenditure survey to estimate a
ESTIMATING MARGINAL UTILITIES WITHOUT AGGREGATION
7
Frischian demand system (demands which depend on prices and marginal utilities). Estimating such a system allows us to more or less directly recover estimates of households' marginal utilities in each round,
1
which in turn can be used as an input to a dynamic analysis.
There is, of course, a vast literature on dierent approaches to estimating demand systems, so it's been surprising to discover that none
of these approaches seems well-suited to our problem. The rst issue
is simply that almost all existing approaches are aimed at estimating Marshallian demands, rather than Frischian.
2
Related, demand
systems which are nicely behaved (e.g., linear in parameters,) in a
Marshallian setting are typically ill-behaved in a Frischian.
This in-
cludes essentially all of the standard demand systems based on a dual
approach (e.g., the AIDS system).
Other existing demand systems can be straight-forwardly adapted to
estimating Frischian demand systems, such as the Linear Expenditure
System (LES), which can be derived from the primal consumer's problem when that consumer has Constant Elasticity of Substitution (CES)
utility. But as we've already noted, such systems are too restrictive,
imposing a linearity in demand which is sharply at odds with observed
demand behavior.
Accordingly, this paper introduces some thoughts regarding the use
of what we call the
Variable Elasticity of Substitution
(VES) demand
system for drawing inferences regarding household welfare.
mand system is essentially a generalization of the well known
Elasticity of Substitution
This de-
Constant
(CES) demand system, but unlike the CES,
it allows for dierent curvatures of sub-utility functions.
There are various reasons to be interested in such a demand system.
First, as the name suggests, if consumers' actual behavior suggests
that elasticities of substitution are
not constant, then the VES demand
system oers a possible description.
Second, the CES demand system not only puts restrictions on the
substitution matrix, but also on the
income elasticity
of demand; as
already noted, this demand system essentially requires that all Engel
curves be linear. There is evidence against such linearity.
1It
would, of course, also be possible to estimate the demand system and the
dynamic model jointly. But since we so often are able to reject the dynamic models
we estimate, joint estimation seems likely to result in a mis-specied system; here,
we prefer to not impose any dynamic restrictions on the expenditure data so as
to allow ourselves to remain comfortably agnostic about what the `right' dynamic
model ought to be.
2Notable exceptions include Browning et al. (1985); Kim (1993) and Blundell
(1998)
8
ETHAN LIGON
Third, but of central importance for our present problem, with the
(possibly) non-linear Engel curves of the VES demand system it's possible to develop parsimonious direct methods for drawing inferences
regarding consumer's marginal utilities of income.
Among other ap-
plications, this may allow us to identify people or households who are
needy, and to also see when and how neediness changes over time.
So, the approach considered here involves three main, interlocking
elements. First, we provide a simple description of aspects of the VES
system. Second, we provide computational methods for
computing
de-
mands and related elements. Such methods are necessary, since Marshallian demands in the VES system have a closed form solution only
in very special cases.
may be useful for
Third, we oer remarks and procedures which
estimating
VES demand systems from dierent sorts
of datasets.
4. Model of Household Behavior
In this section we give a simple description of a Frischian function
λ,
which at the same time maps prices and resources into a welfare
function (higher values mean that the household is in greater need), and
which also serves as the central object for making predictions regarding
future
welfare.
4.1. The household's one-period consumer problem. To x concepts, suppose that in a particular period
of prices for goods
good
xt
pt
t
a household faces a vector
and has budgeted a quantity of the numeraire
to spend on contemporaneous consumption, from which it
derives utility via an increasing, concave, continuously dierentiable
utility function
U.
Within that period, the household uses this budget
c ∈ X ⊆ Rn ,
to purchase non-durable consumption goods and services
solving the classic consumer's problem
(2)
V (pt , xt ) = max
U (c1 , . . . , cn )
n
{ci }i=1
subject to a budget constraint
(3)
n
X
pit ci ≤ xt .
i=1
The solution to this problem is characterized by a set of
conditions which take the form
(4)
Ui (c1 , . . . , cn ) = λt pit
n
rst order
ESTIMATING MARGINAL UTILITIES WITHOUT AGGREGATION
(where
Ui
ith
denotes the
ity function
U ),
partial derivative of the momentary util-
along with the budget constraint (3), with which the
Karush-Kuhn-Tucker multiplier
So long as
U
9
λt
is associated.
is strictly increasing the solution to this problem delivers
a set of demand functions, the Marshallian indirect utility function
V,
and a Frischian measure of the marginal value of additional resources
to the household
λt = λ(pt , xt ).
It is this last object which is of central interest for the purposes of
this note.
By the envelope theorem, the quantity
thus positive but decreasing in
xt ,
λt = ∂V /∂xt ;
it's
so that `neediness' decrease as the
total value of per-period expenditures increase.
4.2. The household's intertemporal problem. Of course, we're interested in the welfare of households in a stochastic, dynamic environment. But it turns out to be simple to relate the solution to the static
one-period consumer's problem to a multi-period stochastic problem;
at the same time we introduce a simple form of (linear) production.
We assume that households have time-separable von NeumannMorgenstern preferences, and that households discount future utility
using a common discount factor
β.
As above, within a period
t,
a
household is assumed to assumed to allocate funds for total expenditures in that period obtaining a total momentary utility described by
the Marshallian indirect utility function
prices, and
xt
are time
t
V (pt , xt ),
where
pt
are time
t
expenditures.
The household brings a portfolio of assets with total value
the period, and realizes a stochastic income
hold decides on investments
bt+1
yt .
Rt bt
for the next period, leaving
consumption expenditures during period
t.
into
Given these, the house-
xt
for
More precisely, the house-
hold solves
max
−t
{bt+1+j }T
j=1
Et
T −t
X
β j V (pt+j , xt+j )
j=0
subject to the intertemporal budget constraints
xt+j = Rt+j bt+j + yt+j − bt+1+j
and taking the initial
bt
as given.
The solution to the household's problem of allocating expenditures
across time will satisfy the Euler equation
∂V
∂V
(pt , xt ) = β j Et Rt+j
(pt+j , xt+j ).
∂x
∂x
But by denition, these partial derivatives of the indirect utility function are equal to the functions
λ
evaluated at the appropriate prices
10
ETHAN LIGON
and expenditures, so that we have
λ(pt , xt ) = β j Et Rt+j λ(pt+j , xt+j ).
(5)
This expression tells us, in eect, that the household's neediness or
λt satises
of λt play a
marginal utility of income
a sort of martingale restriction,
so that the current value
central role in predicting
values
future
λt+j .
When we estimate Frisch demands, we will typically also directly
obtain estimates of the consumer's
these estimated
λt .
And notice that once we have
{λt } in hand restrictions such as (5) are linear
in these
variables. This can simplify estimation, and perhaps also make dealing
with measurement error a comparatively straight-forward procedure.
5. Choosing Subsets of Goods and Services with which to
Estimate Marginal Utilities
If what we're interested in is household resources, why not simply try
to measure total expenditures
xt
directly? This is an approach which
has long been pursued by researchers, and the World Bank Living Standards Measurement Surveys (LSMS) were designed quite explicitly to
try and construct estimates of
all
ing.
xt
(Deaton, 1997). However, capturing
expenditures on nondurables and services is extremely demandServices in particular may be hired or may ow from consumer
durables, or physical assets such as housing, and while the LSMSs try
to collect these data, doing so is expensive and the calculation of these
dicult (Deaton and Zaidi, 2002).
5.1. Separability and Incomplete Demand Systems. An alternative is to attempt to carefully measure only expenditures on particular
kinds of goods or services.
When the momentary utility function is
(strongly) separable, e.g., when it can be written as U (c1 , c2 . . . , cn ) =
U 1 (c1 ) + U 2 (c2 , . . . , cn ) for some functions U 1 and U 2 , then the rst
order condition corresponding to the choice of
c1
in the consumer's
problem becomes simply
λt = U 10 (c1t )/p1t .
Then, provided that one knows the form of the function λ, measuring
λ becomes as simple as measuring the price of and expenditures on the
single good c1t ; it's now longer necessary to measure all expenditures.
Economists often suppose that food is a good of this sort; indeed, Engel's second law asserts that the share of food in total expenditures
is in fact is the best measure of the material standard of living (Engel,
ESTIMATING MARGINAL UTILITIES WITHOUT AGGREGATION
11
1857). But regardless of whether one accepts Engel's assertion regarding food, the assumption of separability allows one to make statements
about household welfare solely on the basis of demand for a particular
(category of ) goods (e.g., Pollak, 1969; Blackorby et al., 1978).
However, treating food (or any other expenditure category) this way
involves making two problematical assumptions:
rst, that dierent
kinds of food are perfect substitutes; and second, that food overall is
neither a substitute for nor a complement with other goods or services.
We consider two dierent ways of addressing this problem; the rst
involving the imposition of a hierarchical structure on the commodity
space, and the second an apologize later approach which involves
simply ignoring `dicult' goods or groups for which separability can
be rejected. We discuss each of these in turn.
5.1.1.
Utility trees.
Our rst idea to address problems of separabil-
ity follows the approach of Brown and Heien (1972), and involves
choosing a particular hierarchical description of the commodity space.
This essentially allows us to group dierent goods so as eliminate dependence between the groups.
ing that a single good
c1
In our example, instead of suppos-
is separable, we nd some
group
which is
separable; i.e., that the utility function can be written in the form
U (c1 , c2 . . . , cn ) = F (U 1 (c1 , c2 , . . . , cm ) + U 2 (cm+1 , . . . , cn )) for some ordering of goods and for some monotonically increasing function
rst order conditions associated with the rst
m
F.
The
goods then take the
form
∂U 1
(c1 , . . . , cm ) = λpi ,
∂ci
i = 1, . . . , m.
Note that the Frisch demands for goods in the rst group depend only
on prices in that rst group. Thus, to measure
λt
it suces to observe
prices and quantities of just (say) the rst group, leaving us with a
system of
m
equations which can be jointly solved to compute
λ.
This idea can be further elaborated to form what Brown and Heien
call an s-branch utility tree, with conditional demands for goods on a
particular branch which depend only on the prices of goods which appear on that same branch. For example, within-period utility might be
1
11
12
2
expressed as U (U (U (c1 , c2 ), U (c3 , c4 , c5 )), U (c6 , . . . , cn )), so that
demand for the
groups (c1 , c2 , c3 , c4 , c5 )
and
(c6 , . . . , cn ) are (weakly)
(c1 , c2 ) and (c3 , c4 , c5 )
separable from each other, and further groups
have demands which are separable
conditional
allocated to the goods which appear in
U
1
.
on total expenditures
12
ETHAN LIGON
Because the set of possible trees is large, organizing the commodity
space into such a tree may require some prior knowledge, but very often
expenditure data has a naturally hierarchical form in any case.
5.1.2.
Ignoring `dicult' goods.
For this approach we suppose that
commodities (or groups, by an obvious extension) are either strongly
separable or not. We then estimate demands for all the goods (groups)
we have,
without
fact that
λ
imposing any cross-equation restrictions (such as the
ought to be the same across goods). We then conduct one
or more specication tests. For example, if our demand equations are
correctly specied, then the residuals from these equations ought not
to vary with prices of other goods, or with total expenditures. Further,
the values of
λ
obtained (perhaps up to a factor of proportionality)
ought not to vary across any correctly specied separable demands for
a particular individual/household in a particular period.
This, this approach suggests a simple, practical recipe: (i) separately
estimate all demands under the null hypothesis of separability; (ii) subject each estimated demand to a battery of specication tests, setting
aside any goods or groups where the null hypothesis of a correct specication can be rejected; and (iii) using only the demands that remain
to draw inferences regarding
λ.
6. The Variable Elasticity System of Demand
The discussion to this point has avoided the fact that we're unlikely to know the marginal functions
adopt a simple parameterization.
Ui ,
and our approach here is to
However, existing well-known pa-
rameterizations such as the constant elasticity of substitution (CES)
or other demand systems such as AIDS or translog which rely on socalled PIGLOG preference structures (Muellbauer, 1976) form aren't
suitable, as they impose unwanted structure on the income elasticities
of demands.
Accordingly, we work with a richer parameterization of utility functions than is usual in applied work, and one which is neither necessarily
PIGLOG nor Gorman-aggregable. In particular, let a household's preferences over
n
dierent consumption goods depend on a momentary
utility function
(6)
U (c1 , . . . , cn ) =
n
X
i=1
Here, the parameters
{γi }
αi
(ci + φi )1−γi − 1
.
1 − γi
n sub-utility
n goods (and
govern the curvature of the
functions associated with consumption of the various
thus the household's risk preferences over variation in consumption of
ESTIMATING MARGINAL UTILITIES WITHOUT AGGREGATION
13
these goods). We assume that γi ≥ 0, and in the usual way use the fact
(xi +φi )1−γi −1
that limγi &1
= log xi to interpret values of γi = 1 as though
1−γi
the corresponding sub-utility function is logarithmic. The parameters
{αi } govern the weight of the n sub-utilities in total momentary utility,
and the parameters {φi } `translate' the commodity space in such a way
to make it simple to accomodate subsistence levels for some goods, or
more generally to control the marginal utility of consumption near zero
for any of the goods.
6.1. Frischian Demands. An important feature of these preferences
is that if dierent goods are associated with curvature parameters (i.e.,
there exists an
(i, j)
such that
γi 6= γj )
then the preferences are not
Gorman-aggregable; indeed, while Marshallian demand functions exist
for these preferences, except for some special cases these Marshallian
demands won't have closed form solutions.
However, a more general approach to characterizing the demand system is available to us. Instead of deriving the Marshallian demands,
we'll instead work with the Frisch demand system. Instead of expressing demands as a function of income and prices, as in the Marshallian
demand system, the Frisch system expresses demands as a function of
prices and the marginal utility of income. Though the result is standard, it's not as well known as it might be, and so we give it here in
the form of the following proposition.
Let V (p, x) denote the indirect utility function associated with the problem of choosing consumption
goods so as to
P
maximize (6) subject to a budget constraint ni=1 pi ci ≤ x, and let
λ(p, x) = ∂V
(p, x). Then the Frisch demand for good i can be written
∂x
as some function ci (p, λ), which is related to the Marshallian demand
via the identity cm
i (p, x) ≡ ci (p, λ(p, x)).
Proposition 1.
For the particular case at hand, Frisch demands are
ci (p, λ) =
(7)
for
i = 1, . . . , n,
αi
λpi
1/γi
− φi
and the Frischian counterpart to the indirect utility
function (which we'll unimaginatively call the `Frischian indirect utility
function') is given by
n
X
1/γi −1 X
n
1
αi
1
1/γi
αi
−
,
(8)
V(p, λ) =
1 − γi
λpi
1 − γi
i=1
i=1
Pn pi αi 1/γi Pn αi
or V(p, λ) = λ
− i=1 1−γi .
i=1 1−γi
pi λ
14
ETHAN LIGON
7. Estimation
Here we describe a recipe for estimating marginal utilities from lim-
ited data. Our recipe uses three main ingredients. The rst of these is a
Frisch demand system, which species consumer demands as functions
of prices and of the marginal utility of income. Working with Frisch demands not only puts the focus of the analysis on the marginal utilities
which are our central interest, but also allows us to estimate theoretically consistent
incomplete
demand systems. This is critical whenever
our data don't include quantities and prices for all goods and services
that the household might consume. Second is a
of the commodity space.
recursive description
This ingredient is important for being able to
specify a demand system in which the quantities and prices of goods we
observe are plausibly independent of the prices of the goods we don't
observe. And third is a new sort of demand system which I'll call the
Variable Elasticity of Substitution
(VES). This ingredient is critical to
identifying the particular goods or services which are most responsive
to changes in the marginal utility of incomeprevious demand systems
such as the LES, CES, or AIDS all imply that all goods are equally
responsive.
The approach to estimation necessarily relies on the sort of data
available; we discuss several possible cases.
7.1. Estimation Using a Cross-Section and Price Data. Suppose
we have cross-sectional data on prices and household or individuallevel quantities consumed for
m ≤ n commodities.
By taking the basic
expression for Frisch demands, and taking the logarithm of both sides,
we obtain
log(cji + φji ) =
(9)
where
j
1
1
1
log αij − log λj − log pi ,
γi
γi
γi
indexes the consumer (be it household or individual). We're
implicitly assuming here that prices are the same for all consumersif
this is not the case and consumer-specic prices are observed, then we
1
would want to replace the last term with −
log pji .
γi
We'd like to use this equation as the basis of our approach to esti1
mation. The rst term,
log αij introduces too many parameters into
γi
the demand system to estimate in a cross-section without additional
j
structure. However, the interpretation of the parameters αi have to do
with the importance of the commodity to the household (if all the γi
j
were equal across commodities and all the subsistence parameters φi
j
were equal to zero, then the αi would be simple expenditure shares),
and it seems reasonable to suppose that these are in turn importantly
ESTIMATING MARGINAL UTILITIES WITHOUT AGGREGATION
15
related to observable household characteristics, such as household size
and composition.
Accordingly, assume for our present purposes that the terms involvj
ing αi are equal to
1
1
log αij = d|i Z j + ji ,
γi
γi
Zj
is a vector of observable characteristics, di is a (commodityj
specic) vector of unknown parameters, and i is a residual term reecting the eect of unobserved characteristics on demand. As a special
where
case of this structure, if
αi
doesn't vary across households, then we
obtain a simple set of good-specic constants
ci .
The second term on the right-hand side of (9) involves the consumer's
j
marginal utility of expenditures, λ . Though in the absence of full insurance this quantity will vary across time, within a period it will be
common across the demands for all
m
observed consumption. Accord-
ingly, we can replace this second term with a household xed eect
multiplied by a parameter bi which can vary by commodity, so that we
1
have −
log λj = bi η j .
γi
This brings us to the third term on the right-hand side of (9); a term
involving the price of the commodity. This is perhaps the most straightforward term to deal with, as it simply involves a single parameter (say
ci )
multiplied by the logarithm of the observed price.
Putting this all together, we obtain a `reduced form' system of equations
(10)
log(cji + φji ) = ai log pi + bi η j + d|i Z j + ci ji .
This is very nearly something that could be estimated by using ordinary
least squares, but for two issues which we tackle in turn.
j
First, the parameters φi aect the left-hand side of the equation in
a non-linear way, and there are a large number of them. One way to
j
deal with this is to treat the φi in a manner similar to our treatment of
j
the αj parameters; namely, to assume that these are linear function of
j
| j
observables, and to replace each φi with some gi Z . Then conditional
on the value of the vector gi , the remaining parameters can be estimated
using least squares. This will yield a collection of residuals which are
dependent on the value of
di ,
namely
j
eji (gi ) = log(cji + gi| Z j ) − dˆ|i Z j − bd
i η − âi log pi .
Using these residuals along with a conditional moment restriction of
the form
j
j
E(ei (di )|Z , pi )
=0
16
ETHAN LIGON
suggests a simple GMM estimator.
Using GMM would also allow us to address the second problem with
using the OLS estimator: namely, that the residuals associated with
the equation for one commodity may include useful information for
estimating the parameters in another equation. This is an issue that
can be addressed through a sensible choice of weighting matrix for the
GMM estimator.
7.2. Estimation Using a Cross-Section Without Price Data. It
may often happen that though we have reliable data on expenditures,
we lack data on prices and quantities. In this case we can choose a reference good (say good 1), and use the observation that the consumer's
marginal rate of substitution between good 1 and good
i
will be equal
to the ratio of prices of these goods. Exploiting our assumed functional
form for utility, taking the logarithm of this necessary condition yields
(11)
log αij −γi log(xji +pi φji )−γi log pi = log α1j −γ1 log(xj1 +p1 φj1 )−γ1 log p1
This implies
(12)
log(xji +pi φji )
1
γ1
γ1
j
j
= (a1 −ai ) Z + log(p1 x1 +p1 φ1 )+
log p1 − log pi + (j1 −ji ).
γi
γi
γi
|
j
Because the right-hand side term involving prices doesn't vary across
households
j,
it can be treated as a good-specic latent variable
πi ,
yielding an estimating equation
(13)
log(xji +pi φji ) = (a1 −ai )| Z j +
γ1
1
log(xj1 +p1 φj1 )++πi + (j1 −ji ).
γi
γi
Here, we can identify the key parameter
1/γi
only up to a factor of pro-
portionality. Nevertheless, this specication can tell us something important about the relative importance of dierent goods in the demand
system. Further, it can serve as the basis of tests of the adequacy of the
variable elasticity demand system for a given set of expenditure data.
By formulating a regression along the lines of (13) and then adding
additional terms which are functions of expenditures on the numeraire
good (e.g., the square of the log of expenditures on the numeraire),
one can test the null hypothesis that the coecients associated with
these additional terms ought to be zero, and reject (or fail to reject)
the underlying specication of the demand system accordingly.
7.3. Estimation Using a Panel. Suppose we have data on expenditures at two dierent periods (but lack data on prices). We want to
ESTIMATING MARGINAL UTILITIES WITHOUT AGGREGATION
17
use these data to estimate the parameters of
xjit = pit
αitj
pit
!1/γi
− pit φjit .
Note the considerable exibility we allow in the
may vary by household, time, and good.
α
parameters, which
We obviously need some
restrictions on these parameters to achieve identication, so we assume
that
| j
j
αitj = ᾱi eδi zt +it ,
so that we're assuming a time-invariant set of population parameters
ᾱk , which individual households deviate because of observables ztj (due
j
to, e.g., household composition) and unobservables it .
j∗
j
j
Now, let xit = xit + pit φit . Taking logarithms of the expression for
the Frischian demand gives us
log xj∗
it
1
1
1
1
1
= 1−
log pit + log ᾱi + δi| ztj − log λjt + jit ;
γi
γi
γi
γi
γi
then dierencing this over periods gives
∆ log xj∗
it
1
1
1
1
∆ log pit + δi| ∆ztj − ∆ log λjt + ∆jit .
= 1−
γi
γi
γi
γi
In reduced form we express this as
|
j
j
j
∆ log xj∗
it = ait + di ∆zt − ci bt + ci ∆it .
Here the
ait
are latent variables reecting the eect of price changes on
ci bjt is an interaction between a
good-specic dummy and a household-time specic eect (which we'd
demand across periods, while the term
interpret as being due a change in its wealth or neediness).
Now, it's fairly obvious that this reduced form is not quite identied,
j
|
j
because the vector space spanned by the ci bt terms also spans the di ∆zt
terms. Four ways forward suggest themselves.
7.3.1.
Use restrictions from a dynamic model.
Households solving the
intertemporal problem described above will try to smooth
λt
over time
(and states). Indeed, if there's full insurance then these will be conj
stant, and the ci bt term will fall out of the regression (Townsend, 1994).
More generally,
λt
will vary over time, but its evolution will be re-
stricted in a way which depends on the markets for insurance, credit,
or storage available to the household. When there are (even imperfect)
credit markets, for example,
λt
will satisfy a martingale restriction im-
plied by the Euler equation, and these restrictions may allow us to
18
ETHAN LIGON
separately identify both
ci
and
di
in the reduced form estimating equa-
tion above.
Assume that only xed household observables aect demand.
7.3.2.
If
j
j
zt = z , then we allow households to have systematically dierent preferences from each other; one household might particularly like beans
rather than rice, for example, but we don't allow time-varying observj
ables to inuence demand. With this assumption ∆zt = 0, so while di
isn't identied, ci is.
7.3.3.
Assume that δi = δ .
An alternative allows time-varying observ-
ables to aect demand, but requires those observables have a similar
inuence on demand for
all
goods. This may make sense, for example,
as a way of modeling the eects of household size on demand. In this
j
case (adding a constant to ∆zt ) we fold identication of the γi into
j
the di vector, so that ci bt = 0. Then we can estimate ratios of the
curvature parameters, since
γi
d| ∆z j
= k| tj .
γk
di ∆zt
This allows us to construct an
n×n
matrix of these ratios.
Because we can only construct ratios of the
γi ,
we can only identify
these individually up to a factor of proportionality (this is a place where
time series restrictions on
the
γi ,
7.3.4.
λt
would help). However, we can still
order
which may be of use.
Using Singular Value Decomposition (SVD).
An elegant and
general approach involves the use of a singular value decomposition
j
of variation in ∆ log xit which can't be accounted for by variation in
(common) prices. We take as our starting point the dierenced version
of our demand equation,
j∗
(14) ∆ log xit
Let
1
N
PN
j=1
1
1
1
1
∆ log pit + δi| ∆ztj − ∆ log λjt + ∆jit .
= 1−
γi
γi
γi
γi
∆ log λjt = ηt .
Then, summing over households, we have
N
N
1
1 X
1
1 X j
j∗
1−
∆ log pit =
∆ log xit + ηt −
∆ .
γi
N j=1
γi
γi N j=1 it
ESTIMATING MARGINAL UTILITIES WITHOUT AGGREGATION
19
Substituting this into (14) and arranging in matrix form yields the
following:
N
1 X
1
j∗
∆ log xit −
∆ log xj∗
ηt − ∆ log λjt
it =
N j=1
γi
Y
n×N (T −1)
= g
w|
n×11×N (T −1)
δ|
1
+ i ∆ztj +
γi
γi
+d
Z|
n×``×N (T −1)
N
1 X j
j
∆it −
∆
N j=1 it
+ e.
The rst term in this matrix equation captures the role that variation
in
λ
plays in explaining variation in expenditures. Because it's com-
puted as the outer product of two vectors, this rst term is at most of
rank one. The second term captures the further role that changes in
` such
¯
observables, then this second term is of at most rank ` = min(`, n − 1).
household observables play in changes in demand; if there are
We proceed by considering the singular value decomposition of
Y = UΣV| , where U and V are unitary matrices, and where Σ
is a diagonal matrix of the singular values of
Y,
ordered from the
largest to the smallest. Then the rank one matrix that depends on
λ is gw| = σ1 u1 v1| , while the second matrix (of at most rank `¯) is
P¯
dZ| = `k=2 σk uk vk| , where σk denotes the k th singular value of Y,
and where the subscripts on u and v indicate the column of the corresponding matrices
1 + `¯ rank
U and V.
approximation to
The sum of these matrices is the optimal
Y,
in the sense that by the Eckart-Young
theorem this is the solution to the problem of minimizing the Frobenius distance between
Y
and the approximation; accordingly, this is
also the least-squares solution (Golub and Reinsch, 1970).
j
|
A representative element of gw is simply 1/γi (ηt − ∆ log λt ). The
j
parameters γi and the factors (ηt − ∆ log λt ) are each identied only up
(r, s). Presuming that the (1,1) element of this matrix
is non-zero, let r be the absolute value of the reciprocal of the rst
element of g, and let s be the absolute value of the reciprocal of the
|
rst element of w. Then rs times the rst column of gw is equal to
[γ1 /γi ], while rs times the rst row of gw| is s[(ηt − ∆ log λjt )].
to scale factors
8. Conclusion
In these notes we've outlined some of the key methodological ingredients needed in a recipe to measure marginal utilities in a manner which
is theoretically coherent, which lends itself to straightforward statistical inference and hypothesis testing, and which is very parsimonious in
its data requirements.
Our goal is to devise procedures to easily
measure
and
monitor
households' marginal utilities over both dierent environments and
!
20
ETHAN LIGON
across time. To this end, we've described an approach which involves
estimating an incomplete demand system of a new sort which features
variable elasticities
of income and substitution across goods.
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ETHAN LIGON
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Appendix A. Application in Uganda
To illustrate some of the methods and issues discussed above, we use
data from two rounds of surveys conducted in Uganda (in 200506 and
200910).
3
Excluding durables, taxes, fees & transfers, there are 99 categories
of expenditure in the data, of which 61 are dierent food items or
categories, and 38 are nondurables or services.
Figure 1 paints a picture of aggregate expenditure shares across these
categories. A glance reveals that shares of aggregates is fairly stable
across the two rounds, with an increase in the share of housing from
8% to 10% the most notable change.
A.1. Shares for All Expenditure Categories.
3Data
on expenditures was provided by Thomas Pavehson, and on income and
household characteristics by Jonathan Kaminski. My thanks to them both for
sharing their work.
ESTIMATING MARGINAL UTILITIES WITHOUT AGGREGATION
23
Table 1. Shares of Aggregate Expenditures in Uganda,
(2005 and 2010) for all goods with shares greater than
1%.
Agg.
Expenditure Item
Imputed rent of owned house
Shares
Mean
Shares
2005
2010
2005
2010
0.078
0.100
0.060
0.067
Matoke (Bunch)
0.050
0.054
0.044
0.050
Sweet potatoes (Fresh)
0.042
0.045
0.051
0.056
Maize (our)
0.040
0.038
0.052
0.049
Medicines etc
0.038
0.041
0.034
0.038
Water
0.033
0.030
0.044
0.033
Food (restaurant)
0.033
0.033
0.030
0.031
Beef
0.033
0.033
0.027
0.028
Sugar
0.031
0.029
0.030
0.029
Beans (dry)
0.031
0.033
0.040
0.044
Taxi fares
0.027
0.023
0.020
0.016
Firewood
0.027
0.030
0.040
0.042
Hospital/ clinic charges
0.026
0.020
0.021
0.018
Cassava (Fresh)
0.024
0.026
0.029
0.034
Fresh Milk
0.022
0.024
0.018
0.021
Air time & services fee for owned xed/mobile phones
0.022
0.035
0.011
0.023
Cassava (Dry/ Flour)
0.020
0.024
0.028
0.032
Rent of rented house
0.019
0.017
0.017
0.015
Rice
0.015
0.015
0.012
0.013
Fresh Fish
0.013
0.012
0.012
0.012
Paran (Kerosene)
0.013
0.011
0.016
0.015
Cooking oil
0.012
0.011
0.015
0.013
Washing soap
0.012
0.012
0.016
0.015
Charcoal
0.012
0.014
0.010
0.011
Barber and Beauty Shops
0.011
0.011
0.008
0.008
Tomatoes
0.011
0.010
0.012
0.011
Petrol, diesel etc
0.011
0.015
0.003
0.006
24
ETHAN LIGON
Figure 1. Aggregate expenditure shares in 2005 and
2010, using all expenditure items.
Table 1 describes the share of aggregate expenditures on dierent
consumption items in each of the two rounds of the survey, for goods
with an expenditures share greater than 1% in 2005.
A.2. Shares for All Food Categories.
ESTIMATING MARGINAL UTILITIES WITHOUT AGGREGATION
Figure 2. Ratio of mean shares to aggregate shares.
Items on the left form a disproportionately large share
of the budget of the rich; items on the right a disproportionately large budget share of the needy.
25
26
ETHAN LIGON
Figure
3. Changes
in
mean
shares
rounds, using all expenditure categories.
across
survey
ESTIMATING MARGINAL UTILITIES WITHOUT AGGREGATION
Table 2. Aggregate expenditure shares in 2005 and
2010, using all food items.
Agg.
Expenditure Item
Shares
Mean
Shares
2005
2010
2005
2010
Matoke (Bunch)
0.086
0.095
0.072
0.080
Sweet potatoes (Fresh)
0.072
0.079
0.079
0.084
Maize (our)
0.069
0.067
0.084
0.075
Food (restaurant)
0.057
0.058
0.050
0.053
Beef
0.057
0.057
0.043
0.044
Sugar
0.054
0.051
0.051
0.048
Beans (dry)
0.053
0.057
0.065
0.069
Cassava (Fresh)
0.041
0.045
0.046
0.052
Fresh Milk
0.039
0.042
0.030
0.035
Cassava (Dry/ Flour)
0.035
0.042
0.041
0.046
Rice
0.026
0.026
0.021
0.021
Fresh Fish
0.023
0.021
0.020
0.019
Cooking oil
0.022
0.020
0.024
0.021
Tomatoes
0.019
0.018
0.020
0.019
Chicken
0.018
0.020
0.012
0.015
Dry/ Smoked sh
0.018
0.018
0.020
0.018
Other Alcoholic drinks
0.016
0.015
0.019
0.020
Millet
0.015
0.012
0.016
0.013
Matoke (Cluster)
0.015
0.003
0.013
0.003
Bread
0.014
0.019
0.010
0.013
Beer
0.014
0.009
0.007
0.006
Ground nuts (pounded)
0.014
0.016
0.014
0.016
Other Fruits
0.013
0.008
0.011
0.008
Other foods
0.013
0.003
0.019
0.004
Maize (cobs)
0.013
0.015
0.013
0.014
Beans (fresh)
0.011
0.017
0.012
0.020
Irish Potatoes
0.010
0.009
0.009
0.009
27
28
ETHAN LIGON
Figure 4. Aggregate expenditure shares in 2005 and
2010, using all food items.
A.3. Shares for All Non-food Non-durable Categories.
ESTIMATING MARGINAL UTILITIES WITHOUT AGGREGATION
Figure 5. Ratio of mean shares to aggregate shares,
using all food items. Items on the left form a disproportionately large share of the budget of the rich; items on
the right a disproportionately large budget share of the
needy.
29
30
ETHAN LIGON
Figure
6. Changes
in
mean
rounds, using all food items.
shares
across
survey
ESTIMATING MARGINAL UTILITIES WITHOUT AGGREGATION
31
Table 3. Aggregate expenditure shares in 2005 and
2010, using all nondurable items.
Agg.
Expenditure Item
Shares
Mean
Shares
2005
2010
2005
2010
Imputed rent of owned house
0.184
0.233
0.162
0.186
Medicines etc
0.090
0.095
0.085
0.099
Water
0.079
0.070
0.129
0.092
Taxi fares
0.064
0.052
0.046
0.036
Firewood
0.064
0.071
0.127
0.139
Hospital/ clinic charges
0.062
0.046
0.047
0.042
Air time & services fee for owned xed/mobile phones
0.052
0.083
0.022
0.057
Rent of rented house
0.044
0.040
0.040
0.032
Paran (Kerosene)
0.031
0.026
0.050
0.050
Washing soap
0.028
0.028
0.049
0.053
Charcoal
0.028
0.034
0.024
0.026
Barber and Beauty Shops
0.027
0.024
0.021
0.021
Petrol, diesel etc
0.025
0.035
0.006
0.011
Tires, tubes, spares, etc
0.021
0.016
0.019
0.015
Maintenance and repair expenses
0.021
0.011
0.012
0.012
Others (Health and Medical Care)
0.021
0.001
0.016
0.002
Boda boda fares
0.020
0.022
0.016
0.020
Electricity
0.019
0.020
0.009
0.010
Imputed rent of free house
0.017
0.017
0.018
0.016
Cosmetics
0.017
0.014
0.023
0.020
32
ETHAN LIGON
Figure 7. Aggregate expenditure shares of nondurables
in 2005 and 2010.
A.4. Shares for Food in Slightly Aggregated Groups.
ESTIMATING MARGINAL UTILITIES WITHOUT AGGREGATION
Figure 8. Ratio of mean shares to aggregate shares, us-
ing all nondurable items. Items on the left form a disproportionately large share of the budget of the rich; items
on the right a disproportionately large budget share of
the needy.
33
34
ETHAN LIGON
Figure
9. Changes
in
mean
shares
rounds, using all nondurable items.
across
survey
ESTIMATING MARGINAL UTILITIES WITHOUT AGGREGATION
Table 4. Aggregate expenditure shares in 2005 and
2010, using slightly aggregated food groups.
Agg.
Expenditure Item
Shares
Mean
Shares
2005
2010
2005
2010
Drinks
0.260
0.275
0.126
0.337
Restaurant meals
0.223
0.294
0.223
0.280
Tobacco
0.210
0.257
0.200
0.232
Matoke
0.076
0.050
0.161
0.042
Fruits
0.049
0.025
0.104
0.023
Sweet potatoes
0.030
0.014
0.103
0.016
Maize
0.020
0.013
0.097
0.014
Eggs
0.019
0.009
0.034
0.001
Vegetables
0.019
0.007
0.032
0.006
Oils
0.018
0.015
0.021
0.007
Fish
0.014
0.009
0.018
0.005
Meat
0.012
0.005
0.034
0.005
Irish Potatoes
0.009
0.005
0.037
0.006
Bread
0.008
0.005
0.036
0.006
Rice
0.008
0.006
0.019
0.006
Infant formula
0.006
0.004
0.012
0.001
Cassava
0.006
0.003
0.007
0.002
Fresh milk
0.003
0.001
0.006
0.001
Millet
0.002
0.001
0.007
0.000
Sorghum
0.002
0.001
0.006
0.001
Sugar
0.001
0.001
0.007
0.007
Beans
0.001
0.000
0.002
0.000
Ground nut
0.001
0.000
0.004
0.007
Salt
0.000
0.000
0.002
0.002
Sim sim
0.000
0.000
0.002
0.002
Coee
0.000
0.000
0.000
0.000
Peas
0.000
0.000
0.000
0.000
Tea
0.000
0.000
0.000
0.000
Other foods
0.000
0.000
0.000
0.000
35
36
ETHAN LIGON
Figure 10. Aggregate expenditure shares in 2005 and
2010, using slightly aggregated food groups.
Table 5. Aggregate expenditure shares in 2005 and
2010, using more aggregated food groups.
Agg.
Shares
Mean
Shares
2005
2010
2005
2010
Drinks
0.260
0.275
0.126
0.337
Restaurant meals
0.223
0.294
0.223
0.280
Stimulants
0.210
0.257
0.058
0.226
Staples 1
0.145
0.087
0.418
0.082
Expenditure Item
Fruits & Vegetables
0.067
0.032
0.137
0.029
Proteins
0.046
0.023
0.087
0.011
Oils, Salt, Sugar
0.020
0.016
0.030
0.016
Staples 2
0.018
0.010
0.056
0.012
Diary
0.010
0.005
0.018
0.002
Legumes, etc.
0.002
0.001
0.008
0.009
Other foods
0.000
0.000
0.000
0.000
Total
1.001
1.0
1.161
1.004
A.5. Shares for Food in More Aggregated Groups.
ESTIMATING MARGINAL UTILITIES WITHOUT AGGREGATION
Figure 11. Ratio of mean shares to aggregate shares,
using slightly aggregated food groups. Items on the left
form a disproportionately large share of the budget of
the rich; items on the right a disproportionately large
budget share of the needy.
37
38
ETHAN LIGON
Figure
12. Changes
in
mean
shares
across
rounds, using slightly aggregated food groups.
survey
ESTIMATING MARGINAL UTILITIES WITHOUT AGGREGATION
Figure 13. Aggregate expenditure shares in 2005 and
2010, using more aggregated food groups.
Table 6. Aggregate expenditure shares in 2005 and
2010, using CIOCOP groups.
Expenditure Item
Agg.
Shares
Mean
Shares
2005
2010
2005
2010
Food
0.419
0.406
0.466
0.382
Alcohol & Tobacco
0.371
0.462
0.578
0.498
Housing & Utilities
0.116
0.078
0.224
0.071
Health
0.043
0.022
0.062
0.022
Other non-food
0.027
0.014
0.056
0.015
Communication
0.014
0.013
0.012
0.010
Transport
0.005
0.003
0.008
0.002
Furnishing & ?
0.003
0.002
0.002
0.001
Year
0.001
0.000
0.002
0.005
Recreation
0.001
0.000
0.001
0.000
Clothing & Footwear
0.000
0.000
0.000
0.000
Education
0.000
0.000
0.000
0.000
A.6. Shares for Using COICOP Categories.
39
40
ETHAN LIGON
Figure 14. Ratio of mean shares to aggregate shares,
using more aggregated food groups.
Items on the left
form a disproportionately large share of the budget of
the rich; items on the right a disproportionately large
budget share of the needy.
ESTIMATING MARGINAL UTILITIES WITHOUT AGGREGATION
Figure
15. Changes
in
mean
shares
across
rounds, using more aggregated food groups.
survey
41
42
ETHAN LIGON
Figure 16. Aggregate expenditure shares in 2005 and
2010, using CIOCOP groups.
ESTIMATING MARGINAL UTILITIES WITHOUT AGGREGATION
Figure 17. Ratio of mean shares to aggregate shares,
using CIOCOP groups. Items on the left form a disproportionately large share of the budget of the rich; items
on the right a disproportionately large budget share of
the needy.
43
44
ETHAN LIGON
Figure
18. Changes
in
mean
rounds, using CIOCOP groups.
shares
across
survey