ESTIMATING THE LINEAR EXPENDITURE SYSTEM

WITH CROSS-SECTIONAL DATA

Kenneth W. Clements, Marc Jim M. Mariano and George Verikios

2020-06

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ESTIMATING THE LINEAR EXPENDITURE SYSTEM

WITH CROSS-SECTIONAL DATA

Kenneth W. Clements, Marc Jim M. Mariano and George Verikios1

Abstract

The linear expenditure system (LES) is a popular model for analysing consumer behaviour in

relation to changes in prices and income. The first part of this paper provides a comprehensive

review of LES, including its positive and negative attributes. Emphasis is placed on the

application to cross-section data where there is no price variation. In such situations, the LES

parameters are under identified unless the value of the income elasticity of the marginal utility

of income (referred to as “Frisch parameter”) is known. We evaluate several sources from the

literature for this parameter value. The second part of the paper is an empirical illustration of

the application of LES with a cross-section of about 10,000 Australian households. To

overcome the aggregation and linearity issues of LES, we disaggregated households into

income quintiles and estimated a separate LES for each quintile. A comparison of the quintile

estimates with the one-consumer case (when the data are pooled) reveals substantial differences

in demand responses that are masked when LES is constrained to have the same parameters

across the income distribution. We also established the further advantage of disaggregation that

the quintile estimates tend to fit the data considerably better than the single LES.

Keywords: Linear expenditure system, consumer behaviour, expenditure elasticities, distributional

analysis, CGE models

JEL Classification: D12, C31, C68

1 UWA Business School, KPMG Economics, and KPMG Economics and Griffith University, respectively.

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1. Introduction

Expenditure elasticities are a convenient way of summarising how income and prices affect the

demand for a particular commodity. Empirical estimates of these elasticities may serve as a

guide in policy decision-making. For instance, understanding how commodity demands change

in response to varying income and prices across households is essential in making adequate

measures to implement government price stabilisation and household support policies.

Moreover, a decomposition of the elasticity estimates across income groups provides insights

on the distributional effects of a policy-induced change in commodity prices or household

income.

Elasticity values are used as an important input to various economic models. A popular example

is in computable general equilibrium (CGE) models where the expenditure elasticities are

exogenously specified to calibrate demand equations. These expenditure elasticities together

with other parameters influence the outcomes of policy simulations. A more advanced use of

these elasticities is in microsimulation models. A prerequisite in these models is the

decomposition of the elasticities into income groups in order to capture distributional effects.

For instance, for five developing countries (Brazil, Pakistan, Tanzania, Uruguay, Vietnam), the

MIRAGE model of the world economy disaggregates the representative household into 40

households grouped according to income and consumption structures (Bouët et al., 2011). The

KPMG-CGE model of the Australian economy disaggregates the household sector into 10,046

households and uses quintile estimates of expenditure elasticities to represent the demand

behaviour of these households (Verikios et al., 2020). Other microsimulation studies that have

used disaggregated household demands in CGE models include Decaluwé et al.(1999),

Cogneau et al. (2000), Cockburn (2001), Cororaton (2003), Bussolo and Lay (2003), and Jensen

and Tarr (2003).

The nature of econometric estimation of demand elasticities varies widely in the existing

literature due to differences in functional form, commodity aggregation, type of data, and

sample stratification used in the estimation procedure. As emphasized by Shoven and Whalley

(1992) and McKitrick (1998), the appropriate estimation of these key behavioural parameters

affects the empirical validity of models in which these parameters are used. A number of studies

(Hertel (1985), Despotakis and Fisher (1988), Robinson et al. (1991), McKitrick (1998), and

Ho et al. (2020)) have drawn attention to the significant bearing of the choice of functional form

upon simulation results. Arndt et al. (2002) noted that the inconsistency of the household

demand specification between partial equilibrium models in which the parameters are being

estimated and the general equilibrium models in which the elasticities are being used, along

with the scarcity of data, have been cited by CGE modellers as challenging barriers to the

estimation and application of these parameters.

In this paper we provide a parametric estimation of the demand elasticities in which the

structure of the demand system being estimated is consistent with a functional form used in

CGE models. Shoven and Whalley (1984) noted that the selected functional form should be

homogeneous of degree zero, continuous and produce a demand system that satisfies Walras’s

Law. These constraints have compelled CGE modellers to choose functional forms such as the

Cobb-Douglas (C-D) function, the constant elasticity of substitution (CES) function, or the

linear expenditure system (LES) (Pauw, 2003). The C-D utility function displays constant

average budget shares but restricts the price and expenditure elasticities to all equal to one.

These unitary restrictions has been regarded as a major drawback of the C-D function since

unitary elasticities are not consistent with empirical evidence. The CES function allows for the

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possibility of non-unitary price elasticities. This property provides greater flexibility in

choosing different substitution possibilities across commodities especially when employing

nested CES functions (Partridge and Rickman, 1998). The CES function, however, still restricts

the expenditure elasticities to unity which implies that budget shares of commodities are

independent of the level of income. To overcome this limitation, the LES generalizes the C-D

utility function by imposing positive subsistence consumption in the LES functional form

(Boer, 2009).

In this paper we employ the LES to estimate the expenditure and price elasticities of household

consumption. We use a large cross-sectional dataset in our estimation consisting of 10,046

households. The LES has been recognized as an appropriate tool to estimate large systems of

demand equations (Braithwait, 1977 & 1980; Capps, 1983). Within the cross-sectional context,

the LES estimation assumes that all households face the same commodity prices. This

assumption simplifies the data requirement for estimating the LES parameters given the paucity

of price data. A major challenge of this approach, however, is how to identify the LES

parameters when there is no variation in prices. In this paper we evaluate several sources of

additional information that can be used for identification. Then, we demonstrate an application

of the LES approach using an up-to-date micro household data in Australia. We estimate the

expenditure elasticities for 20 commodities and then derive the own-price and cross-price

elasticities as well as the commodity budget shares to provide a detailed analysis of household

consumption patterns. Furthermore, we estimate the LES parameters for five households types

grouped according to income. The benefit of these quintile estimates is twofold: (1) it provides

a better understanding of the heterogeneity of consumption behaviour across income levels, and

(2) it provides a valuable input for CGE models with microsimulation capabilities.

The remainder of this paper is outlined as follows. Section 2 provides a detailed exposition of

the theoretical underpinnings of the LES including a review of its strengths and weaknesses.

Section 3 discusses issues related to aggregating demand equations over commodities and

consumers. The stringent requirements for consistent aggregation point to the advantage of

using disaggregated data. Section 4 provides an empirical application of estimating the LES

using Australian data. More specifically, we describe the consumption pattern of Australian

households across commodities and then we present our estimates of expenditure and price

elasticities. Section 5 deals with the implications of disaggregating the LES estimates into

quintiles. Section 6 wraps up this paper with some concluding remarks.

2. The Linear Expenditure System

The linear expenditure system (LES, Stone, 1954) expresses expenditure on good i (𝑝𝑖𝑞𝑖 , 𝑖 = 1,⋯ , 𝑛) as a linear function of the prices (𝑝𝑖) and income (𝑀):

𝑝𝑖𝑞𝑖 = 𝑝𝑖𝛾𝑖 + 𝛽𝑖 (𝑀 −∑𝑝𝑗𝛾𝑗

𝑛

𝑗=1

) , 𝑖 = 1,⋯ , 𝑛. (1)

Here, M = ∑ 𝑝𝑖𝑛𝑖=1 𝑞𝑖 is total expenditure, but is conventionally referred to as “income”. As the

first term on the right-hand side of equation (1), 𝑝𝑖𝛾𝑖 , is expenditure on good i unrelated to

income, it can be interpreted as the cost of “subsistence consumption” of the good (assumed to

be positive). Thus, according to (1), the consumer first spends ∑ 𝑝𝑗𝛾𝑗𝑛𝑗=1 to satisfy all

subsistence requirements. The remaining amount, 𝑀 − ∑ 𝑝𝑗𝛾𝑗𝑛𝑗=1 , is called “supernumerary

income”, and a fraction 𝛽𝑖 of this is spent on good i. This 𝛽𝑖 is known as the 𝑖𝑡ℎ marginal share,

with ∑ 𝛽𝑖 = 1,𝑛𝑖=1 0 < 𝛽𝑖 < 1, and answers the question, what fraction of a one-dollar rise in

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income is spent on the good? The utility function lying behind (1) is the Stone-Geary,

𝑢(𝑞1,⋯ , 𝑞𝑛) = ∑ 𝛽𝑖 log(𝑞𝑖 − 𝛾𝑖),𝑛𝑖=1 𝑞𝑖 > 𝛾𝑖 . This takes the Cobb-Douglas form if 𝛾𝑖 = 0, 𝑖 =

1,⋯ , 𝑛. Comprehensive accounts of LES are contained in Goldberger (1987) and Powell

(1974).

The budget share is the fraction of income spent on the good, defined as 𝑤𝑖 =𝑝𝑖𝑞𝑖

𝑀; these shares

are all positive fractions and have a unit sum. Under LES,

𝑤𝑖 =𝑝𝑖𝛾𝑖𝑀

+ 𝛽𝑖 (1 −∑𝑝𝑗𝛾𝑗𝑀

𝑛

𝑗=1

) = 𝑠𝑖 + 𝛽𝑖 (1 −∑𝑠𝑗

𝑛

𝑗=1

) = 𝑠𝑖 + 𝛽𝑖𝑟, (2)

where 𝑠𝑖 =𝑝𝑖𝛾𝑖

𝑀 is the fraction of income devoted to the cost of subsistence consumption of good

i and 𝑟 = 1 − ∑ 𝑠𝑗𝑛𝑗=1 =

𝑀−∑ 𝑝𝑗𝛾𝑗𝑛𝑗=1

𝑀 is the supernumerary ratio, the ratio of supernumerary

income to income. Define the 𝑛 × 1 vector of prices 𝒑 and 𝜸 as the 𝑛 × 1 vector of subsistence

parameters, so that 𝒑′𝜸 = ∑ 𝑝𝑗𝛾𝑗𝑛𝑗=1 is the total cost of subsistence. Then, 𝑤𝑖 = (

𝒑′𝜸

𝑀)𝑝𝑖𝛾𝑖

𝒑′𝜸+

(1 −𝒑′𝜸

𝑀) 𝛽𝑖 , or

𝑤𝑖 = (1 − 𝑟)𝑝𝑖𝛾𝑖𝒑′𝜸

+ 𝑟𝛽𝑖 .

This interesting result reveals the budget share is itself a weighted average of two other shares

that are both constants (when prices remain unchanged), viz., 𝑝𝑖𝛾𝑖

𝒑′𝜸, the share of the good in the

total cost of subsistence, and 𝛽𝑖 , the marginal share. When the consumer is poor, most of income

is absorbed by subsistence, so the supernumerary ratio 𝑟 ≈ 0 and the budget share of good i

approximates 𝑝𝑖𝛾𝑖

𝒑′𝜸. As income grows, 𝑟 rises and 𝑤𝑖 moves away from subsistence towards its

marginal share 𝛽𝑖. This implies that if the good is a necessity, so its share falls with higher

income, then 𝑝𝑖𝛾𝑖

𝒑′𝜸> 𝛽𝑖 , and vice versa for a luxury. Loosely, this says necessities have relatively

large subsistence components, while those of luxuries are small.

The logarithmic differential of (1) is

𝑑(log 𝑝𝑖) + 𝑑(log 𝑞𝑖) =𝑠𝑖𝑤𝑖𝑑(log 𝑝𝑖) +

𝛽𝑖𝑤𝑖𝑑(log𝑀) −

𝛽𝑖𝑤𝑖∑𝑠𝑗

𝑛

𝑗=1

𝑑(log 𝑝𝑗),

where, as before, 𝑠𝑖 =𝑝𝑖𝛾𝑖

𝑀. Collecting terms and defining 𝛿𝑖𝑗 as the Kronecker delta

(𝛿𝑖𝑗 = 1 if 𝑖 = 𝑗, 0 otherwise), this becomes

𝑑(log 𝑞𝑖) =𝛽𝑖𝑤𝑖𝑑(log𝑀) +∑[𝛿𝑖𝑗 (

𝑠𝑖𝑤𝑖− 1) −

𝛽𝑖𝑤𝑖𝑠𝑗]

𝑛

𝑗=1

𝑑(log 𝑝𝑗).

This shows that 𝜂𝑖 =𝛽𝑖

𝑤𝑖 is the income elasticity of good i and 𝜂𝑖𝑗

∗ = 𝛿𝑖𝑗 (𝑠𝑖

𝑤𝑖− 1) −

𝛽𝑖

𝑤𝑖𝑠𝑗 is the

(𝑖, 𝑗)𝑡ℎ Marshallian (or uncompensated) price elasticity. The Slutsky equation is 𝜂𝑖𝑗 = 𝜂𝑖𝑗∗ +

𝑤𝑗𝜂𝑖 , where 𝜂𝑖𝑗 the corresponding Slutsky (compensated) elasticity. Thus, in LES,

𝜂𝑖𝑗 = 𝛿𝑖𝑗 (𝑠𝑖𝑤𝑖− 1) +

𝛽𝑖𝑤𝑖(𝑤𝑗 − 𝑠𝑗). (3)

According to the homogeneity principle, the quantity demanded of each good depends on real

income and relative prices. Thus, an equiproportional change in all prices and (nominal) income

will leave all demands unaffected as relative prices and real income are constant. This implies

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that ∑ 𝜂𝑖𝑗 = 0𝑛𝑗=1 . To show LES satisfies this property, sum both sides of equation (3) over 𝑗 =

1,⋯ , 𝑛:

∑𝜂𝑖𝑗 =𝑠𝑖𝑤𝑖− 1 +∑

𝛽𝑖𝑤𝑖(𝑤𝑗 − 𝑠𝑗)

𝑛

𝑗=1

𝑛

𝑗=1

=1

𝑤𝑖{𝑠𝑖 + 𝛽𝑖 (1 −∑𝑠𝑗

𝑛

𝑗=1

)}− 1 =𝑤𝑖𝑤𝑖− 1 = 0,

where the third step is based on equation (2).

Panels A to G of Table 1 present a convenient summary of the notation, LES and the main

implications of that system. Panel H will be discussed subsequently.

2.1. Attributes of LES, Good and Bad

If citations are a measure of success, then with more than 1,800 cites to the original Stone

(1954) article alone, LES has been hugely successful.2 Partly this is due to its status in history:

It was the first demand system satisfying the “triad” requirements of microeconomics of the

budget constraint, homogeneity of degree zero and symmetry of the substitution effects, as well

being consistent with Engel’s law (the income elasticity of food demand is less than unity).3

There are several other reasons accounting for the prominence of LES:

1. Linearity in variables. As its name indicates, LES expresses expenditures as a linear

function of income and the prices. This leads to straightforward algebraic

manipulation, as well as ease of use when the system is embedded as part of a larger

model.

2. Clear interpretation. As mentioned above, when the 𝛾𝑖’s are positive, they can be

interpreted as subsistence quantities, the minima needed to generate utility. According

to LES, total expenditure is first allocated to subsistence and then to supernumerary

requirements. The coefficients attached to supernumerary income, the 𝛽𝑖’s, are

marginal shares, and the ratio of these to the corresponding budget shares are the

income elasticities. There is no problem when 𝛾𝑖 < 0, but the subsistence

interpretation is not applicable.

3. Parsimony of parameters. In general, the 𝑛 × 𝑛 matrix of compensated price slopes of

the n demand functions is symmetric, negative semi-definite with 1

2𝑛(𝑛 − 1) free

elements. Adding the n income responses gives a total of 1

2𝑛(𝑛 + 1) free parameters,

which for 𝑛 = 10 is 55. By contrast, LES contains n 𝛾𝑖’s and (𝑛 − 1) 𝛽𝑖’s (one is

constrained by ∑ 𝛽𝑖 = 1𝑛𝑖=1 ), or 2𝑛 − 1 = 19 for 𝑛 = 10. There is thus a substantial

economy of parametrization in LES, which is attractive for econometric estimation,

especially for medium to large-size systems and/or when the model is applied to data

with modest variability.

4. Mildly nonlinear in the parameters. As the marginal shares and 𝛾𝑖’s appear as a

product in LES, nonlinear optimisation approaches are needed for estimation. But

modern software can handle this challenge fairly easily.

The above attractive features come at some cost, however – there are no free lunches. The main

limitations of LES are:

1. Additive utility. The underlying utility function is additive in the n goods, meaning that

each marginal utility depends only on consumption of the good in question, not the

others. This is usually taken to imply that LES can be applied to the consumption of

2 The citation count is from Google, March 2020. 3 Cobb-Douglas and CES also satisfy the triad, but as all income elasticities are unity, violate Engel’s law.

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broadly defined commodities only, not more narrow ones that could have utility

interactions with others. Additive utility leads to the following two restrictions on

demand.

2. Income effects. All goods are normal, none inferior. This restriction is possibly a mild

one for all but a few goods.

3. Price substitution.

i. Goods are substitutes. On the basis of the income-compensated substitution

effect, all pairs of goods are substitutes, there can be no complementarity. This

means the Slutsky cross-price elasticities are all positive, as shown in panels F

and G of Table 1.

ii. Price inelasticity. The Marshallian own-price elasticity (which combines the

income and substitution effects) is less than unity in absolute value when the

associated 𝛾𝑖 > 0, as is usually the case (panel D of Table 1).

4. Functional form. Linearity is one of the above advantages of LES, but linear Engel

curves is a restrictive property when the range of income is substantial.

All models have their own strengths and weaknesses; they are only imperfect approximations

to actual behaviour. The LES is not perfect, but when applied with care, the model can shed

light on the determinants of consumption patterns.

2.2. Are all Parameters Identified?

Write equation (1) as

𝑝𝑖𝑞𝑖 = 𝛼𝑖 + 𝛽𝑖𝑀, 𝑖 = 1,⋯ , 𝑛 (4)

where

𝛼𝑖 =∑(𝛿𝑖𝑗−𝛽𝑖)

𝑛

𝑗=1

𝑝𝑗𝛾𝑗 , 𝑖 = 1,⋯ , 𝑛. (5)

As mentioned above, ∑ 𝛽𝑖 = 1.𝑛𝑖=1 Accordingly, the intercepts defined by (5) satisfy

∑ 𝛼𝑖 = 0.𝑛𝑖=1 This restriction, together with ∑ 𝛽𝑖 = 1𝑛

𝑖=1 , ensure that the budget constraint,

∑ 𝑝𝑖𝑞𝑖𝑛𝑖=1 = 𝑀, is satisfied by system (4).

Consider a household survey in which all households face the same prices. These data could be

used with system (4) to estimate the n marginal shares, 𝛽1, ⋯ , 𝛽𝑛 , as well as the n intercepts,

𝛼1,⋯ , 𝛼𝑛 . The question is, can we then recover from these estimates the n subsistence

parameters, 𝛾1, ⋯ , 𝛾𝑛, by using the relationship (5)? Answer: No.

To establish this negative result, define the 𝑛 × 1 vectors 𝜶 = [𝛼1,⋯ , 𝛼𝑛]′, �̃� =

[𝑝1𝛾1, ⋯ , 𝑝𝑛𝛾𝑛]′, and 𝜷 = [𝛽1,⋯ , 𝛽𝑛]

′. We can then write (5) in vector form as 𝜶 =(𝑰 − 𝜷𝜾′)�̃�, where 𝑰 is the 𝑛 × 𝑛 identity matrix and 𝜾′ = [1,⋯ ,1], or

𝜶 = 𝑨�̃�, 𝑨 = 𝑰 − 𝜷𝜾′. The A matrix has zero column sums, that is,

𝜾′𝑨 = 𝜾′𝑰 − 𝜾′𝜷𝜾′ = 𝜾′ − 𝜾′ = 𝟎′, where the second step follows from ∑ 𝛽𝑖 = 1,

𝑛𝑖=1 or 𝜾′𝜷 = 1. This means 𝑨 is singular and �̃�

cannot be recovered as 𝑨−1𝜶. The basic problem is that there are only 𝑛 − 1 independent

values of the intercepts, as one is constrained by ∑ 𝛼𝑖 = 0,𝑛𝑖=1 which provide insufficient

information to determine the n subsistence expenditures. These matters have been discussed by

Howe (1975), Pollak and Wales (1978) and Powell (1973).

The next several sub-sections present alternative approaches to identifying the parameters of

LES.

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2.3. A Zero Restriction

Suppose we now add another “good”, to be labelled the (𝑛 + 1)𝑡ℎ, and apply LES. Denote the

marginal share of the new good by 𝛽𝑛+1 > 0, and those of the pre-existing ones by 𝛽1∗, ⋯ , 𝛽𝑛

∗ , so the vector of all 𝑛 + 1 shares is 𝜷 = [𝛽1

∗, ⋯ , 𝛽𝑛∗ , 𝛽𝑛+1]

′ = [𝜷∗′, 𝛽𝑛+1]′. As before, the

marginal shares have a unit sum, 𝜾′𝜷 = 1, or 𝜾′𝜷∗ = 1 − 𝛽𝑛+1. Suppose also we believe the subsistence parameter for the new good is zero, that is, 𝛾𝑛+1 = 0. Then, LES takes the form

𝑝𝑖𝑞𝑖 =

{

𝛼𝑖

∗ + 𝛽𝑖∗𝑀, 𝛼𝑖

∗ =∑(𝛿𝑖𝑗 − 𝛽𝑖∗)

𝑛

𝑗=1

𝑝𝑗𝛾𝑗 , 𝑖 = 1,⋯ , 𝑛,

𝛼𝑛+1 + 𝛽𝑛+1𝑀, 𝛼𝑛+1 = −𝛽𝑛+1∑𝑝𝑗𝛾𝑗

𝑛

𝑗=1

, 𝑖 = 𝑛 + 1.

(6)

As usual, “income” is interpreted as total “expenditure”, 𝑀 = ∑ 𝑝𝑖𝑞𝑖𝑛+1𝑖=1 . Denoting the vector

of intercepts in (6) by 𝜶 = [𝛼1∗,⋯ , 𝛼𝑛

∗ , 𝛼𝑛+1]′ = [𝜶∗′, 𝛼𝑛+1]

′ , and the vector of subsistence

expenditures by �̃� = [𝑝1𝛾1, ⋯ , 𝑝𝑛𝛾𝑛]′, then, similar to above, there is the relationship:

𝜶∗ = 𝑨∗�̃�, 𝑨∗ = 𝑰 − 𝜷∗𝜾′. Each column sum of the matrix 𝑨∗is equal to 𝛽𝑛+1 as

𝜾′𝑨∗ = 𝜾′ − 𝜾′𝜷∗𝜾′ = 𝜾′ − (1 − 𝛽𝑛+1)𝜾′ = 𝛽𝑛+1𝜾

′. Thus, in contrast to before, 𝑨∗is not singular, and the subsistence parameters can be recovered

as

�̃� = (𝑨∗)−1𝜶∗. The above shows that all parameters of LES are identified by applying the model to the sub-

system comprising the first n of the 𝑛 + 1 goods and by assuming that 𝛾𝑛+1 = 0. This simple

result is potentially useful, but it raises the obvious question, what is the source of this zero

restriction? As established by Howe (1975), taking expenditure on “good” 𝑛 + 1 to be savings

and setting 𝛾𝑛+1 = 0 produces the extended linear expenditure system (ELES, Lluch, 1973).

The “income” term in ELES is literally income, total expenditure on the n goods plus savings.

The slope coefficient 𝛽𝑖∗ in (6) is then interpreted as the 𝑖𝑡ℎ marginal propensity to spend out of

income, while 𝛽𝑖∗

∑ 𝛽𝑗∗𝑛

𝑗=1 is the corresponding marginal share.4 The zero restriction of this approach

provides the one additional piece of information required for identification.

2.4. The Cost of Subsistence

The parameters of LES can also be identified once the cost of subsistence is known. Using

minimum-cost diets, Howe (1977) determines subsistence expenditure for food, thereby

identifying LES. As will be shown in this sub-section, within the LES framework, the total cost

of subsistence is implied by a measure of the income-curvature of the utility function that is

related to the overall degree of substitutability among goods.

As mentioned above, the utility function corresponding to LES is 𝑢(𝑞1,⋯ , 𝑞𝑛) =

∑ 𝛽𝑖 log(𝑞𝑖 − 𝛾𝑖)𝑛𝑖=1 . The marginal utility of good i is

𝜕𝑢

𝜕𝑞𝑖=

1

𝑞𝑖−𝛾𝑖. For this to be positive, 𝑞𝑖 >

𝛾𝑖 , or 𝑝𝑖𝑞𝑖 > 𝑝𝑖𝛾𝑖 . Summing over 𝑖 = 1,⋯ , 𝑛, this implies

𝑀 > 𝒑′𝜸, (7)

4 Powell (1973) provides a detailed analysis of the close relationship between LES and ELES and related econometric issues.

8

where M is income and 𝒑′𝜸 = ∑ 𝑝𝑖𝑛𝑖=1 𝛾𝑖 is the cost of subsistence. Income exceeding

subsistence means supernumerary income, 𝑀 − 𝒑′𝜸, is positive.

For a budget-constrained maximum, each marginal utility is proportional to the corresponding

price, that is, 𝛽𝑖

𝑞𝑖−𝛾𝑖= 𝜆𝑝𝑖 , 𝑖 = 1,⋯ , 𝑛, where 𝜆 is the marginal utility of income

𝜕𝑢

𝜕𝑀.

Multiplying through by 𝑞𝑖 − 𝛾𝑖 and then summing over 𝑖 = 1,⋯ , 𝑛 gives ∑ 𝛽𝑖𝑛𝑖=1 = 𝜆(𝑀 −

𝒑′𝜸). As ∑ 𝛽𝑖𝑛𝑖=1 = 1, 𝜆 =

1

𝑀−𝒑′𝜸> 0, the reciprocal of supernumerary income. Since

𝜕 𝜆

𝜕𝑀=

−𝜆2 < 0, the income elasticity of the marginal utility of income is 𝜕 𝜆

𝜕𝑀

𝑀

𝜆= −

𝑀

𝑀−𝒑′𝜸 , the

negative inverse of the supernumerary ratio. This elasticity, to be denoted by 𝜔, shall be

referred to as the “Frisch parameter”:

𝜔 = −𝑀

𝑀 − 𝒑′𝜸. (8)

Under condition (7), the marginal utility of income is positive (𝜆 =1

𝑀−𝒑′𝜸> 0, ), but declines

with increasing income, according to (8). The larger is the supernumerary ratio, the smaller the

Frisch parameter in absolute value – that is, a rise in income causing the supernumerary ratio

to also rise leads to a decrease in |𝜔|.5 Rearrangement of equation (8) yields a simple expression for the cost of subsistence as a

proportion of income 𝒑′𝜸

𝑀= 1 +

1

𝜔. (9)

In view of condition (7), 𝜔 < −1, which ensures the right-hand side of (9) is a positive fraction.

The Frisch parameter is a measure of the curvature of the indirect utility function and −1

𝜔 is the

average elasticity of substitution 𝜎 (Powell, 1992). Panel H of Table 1 summarises the above

utility aspects of LES.

Equation (9) reveals that once the value of 𝜔 is known, the subsistence share of income can be

determined; and given income, we can infer 𝒑′𝜸. Writing equation (5) as 𝛼𝑖 = 𝑝𝑖𝛾𝑖 − 𝛽𝑖𝒑′𝜸,

the value of subsistence expenditure for each good can then be recovered from the estimated

intercept and marginal share as 𝑝𝑖𝛾𝑖 = 𝛼𝑖+𝛽𝑖𝒑

′𝜸, 𝑖 = 1,⋯ , 𝑛. (10)

Adding both sides of this equation over 𝑖 = 1,⋯ , 𝑛 gives

∑𝑝𝑖𝛾𝑖 =∑𝛼𝑖 +

𝑛

𝑖=1

(∑𝛽𝑖

𝑛

𝑖=1

)

𝑛

𝑖=1

𝒑′𝜸.

For ∑ 𝑝𝑖𝛾𝑖 =𝑛𝑖=1 𝒑′𝜸 to hold, it must be the case that ∑ 𝛼𝑖 = 0 and∑ 𝛽𝑖

𝑛𝑖=1

𝑛𝑖=1 = 1. These

conditions were discussed above, and are satisfied automatically by model (4). Accordingly,

the approach is internally consistent. Note that as the units of 𝛾𝑖 are the same as those of 𝑞𝑖 , the

units of 𝑝𝑖𝛾𝑖 are dollars per period, the same as expenditure.

5 A note on notation and nomenclature: The disparate conventions used in the literature can be a source of confusion. Frisch

(1959) denotes the marginal utility of income by 𝜔, and its income elasticity by �̌�, which he calls the “money flexibility”. In the

context of the differential approach, Theil (1975/76) calls 1/𝜔 the “income flexibility” and denotes it by 𝜙. Below, 𝜔 denotes the income elasticity of the marginal utility of income, (𝜕 𝜆/𝜕𝑀)(𝑀/𝜆), and following Lluch et al. (1977), we refer to it as “the Frisch

parameter”, although this is not a constant parameter in LES. Lluch et al. (1976) denote the supernumerary ratio of LES, (𝑀 −𝒑′ 𝜸)/𝑀, by −𝜙, so in that context 𝜔 = 𝜙−1. Clearly, this 𝜙 and Theil’s are both the reciprocal of the income elasticity of the marginal utility of income, the only difference being the former refers to LES, the latter to general differential demand equations.

9

2.5. The Frisch Parameter

To use equations (9) and (10) for identification, we need the value of the Frisch parameter 𝜔.

Where do we turn?

LES implies that 𝜔 is not constant, but dependent upon income and prices. From equation (8),

the income elasticity of 𝜔 is 𝜕 log|𝜔|

𝜕 log𝑀= 1 + 𝜔 < 0. (11)

The negative value of this elasticity follows from condition (7), which implies 𝜔 < −1. Thus,

𝜔 declines in absolute value as income rises. As stated above, the marginal utility of income is

always positive and always declines with additional income (that is, 𝜔 < 0). But according to

equation (11), the decline in |𝜔| is damped as the consumer becomes richer. Again from

equation (8), the price elasticity of the Frisch parameter is 𝜕 log|𝜔|

𝜕 log 𝑝𝑖= −𝜔

𝑝𝑖𝛾𝑖𝑀

> 0, 𝑖 = 1,⋯ , 𝑛.

Thus, the effect on |𝜔| of a price increase has the opposite sign to an income increase. As

∑𝜕 log|𝜔|

𝜕 log 𝑝𝑖= −𝜔

𝒑′𝜸

𝑀

𝑛𝑖=1 = −(1 + 𝜔), which in combination with (11), shows that an

equiproportional rise in income and all prices leaves Frisch unchanged. This, of course, simply

reflects the homogeneity of expression (8).

Thus to apply (9) and (10), should the LES framework be strictly followed and 𝜔 be taken to

be a function of income and prices? Or might a local approximation of taking 𝜔 to be a constant

over the relevant range be satisfactory? There can be no hard and fast answers in the abstract,

as the issue would seem to be an empirical one to be resolved with reference to the particular

dataset being used. Nonetheless, some guidance is available from three sources. To the extent

there are variations in 𝜔, it seems reasonable that income changes would be the dominant

influence. Accordingly, in what follows we shall ignore any impact of prices.

2.5.1. Source I, Frisch’s Conjecture

Frisch (1959, p. 189) famously speculated how the income elasticity of the marginal utility of

income might vary with the affluence of consumers. In what has become known as the “Frisch

conjecture”, he viewed 𝜔 as taking a large negative value for the extremely poor and then

moving in the direction of 0 as consumers become more affluent. In his words:

We may, perhaps, assume that in most cases the [Frisch parameter] has values of the order of

magnitude given below.

𝜔 = −10 for the extremely poor and apathetic part of the population.

𝜔 = −4 for the slightly better off but still poor part of the population with a

fairly pronounced desire to become better off.

𝜔 = −2 for the middle income bracket, “the median part” of the population.

𝜔 = −0.7 for the better off part of the population.

𝜔 = −0.1 for the rich part of the population with ambitions towards

“conspicuous consumption”.

The above direction of change in 𝜔 as income rises agrees with that implied by LES --|𝜔| falls

in both cases. Although Frisch did not work within the LES framework, the combination of his

conjecture with LES provides some interesting implications. Given 𝜔 and its change, the

income change implied by equation (11) is 𝑑(log𝑀) =𝑑(log|𝜔|)

1+𝜔. For consumer groups 𝑔 and

𝑔 + 1, this difference can be evaluated as

10

log𝑀𝑔+1𝑀𝑔

=log(𝜔𝑔+1 𝜔𝑔⁄ )

1 + �̅�, (12)

where �̅� =1

2(𝜔𝑔+1+𝜔𝑔). For the comparison of group 𝑔 + 1 with 𝑔, the use of the average

value of 𝜔 in equation (12) seems natural.

Table 2 uses equation (12) to obtain the income associated with the values of 𝜔 suggested by

Frisch.6 The final column of the table shows that income grows by 16 percent in moving from

the extremely poor to the slightly better off, while those in the middle are 65 percent more

affluent than the extremely poor. These implications might seem not completely unreasonable

and provide some support for using the Frisch conjecture as a guide to specifying 𝜔. On the

other hand however, there are three problems. First, the middle-poor income gap appears to be

too narrow. Second, Frisch’s values of 𝜔 for the better off and the rich violate the LES

condition (7), which means that the approach breaks down for these groups. Third, the basis for

Frisch’s values is not clear-cut. That is not to say the Frisch conjecture is based on no evidence.

The above quotation from Frisch follows immediately after his discussion of values of 𝜔

obtained by Johansen with Norwegian data from the 1950s. Frisch (1959, p. 189) gives three

values of 𝜔: -1.94 (derived from the demand elasticities for agricultural products), -2.13

(industrial foods and beverages) and -1.85 (land and air transport). He comments “It is truly

remarkable that data taken from such different sources and giving such different budget

proportions and Engel elasticities, give such consistent values of [𝜔]. A compromise value of

about -2 for the money flexibility [𝜔] ought to be a reasonable estimate.” Thus, Frisch’s value

for the middle income bracket of -2 is consistent with this evidence, but the source for the other

values is not revealed. Relatedly, the range of his values could be questioned: In going from -

10 to -0.1, these values differ by a factor of 100, which for an elasticity, is huge. It seems fair

to conclude that, taken as a whole, the Frisch conjecture is more impressionistic than

econometric.

2.5.2. Source II, the Lluch-Powell-Williams Book

A widely used source of values of the Frisch parameter 𝜔, especially in CGE modelling, is the

book by Lluch, Powell and Williams (1977).7

Lluch et al. (1977) estimated the extended linear expenditure system (ELES, Lluch, 1973) with

time-series data. In this model, 𝜔 = −𝑀

𝑀−𝒑′𝜸< 0 , which is the same as for the linear

expenditure system, equation (8). For a given value of 𝒑′𝜸, 𝜔 falls absolutely as income (M)

rises, according to equation (11). But as Lluch et al. (1977) estimate ELES separately for each

of a number of countries with time-series data, 𝒑′𝜸 and 𝜔 take their own values for each

country, so that there is no algebraic relationship between the value of the Frisch parameter

and income across countries. There is, however, an empirical relationship identified by Lluch

et al. For 14 countries, they regressed their estimates of the logarithm of −𝜔 on the logarithm

of GDP per capita to give (p. 76)

−𝜔 = 36 × GDP−0.36. (13)

6 An alternative to equation (12) might be to integrate equation (11) to give |𝜔| = 𝑘𝑀1+𝜔 , where k is an arbitrary constant.

Rearranging, 𝑀 = (|𝜔|

𝑘)

1

1+𝜔. Given 𝜔 (from Frisch), this is one equation in two unknowns, M and k. Thus, similar to setting 𝑀1 =

100 in Table 2, some income normalisation would be needed for this alternative. 7 According to Google Scholar, there are more than 500 citations to this book.

11

Thus, 𝜔 declines in absolute value as income rises, which agrees with the Frisch conjecture.

While the income elasticity is modest at -0.36, with a standard error is 0.10 it is highly

significant.

Figure 1 gives the associated plot: Note the wide range of incomes on the horizontal axis. We

can calculate from the data given in Lluch et al. (1977, Table 4.4, p. 75) that the interquartile

range of income is about 40 percent more than the median. The negative relationship between

−𝜔 and income is clear from the plot, but there seems to be several substantial outliers (such

as countries numbered 5, 7-10) and there may be a question about the linearity assumption. But

still, with only 14 observations, there is probably not much scope for alternatives.

The estimates of 𝜔 used to obtain equation (13) are based on data from the 1950s and 60s. We

shall use that equation to give 𝜔-values for a number of countries in recent times. Lluch et al.

(1977) measure GDP in terms of US dollars at 1970 prices. Starting with GDP in 2018 in US

dollars of 2018, in order to apply (13) we need to convert to GDP in 1970 prices; this would

obviously yield real GDP in 2018 expressed in 1970 prices We shall carry out this conversion

by making some rough guesses at the appropriate US inflation rate. There are obvious

uncertainties involved in this process. The GDPs in 2018 US dollars involve the conversion

from domestic currency units into US dollars: It is not clear what exchange rate should be used

-- the market rate at mid-year, end-of-year, the average over the year, PPP, or any other of a

host of alternatives. And even the choice of the inflation rate is not straightforward -- what

deflator should be used and does it adequately account for quality change, new goods and

substitution?

Table 3 contains the results, as well as the details of the calculations. Bearing in mind the above

issues, these results are purely illustrative of the approach. Several comments can be made

about the projections of −𝜔 in columns 5 and 6 of Table 3.

1. As expected, the results are sensitive to the underlying assumption regarding income.

When inflation is taken to be higher, the associated income is lower and –𝜔 is

correspondingly higher: The column 6 values of −𝜔 (based on inflation of 5 percent p.

a.) are more than 50 percent larger than those of column 5 (2 percent inflation).

2. The projections can be compared with those of the Frisch conjecture. As mentioned,

the decline in −𝜔 is consistent with Frisch. For the poorest country, Burundi, under

the low-income assumption, −𝜔 = 11.12 (first entry of column 6). This is not far

from Frisch’s value of 10 for the extremely poor. Under the higher-income

assumption, the value for richest country (Monaco) is −𝜔 = 0.64, which seems some

way away from Frisch for the rich of 0.1. However, these values refer to the extremes

of the income distribution and by the nature of extremes, these are not representative

of the population. Arguably, percentiles would be more appealing, and these are

summarised in Table 4. This comparison establishes that the only reasonably close

agreement with Frisch occurs for the median under low inflation. For other parts of the

income distribution, the two sources seem to depart substantially. It should be

acknowledged, however, while this comparison is interesting, not too much should be

made of it as Frisch’s values should not necessarily be taken as the benchmark against

which all others are to be assessed. Frisch’s conjecture seems to have an impressionist

feel to it, as argued above, and should not be accorded an exalted status.

3. The values of –𝜔 for the high-income countries of panel B of Table 3 do not vary a

great deal. The Frisch parameter for Singapore, the richest country in the panel, is only

12

about 10 percent less than that of Canada, the least rich.8 Note also the column 5

values of –𝜔 for the four richest countries (which includes Australia) are below 1,

which is not consistent with the LES condition (7). The same problem occurs for

Monaco (in the same column).

4. The cross-country dispersion of GDP has risen substantially since 1970, the year used

to derive equation (13) by Lluch et al. (1977). As noted above, in 1970 the

interquartile range of GDP was about 40 percent more than the median. By 2018 this

had risen to about 170 percent. The 1970 gap of 40 percent is itself considerable, so

the increase to 170 percent is indeed substantial. Thus, a qualification is required

regarding the possible fragility of the extrapolations to a period of much more

disparate incomes, extrapolations that might go beyond what could be reasonably

considered the “relevant range” applicable to the original regression results.

There is much to admire in the Lluch et al. (1977) book. Its boldness of concept, breadth and

clarity make it a landmark in applied demand analysis; it is easy to understand why it has been

so influential. The book provides a valuable source for the Frisch parameter. But due to the

issues noted above, the projections of Table 3 are to be interpreted as only indicative of how

the Lluch et al. results might be used in a contemporary setting. More detailed consideration of

the underlying income projections is needed before that approach could be used in more

substantial applications.

2.5.3. Source III, Set the Frisch Parameter to -2

In view of the issues noted above, a case can be made for the simpler approach of taking 𝜔 to

be constant and equal to -2. While this approach contradicts the range of values conjectured by

Frisch (not such a bad thing by itself), it agrees with his value for the “median part” of the

population. This section briefly reviews previous studies and concludes that while the evidence

in favour of 𝜔 = −2 is not overwhelming, this is a reasonable value to use.

In LES the Frisch “parameter” is a variable. By contrast, in the Rotterdam model, due to Barten

(1964) and Theil (1965), the reciprocal of Frisch parameter (denoted by 𝜙) is parametrized as

a constant. Clements and Si (2017) estimate the Rotterdam model with time-series data for 9

goods as a separate system for each of 37 OECD countries and Table 5 contains the 𝜙 -

estimates. These seem to be more or less scattered randomly around the mean of -0.54. Using

several tests, Clements and Si (2017) find that these estimates are unrelated to income.

Clements and Si (2017, 2018) also apply a related approach to cross-country data for 176

countries from the International Comparison Program (World Bank, 2015). This involves

estimating 𝜙 via a cross-commodity regression. As can be seen from Table 6, the average 𝜙-

value is -0.52, with only the top income quartile (where 𝜙 = −0.22) apparently substantially

different.

These findings indicate 𝜙 has a central value of about −½, or 𝜔 =1

𝜙= −2. This 𝜙-value is

also in broad agreement with earlier estimates, as reviewed by Clements and Zhao (2009, pp.

228-29), obtained with approaches similar to those described above. Still earlier estimates were

surveyed by Brown and Deaton (1972, p. 1206), who state “there would seem to be fair

agreement on the use of a value for 𝜙 around minus one half”.

There are several other approaches to estimating 𝜙:

8 According to Table 3, on a GDP per capita basis, Singapore is about 40 percent richer than the Canada. Thus, from equation

(13), the ratio of the Frisch parameters for these two countries is about 1.4−0.36 = 0.9, or Singapore’s is about 10 percent less, in accordance the values of Table 3.

13

1. The income-tax schedule. Assuming that the tax structure reflects the community’s

collective attitude to the “fair” distribution of after-tax income, 𝜙 can be estimated

from the country’s income-tax schedule with the equal sacrifice model. Using this

approach, Evans (2005) finds the average of estimates of − 1𝜙 from 20 OECD countries

to be about 1.4, so 𝜙 is about -0.71. An open issue is the underlying basis for the tax

schedule: Can it really be interpreted as an explicit statement of the people’s collective

values, or is the schedule set politically/administratively, or some other equally

opaque process?

2. Subjective happiness. Using survey data on happiness as a type of direct measure of

utility, Layard et al. (2008) estimate − 1𝜙 to range from 1.34 to 1.19, with a combined

estimate of 1.26, implying 𝜙 ∈ [−0.75,−0.84], and 𝜙 = −0.79 corresponding to the

combined estimate. These values are (absolutely) larger than −½, although it seems

questionable whether reported subjective happiness is synonymous with utility of the

utility-maximising consumer.9

3. The life-cycle model and choice under uncertainty. According to the life-cycle model,

− 1𝜙 is the coefficient in a regression of the growth in consumption on the real interest

rate (Hansen and Singleton, 1982). In the context of choice under uncertainty, the

Frisch parameter is related to the degree of risk aversion, and insurance data can

provide another way to estimate 𝜙. Groom and Maddison (2019) uses these two ways,

as well as the equal sacrifice model, the happiness approach and the Rotterdam model.

Their combined estimate of 1

𝜙 is about −1.5, or 𝜙 ≈ −0.67, which is unlikely to

differ significantly from −0.5. 10

As discussed above, Lluch et al. (1977) find that –𝜔 declines with income, which is supportive

of the Frisch conjecture. DeJanvry et al. (1972) and Gao (2012, p. 106) also find the same thing.

As these results conflict the above evidence that 𝜙 ≈ −1

2, or –𝜔 ≈ 2, there is uncertainty

surrounding the constancy or otherwise of the Frisch parameter. Bear in mind this parameter is

the income elasticity of the marginal utility of income, which involves the second derivative of

the utility function; and how it changes refers to the third derivative. As such higher-order

effects are usually difficult to estimate precisely, the uncertainty around the status of the Frisch

parameter is not surprising. Nevertheless, if a single value is required, –𝜔 ≈ 2 would appear

to be a reasonable choice, although one cannot be dogmatic. In terms of LES, –𝜔 ≈ 2 implies

subsistence costs one-half of income [see equation (9)], which again appears reasonable.

The Appendix analyses some further implications for the utility function of the choice of the

value of 𝜔. Two other issues are also included in the Appendix: (i) A discussion of another way

to identify the parameters of LES; and (ii) how demographic factors can be incorporated into

LES.

9 Following Layard et al. (2008), Gandelman and Hernández-Murillo (2015) estimate the coefficient of relative risk aversion for 75

countries by regressing happiness on income. They find that the coefficient of relative risk aversion varies between 0 and 3,

and tends to be closely scattered around 1. Although they do not note it, in principle, this coefficient is equivalent to − 1

𝜙. When

− 1

𝜙= 1, 𝜙 = −1, which is some distance away from −

1

2 . That could possibly be explained again by the difficulty in identifying

the concept of self-reported personal happiness with utility. 10 Stern (1977) provides an earlier review of the measurement of 𝜙.

14

3. Aggregation

This section discusses issues related to aggregating demand equations over commodities and

consumers.

3.1. Aggregating Commodities

As before, the linear expenditure system is 𝑝𝑖𝑞𝑖 = 𝑝𝑖𝛾𝑖 + 𝛽𝑖(𝑀 − ∑ 𝑝𝑗𝛾𝑗𝑛𝑗=1 ), 𝑖 = 1,⋯ , 𝑛.

Suppose the n good are combined into G groups, denoted by 𝑺1, ⋯ , 𝑺𝐺 . The alcoholic beverages

group comprising beer, wine and spirits could be one group. LES can then be expressed as

𝑝𝑖𝑞𝑖 = 𝑝𝑖𝛾𝑖 + 𝛽𝑖(𝑀 − ∑ 𝑝𝑗𝛾𝑗𝑛𝑗=1 ), 𝑖 ∈ 𝑺𝑔, 𝑔 = 1,⋯ , 𝐺, which we write as

𝑣𝑖 = 𝛾𝑖∗ + 𝛽𝑖 (𝑀 −∑ ∑ 𝛾𝑗

∗

𝑗∈𝑺𝑔

𝐺

𝑔=1

) , 𝑖 ∈ 𝑺𝑔 , 𝑔 = 1,⋯ , 𝐺. (14)

Here, 𝑣𝑖 = 𝑝𝑖𝑞𝑖 is expenditure on good 𝑖 ∈ 𝑺𝑔 and 𝛾𝑖∗ = 𝑝𝑖𝛾𝑖 is the cost of subsistence of that

good.

Let 𝑉𝑔 = ∑ 𝑣𝑖𝑖∈𝑺𝑔 be total expenditure on group g and Γ𝑔∗ = ∑ 𝛾𝑖

∗𝑖∈𝑺𝑔 be the subsistence cost of

this group. Then, adding both sides of equation (14) over 𝑖 ∈ 𝑺𝑔 gives

𝑉𝑔 = Γ𝑔∗ + Β𝑔 (𝑀 −∑Γℎ

∗

𝐺

ℎ=1

) , 𝑔 = 1,⋯ , 𝐺, (15)

where Β𝑔 = ∑ 𝛽𝑖𝑖∈𝑺𝑔 is the marginal share of group g. According to (15), expenditure on group

g depends on subsistence expenditure and income in exactly the same form as in system (14),

which refers to members of the group. If expenditure on beer, wine and spirits is each governed

by LES, then so is total expenditure on alcohol, which can be expressed by saying LES

aggregates consistently. This means that if we apply LES to broad aggregates (such as food,

alcoholic beverages, clothing, etc.), the coefficients are aggregates of those of the members of

the group, the micro coefficients. It also means the level of aggregation of goods can be easily

varied.

Additionally, expenditure on members of the group can be expressed as depending on within-

group subsistence and total expenditure on the group. To show this, eliminate M from equations

(14) and (15) by rearranging (15) as 𝑀 =𝑉𝑔−Γ𝑔

∗

Β𝑔+∑ Γℎ

∗𝐺ℎ=1 , and then substituting for M in (14).

This yields

𝑣𝑖 = 𝛾𝑖∗ + 𝛽𝑖

′ (𝑉𝑔 − ∑ 𝛾𝑗∗

𝑗∈𝑺𝑔

) , 𝑖 ∈ 𝑺𝑔 , 𝑔 = 1,⋯ , 𝐺, (16)

where 𝛽𝑖′ =

𝛽𝑖

Β𝑔 is the within-group marginal share, the fraction of an additional $1 of group

expenditure allocated to 𝑖 ∈ 𝑺𝑔 . As (16) pertains to within-group variables only, it can be

described as a system of conditional expenditure equations. Further, as (16) is of the LES form,

we can say that this model has the additional property of consistent disaggregation.11

11 A clear statement of the above material, expressed slightly differently, is contained in Powell (1974, pp. 39-40), who attributes

the results to Stone (1970, pp. 85-86). Powell points to a potential troubling consistency issue with the implied price index of

group g, defined as 𝑃𝑔 = ∑ (𝛾𝑖 ∑ 𝛾𝑗𝑗∈𝑺𝑔⁄ ) 𝑝𝑖 ,𝑖∈𝑺𝑔 so that ∑ 𝑝𝑖𝛾𝑖𝑖∈𝑺𝑔 = 𝑃𝑔Γ𝑔, with Γ𝑔 = ∑ 𝛾𝑖 .𝑖∈𝑺𝑔

15

Systems (15) and (16) can be interpreted in terms of a two-stage decision process. The

consumer first allocates total expenditure (income) to the G broad aggregates according to the

group expenditure equations (15). Second, the previously determined expenditure on group g,

𝑉𝑔, is then allocated to the members of the group according to (16). This is an intuitively

appealing interpretation.

3.2. Aggregating over Consumers

We shall illustrate the key issues involved in aggregating over consumers by using two

examples.

First, suppose consumer 𝑐 (𝑐 = 1,⋯ , 𝐶) has income 𝑚𝑐 and spends 𝑣𝑐 on some commodity

according to a linear Engel curve: 𝑣𝑐 = 𝛼𝑐 + 𝛽𝑐𝑚𝑐 . (17)

It is to be noted that as the coefficients 𝛼𝑐 and 𝛽𝑐 differ over consumers, heterogeneity is

allowed for. Thus, equation (17) for 𝑐 = 1,⋯ , 𝐶 represents the individual equations. The

corresponding aggregate equation refers to the relation between total expenditure on the good

by all C individuals and their income. Expressed on a per capita basis, this is: 𝑣 = 𝐴 + 𝐵𝑚, (18)

where 𝑣 =1

𝐶∑ 𝑣𝑐𝐶𝑐=1 and 𝑚 =

1

𝐶∑ 𝑚𝑐𝐶𝑐=1 are per capita expenditure and income, respectively;

and A and B are coefficients. What is the relation between these coefficients and those of

equation (17)? Put another way, if we use aggregate data to estimate (17), what, if anything,

can be inferred regarding the individual coefficients?

Equation (18) can be obtained from (17) by simply adding both sides of the latter over 𝑐 =1,⋯ ,𝐶 and then dividing by C:

𝑣 =1

𝐶∑𝛼𝑐

𝐶

𝑐=1

+1

𝐶∑𝛽𝑐𝑚𝑐 =

𝐶

𝑐=1

𝛼 + (∑𝑚𝑐

𝑀𝛽𝑐

𝐶

𝑐=1

)𝑚, (19)

where 𝛼 is the mean of the intercepts of the individual equations and 𝑀 = ∑ 𝑚𝑐𝐶𝑐=1 = 𝐶𝑚 is

total income. It can be seen that the slope coefficient of (19), ∑𝑚𝑐

𝑀𝛽𝑐

𝐶𝑐=1 , is a weighted average

of the individual slopes, where the weights are the income shares, 𝑚1

𝑀, ⋯,

𝑚𝐶

𝑀; these weights are

all positive and have a unit sum. Accordingly, the aggregated coefficient coincides with the

slope of equation (18), but the difficulty is ∑𝑚𝑐

𝑀𝛽𝑐

𝐶𝑐=1 will only be a constant coefficient under

the unattractive condition that all incomes change proportionately.

Consider the situation when the income slope of each individual takes the same value. When

𝛽𝑐 = 𝛽, the aggregate coefficient obviously coincides with the common individual slope as

𝐵 = ∑𝑚𝑐

𝑀𝛽𝑐 = 𝛽.

𝐶𝑐=1 In this case, aggregation is perfect. Due to its linearity, this result is

applicable to the LES. To see this, assume all consumers face the same prices and that the

marginal shares of LES are the same across consumers, while the subsistence quantities differ,

that is, 𝑝𝑖𝑞𝑖𝑐 = 𝑝𝑖𝛾𝑖𝑐 + 𝛽𝑖(𝑀𝑐 −∑ 𝑝𝑗𝛾𝑗𝑐𝑛𝑗=1 ). The average over the C consumers is

𝑝𝑖1

𝐶∑𝑞𝑖𝑐

𝐶

𝑐=1

= 𝑝𝑖1

𝐶∑𝛾𝑖𝑐

𝐶

𝑐=1

+ 𝛽𝑖 (1

𝐶∑𝑀𝑐

𝐶

𝑐=1

−∑𝑝𝑗1

𝐶∑𝛾𝑗𝑐

𝐶

𝑐=1

𝑛

𝑗=1

).

16

This is just LES in per capita terms, viz., 𝑝𝑖𝑞𝑖 = 𝑝𝑖𝛾𝑖 + 𝛽𝑖(𝑀 − ∑ 𝑝𝑗𝛾𝑗𝑛𝑗=1 ). Under the

conditions stated above, LES aggregates perfectly.12 However, it should be noted, the condition

that the marginal shares are the same across consumers is a strong one.

The second example is a nonlinear Engel curve, a quadratic of the form 𝑣𝑐 = 𝛼𝑐 + 𝛽𝑐𝑚𝑐 +𝛿𝑐𝑚𝑐

2. Aggregating as before, we have

𝑣 =1

𝐶∑𝛼𝑐

𝐶

𝑐=1

+1

𝐶∑𝛽𝑐𝑚𝑐 +

1

𝐶∑𝛿𝑐𝑚𝑐

2 =

𝐶

𝑐=1

𝐶

𝑐=1

𝛼 + (∑𝑚𝑐

𝑀𝛽𝑐

𝐶

𝑐=1

)𝑚

+ (∑𝑚𝑐2

∑ 𝑚𝑑2𝐶

𝑑=1

𝛿𝑐

𝐶

𝑐=1

)∑ 𝑚𝑑

2𝐶𝑑=1

𝐶.

The term linear in income has the same form as before. Defining the variance of income as

var(𝑚𝑐) =1

𝐶∑(𝑚𝑑 −𝑚)

2

𝐶

𝑑=1

=∑ 𝑚𝑑

2𝐶𝑑=1

𝐶−𝑚2,

the quadratic term becomes

(∑𝑚𝑐2

∑ 𝑚𝑑2𝐶

𝑑=1

𝐶

𝑐=1

𝛿𝑐)∑ 𝑚𝑑

2𝐶𝑑=1

𝐶= 𝜆{var(𝑚𝑐) + 𝑚

2}, 𝜆 =∑𝑚𝑐2

∑ 𝑚𝑑2𝐶

𝑑=1

𝐶

𝑐=1

𝛿𝑐.

Thus, the aggregate equation takes the form

𝑣 = 𝛼 + (∑𝑚𝑐

𝑀𝛽𝑐

𝐶

𝑐=1

)𝑚 + 𝜆{var(𝑚𝑐) + 𝑚2}.

Aggregate expenditure now depends explicitly on the dispersion of income, as well as the

squared value of per capita income. The coefficient of the dispersion term, {var(𝑚𝑐) + 𝑚2}, 𝜆,

is a weighted average of the individual coefficients, 𝛿1,⋯ , 𝛿𝐶 . The weights are now more

complex as they are proportional to the squares of individual incomes.

These two simple examples illustrate the difficulties that can arise in inferring individual

behaviour from aggregated data; in other words, taking the coefficients of the aggregate

equation to be constants may be seriously misleading. This suggests there might be substantial

benefits from using disaggregated data, as in the next section.13

4. Empirical implementation of LES

4.1. The Data

Our main source of data is the 2015-16 Household Expenditure Survey (HES), carried out by

the Australian Bureau of Statistics (ABS) through interviews from the usual residents of private

dwellings in urban and rural areas of Australia.14 We use the 10,046 households from HES

12 Goldberger (1987, pp. 65-67) establishes the perfect aggregation of LES when all parameters are the same across consumers.

As shown above, the result requires only the marginal shares be the same. 13 As the above analysis is algebraic, it can be described as the “exact” approach to aggregation. Thus, the aggregate equation

has exactly the same form as the individual ones when the income distribution of income remains unchanged. An alternative is the statistically based “convergence approach”. According to this approach, if certain moments converge to zero as the number of consumers increases indefinitely, the form of aggregate equation is “close” to the individual ones as the aggregate coefficients are approximately constant. In contrast to the exact approach, this result requires no assumptions regarding the income distribution. Arguably, the convergence approach is less rigid. See Theil (1971, 1975/76) for details.

14 Usual residents were residents who regarded the dwelling as their own or main home.

17

identified by the ABS as overlapping with the ABS Survey of Income and Housing.15 Table 7

shows the sample distribution across capital cities and states. Of the sampled households, about

46 per cent are from New South Wales and Victoria, while the remaining 54 per cent are from

other states and territories.

HES provides spending on 693 disaggregated goods and services based on the Household

Expenditure Classification (HEC). We aggregated the 693 into 114 commodities based on the

Input-Output Product Group (IOPG) classification.16 Then, we further aggregated the IOPG

data to a more manageable 20 sectors, which is patterned on the 1-digit commodity

classification used in national accounts. Table 8 summarises the mapping of IOPG items into

the 20 sectors.

We estimate the LES using the pooled dataset (that is, with all 10,046 observations) and by

“income” quintiles, with income interpreted as total expenditure on commodities (i.e., net of

savings income). Also, to put households of different size on more or less the same basis, we

use equivalised expenditures everywhere. 17 We also adjust the few cases of negative

expenditure by replacing them with the corresponding mean expenditure of households

belonging to the same income group.

Table 9 presents descriptive statistics of expenditure across commodities and income groups.

Overall, Housing is the biggest single item among households contributing about a fifth of total

expenditure (17.1%), followed by Insurance (10.6%), Food (9.9%), and Light Goods

Manufactures (7.5%). Regarding the patterns across household types, Figure 2 plots the decile-

average budget shares (expenditures as proportions of income) against total expenditure for

each commodity. It is apparent that the shares of Food, Heavy Goods Manufactures, Utilities,

Communications, and Public Administration decrease as income increases. On the other hand,

the budget share increases with income for Hotels and Restaurants, Private Transport, Finance,

Education, Health and Other Services. The shares of Housing, Insurance and Professional

Services appear to be fairly steady across income.

4.2. The One-Consumer Case

As discussed above, the linear expenditure system (LES) expresses expenditure on good i (𝑝𝑖𝑞𝑖 , 𝑖 = 1,⋯ , 𝑛) as a linear function of the prices (𝑝𝑖) and income (𝑀):

𝑝𝑖𝑞𝑖 = 𝑝𝑖𝛾𝑖 + 𝛽𝑖 (𝑀 −∑𝑝𝑗𝛾𝑗

𝑛

𝑗=1

) , 𝑖 = 1,⋯ , 𝑛 (20)

where M = ∑ 𝑝𝑖𝑛𝑖=1 𝑞𝑖 is “income”. The constant parameters are 𝛾𝑖 , the subsistence quantity of

the good and 𝛽𝑖 the 𝑖𝑡ℎ marginal share, with ∑ 𝛽𝑖 = 1,𝑛𝑖=1 0 < 𝛽𝑖 < 1. For a cross-section

application with no price variability, LES becomes:

𝑝𝑖𝑞𝑖 = 𝛼𝑖 + 𝛽𝑖𝑀, 𝑖 = 1,⋯ , 𝑛, (21)

with ∑ 𝛼𝑖 = 0𝑛𝑖=1 and ∑ 𝛽𝑖 = 1

𝑛𝑖=1 . System (21) can be estimated with n separate OLS

regressions of 𝑝𝑖𝑞𝑖 on M. Columns 2 and 3 of Table 10 give the estimates for the 𝑛 = 20 data

15 For details of this overlap, see https://www.abs.gov.au/ausstats/abs@.nsf/Lookup/by%20Subject/6503.0~2015-

16~Main%20Features~Sampling~13. 16 The concordance between the HEC and IOPG classifications is available at ABS’ website: cat. 5209.0.55.001, Table 40.

Industry and product concordances. 17 Equivalised expenditure is expenditure by the household divided by the number of 'equivalent adu lts’, which takes into account

household size and composition, and adjusts for economies of scale that arise from the sharing of dwellings.

18

described above. All coefficients are significant. The largest marginal share is for housing at

0.16, indicating that 16 cents of a one-dollar rise in income is spent on this good.

The marginal shares are obtained from (21), but the subsistence parameters are not identified,

as discussed previously. The cost of subsistence can be obtained once the value of the Frisch

parameter ω is known by using

𝑝𝑖𝛾𝑖 = 𝛼𝑖+𝛽𝑖𝑀(1 +1

𝜔) , 𝑖 = 1,⋯ ,20. (22)

Here, 𝛼𝑖 and 𝛽𝑖 are the intercept and slope coefficients from equation (21). We use in (22) the

estimates of the coefficients, mean income and 𝜔 = −2, and column 6 of Table 10 contains the

results in the form of the cost of subsistence as a proportion of income, 𝑝𝑖𝛾𝑖

𝑀. Column 7 expresses

subsistence cost of i as a fraction of the total cost of subsistence, 𝑝𝑖𝛾𝑖

𝒑′𝜸 with 𝒑′𝜸 = ∑ 𝑝𝑗𝛾𝑗

20𝑗=1 . For

convenience, this table reproduces the budget shares (column 4).

System (20) can be expressed in share form as 𝑤𝑖 = 𝑠𝑖 + 𝛽𝑖𝑟 = (1 − 𝑟)𝑝𝑖𝛾𝑖

𝒑′𝜸+ 𝑟𝛽𝑖 , where 𝑠𝑖 =

𝑝𝑖𝛾𝑖

𝑀 and 𝑟 =

𝑀−∑ 𝑝𝑗𝛾𝑗20𝑗=1

𝑀 is the supernumerary ratio. When income rises, so does r and the

change in the budget share of good i is

𝑑𝑤𝑖 = (𝛽𝑖 −𝑝𝑖𝛾𝑖

𝒑′𝜸) 𝑑𝑟, ∑ 𝑑𝑤𝑖

20𝑖=1 = 0. (23)

This shows that the budget share rises (falls) with income when the marginal share exceeds (is

less than) the subsistence share. In this situation, the good is a luxury (necessity). Table 10

shows that food, utilities and communications have a subsistence share substantially above their

marginal share. Although (23) applies when prices are constant, these results are consistent with

the changes in the plots of the budget shares in Figure 2 (they are also consistent with the

estimated income elasticities to be discussed below).

The income elasticities are 𝛽𝑖

𝑤𝑖, which we evaluate with means of the budget shares. As shown

in column 8 of Table 10 and Figure 3, food, utilities and communication have the lowest

elasticities. Higher elasticities are observed for finance, private vehicles, apparel, recreation and

hotel and restaurants.

Next, we calculate the Marshallian (or uncompensated) price elasticities:

𝜂𝑖𝑗∗ = 𝛿𝑖𝑗 (

𝑠𝑖𝑤𝑖− 1)−

𝛽𝑖𝑤𝑖𝑠𝑗 , 𝑖, 𝑗 = 1,⋯ ,20,

and the Slutsky (compensated) counterparts:

𝜂𝑖𝑗 = 𝛿𝑖𝑗 (𝑠𝑖𝑤𝑖− 1) +

𝛽𝑖𝑤𝑖(𝑤𝑗 − 𝑠𝑗), 𝑖, 𝑗 = 1,⋯ ,20,

where 𝛿𝑖𝑗 is the Kronecker delta (𝛿𝑖𝑗 = 1 if 𝑖 = 𝑗, 0 otherwise). Using mean budget shares,

the 20 × 20 matrices are given in the Appendix, and columns 9 and 10 of Table 10 summarises

these with the own-price elasticities. These are all less than one in absolute value, as expected.

The Marshallian elasticities are all larger than the Slutsky versions, but the differences are not

substantial. As the difference is 𝜂𝑖𝑖 − 𝜂𝑖𝑖∗ = 𝛽𝑖 , the reason for the similarity is that many of the

marginal shares tend to be modest when there are 20 goods.

19

4.3. Quintile Estimates

We now turn to the LES estimates for quintiles. Appendix T3 gives the full results and as these

have exactly the same format as before, we can just summarise them here. Overall, the estimates

of the slope coefficients have small standard errors and most coefficients are significantly

different from zero. The middle income group performs rather poorly relative to others, with

the slopes of food and communications not statistically significant from zero.

The estimates of subsistence and marginal shares are plotted in panels A and B of Figure 4. As

expected, the subsistence of most households is dominated by food and housing. Low income

households also devote a larger portion of their subsistence budget to utilities. On the other

hand, subsistence-intensive goods for richer households are finance, hotels and restaurants, and

private vehicles (all of which have larger subsistence shares than those of poorer households).

The subsistence shares here are of the form 𝑝𝑖𝛾𝑖

𝒑′𝜸 , 𝑖 = 1,⋯ ,20, which are treated as constants in

the one-consumer case. Panel A reveals that food accounts for about 20 percent of subsistence

for poor households and this falls by about one-half to 10 percent for the rich, and so is anything

but constant across the income distribution. The subsistence share of housing also falls and has

a similar pattern to food. Key expenditure items with high marginal shares are: housing and

food for the lowest quintile; housing and finance for the middle income groups; and housing

and insurance for the highest quintile. Here again, there is substantial dispersion across income

groups, especially for food and housing.

The corresponding income elasticities of panel C of Figure 4 and Table 11 simply reflect the

above pattern of the subsistence and marginal shares. Food and housing are important

commodities to consumers (as judged by their shares in total expenditure). Interestingly, as

income rises there is a noticeable decrease in the income elasticity of food, which will be

discussed in the following section. The income elasticity of housing for the one consumer case

is 0.954. With quintiles, this elasticity is above unity for both the lowest and highest income

groups, while the middle-income groups have lower elasticities (between 0.688 and 0.922).

This illustrates the richness of the disaggregated approach that is not apparent in the one-

consumer case. The own-price elasticities in Table 11 exhibit a general tendency for the

elasticities for the first three income groups to be higher relative to the pooled elasticities (of

Table 10).

To summarise, disaggregation captures important heterogeneity of demand responses.

Although the one-consumer case provides a general insight of the role of income and prices,

there are certain characteristics of households significantly affecting demand responses that can

be unveiled by allowing the parameters of LES to differ across the income distribution. The

next section contains a further elaboration of the value-added of this type of disaggregation.

5. Implications of Disaggregation

In LES the marginal shares, 𝛽𝑖 , are constant, so the Engel curves are linear. It stretches

credibility that affluent consumers would spend the same fraction of additional income on each

good as would the poor.18 Moreover, as the income elasticity of good i is 𝛽𝑖

𝑤𝑖, this elasticity is

inversely proportional to the budget share, 𝑤𝑖 . The budget share of a necessity falls with

increased income, so the income elasticity must rise in LES. Thus, higher income causes the

18 This issue with functional form when there is substantial income variation was mentioned in Section 2.1.

20

food income elasticity to move in the direction of unity -- food is less of a necessity (or more

of a luxury) for the rich, as compared to the poor. Is this unattractive implication of LES

overcome by disaggregating consumers into quintiles?

Figure 5 contrasts the behaviour of the food income elasticity in two cases, (i) with the constant-

𝛽𝑖 obtained from the single-consumer estimates of LES; and (ii) when quintiles are used.

Panel A plots the budget share of food against income, while panel B contains the

corresponding marginal shares -- the horizontal curve for the single-consumer case and the

(mostly) declining curve for quintiles. The budget share declines, but with quintiles, the

marginal share (mostly) falls faster, leading to the corresponding income elasticity also falling,

as shown in panel C. However in the single-consumer case, the elasticity rises with income (the

dotted line in panel C), so much so that it exceeds unity at the top quintile – food is a luxury for

the rich! Evidently, the quintile case deals satisfactorily with the problematic behaviour of the

food income elasticity in LES.

Panel D of Figure 5 gives a further perspective on food by plotting the approximate Engel curve

implied by the quintile estimates. This curve is clearly non-linear, with the slope declining as income

rises, in contrast to the one-consumer case, also shown in the figure.

We can also analyse the appropriateness of disaggregating consumers by examining the

goodness of fit of the models. LES deals with the allocation of total expenditure to the

individual commodities, that is, it is an allocation model. As such, it is natural to think of the

shares of the total accounted for by each good. Thus, one way to evaluate the degree to which

LES explains the data is to examine the budget shares, which are all positive with a unit sum.

LES with a disturbance terms, 휀𝑖 , is

𝑝𝑖𝑞𝑖 = 𝑝𝑖𝛾𝑖 + 𝛽𝑖 (𝑀 −∑𝑝𝑗𝛾𝑗

𝑛

𝑗=1

) + 휀𝑖 , 𝑖 = 1,⋯ ,20,

or using ̂ to denote an estimate,

𝑝𝑖𝑞𝑖 = 𝑝𝑖�̂�𝑖 + �̂�𝑖 (𝑀 −∑𝑝𝑗�̂�𝑗

𝑛

𝑗=1

) + 휀�̂� .

In order for the sum of both sides to be M, the residuals must satisfy ∑ 휀�̂�𝑐 = 0.𝑛𝑖=1 Thus, fitted

expenditure is observed expenditure less the residual, 𝑝𝑖�̂�𝑖 = 𝑝𝑖𝑞𝑖 − 휀�̂� . Dividing by M, the

fitted share of good i is �̂�𝑖 = 𝑤𝑖 −�̂�𝑖

𝑀. As LES is estimated with observations on C consumers

(households), the fitted shares and the corresponding new residuals are

�̂�𝑖𝑐 = 𝑤𝑖𝑐 −휀�̂�𝑐𝑀𝑐 , 𝑤𝑖𝑐 − �̂�𝑖𝑐 =

휀�̂�𝑐𝑀𝑐

, 𝑐 = 1,⋯ , 𝐶.

As ∑ 휀�̂�𝑐 = 0,20𝑖=1 ∑ �̂�𝑖𝑐 = 1.

20𝑖=1 The fit of the model can be judged by a comparison of �̂�𝑖𝑐 with 𝑤𝑖𝑐 .

A convenient way to summarise the fit of the whole model is the weighted sum of squared residuals:

𝑅𝑆𝑆𝑐 =∑𝑤𝑖𝑐 × 𝑅𝑆𝑆𝑖𝑐

20

𝑖=1

, 𝑅𝑆𝑆𝑖𝑐 = (휀�̂�𝑐𝑀𝑐)2

.

Averaging over observations, we have:

𝑅𝑆𝑆𝑖 =1

𝐶∑𝑅𝑆𝑆𝑖𝑐 ,

𝐶

𝑐=1

𝑅𝑆𝑆 =1

𝐶∑𝑅𝑆𝑆𝑐

𝐶

𝑐=1

.

21

These values are displayed in Table 12. Column 2 of the last row shows that for the pooled

model (the one-consumer case), the root-mean squared error × 100 is 0.6024, or about less than

two-thirds of a percentage point of the shares, which seems a credible outcome. However, going

across the last row, we see that disaggregation into quintile leads to a further reduction of

between 5 to 7 points for the first 4 quintiles. Only for the top quintile is the performance of the

disaggregated model worse than that of the pooled one. The table also reveals that

disaggregation leads to better fits for the majority of the individual commodities -- in 16 of the

20 cases, three or more quartiles have root-mean squared errors lower than that of the pooled

model.

Lastly, another ground for disaggregation is the variability of the income elasticities across

household and commodity types. Disaggregation would matter for the income elasticties if (i)

the variance of income elasticities is higher for quintiles relative to the pooled estimate, and (ii)

the variances of the quintile elasticities for each commodity are large. Figure 16 presents the

standard deviations of the income elasticties. Panel A shows a substantially higher dispersion

of income elasticities from the quintile estimates (std. dev. = [0.32 – 0.82]) relative to the pooled

estimate (std. dev. = 0.20). Panel B also shows a large dispersion of the quintile elasticities

across the 20 commodities with standard deviations ranging from 0.07 to 1.09. This higher

dispersion implies that disaggregating the LES estimates into more household types would

better capture the non-monotonic relationship between income and consumption. As

emphasised by Ho et al. (2020), capturing the heterogeneous behaviour across household types

is a necessary condition for achieving improvements in consumption modelling and for

evaluating the distributional impact of policies or economic development.

6. Concluding Comments

The linear expenditure system (LES, Stone, 1954) is a popular demand model, especially in

CGE modelling. The elegant simplicity of the model no doubt accounts for much of its

widespread use. But simplicity comes at the cost of restricted price responses because LES is

based on an additive utility function. With utility interactions between goods ruled out, LES is

applicable to broad aggregates only, where there is likely to be only modest substitutability.

Another issue is the very linearity of LES -- its linear Engel curves in particular. Linearity can

be justified as a local approximation, but this is not satisfactory when the range of observations

is wide, that is, when there is substantial variation in the data. In short, while LES is a valuable

tool, care is needed in using it.

This paper provides a comprehensive review of LES, including its positive and negative

attributes. Emphasis is placed on the application to cross-section data where there is no price

variation. In such situations, LES parameters are under identified and require one additional

piece of information. Identification can be achieved conveniently once the value of the “Frisch

parameter” is known -- the income elasticity of the marginal utility of income. We evaluate

several sources for this parameter value. Also dealt with is aggregation over commodities and

consumers.

The second part of the paper is an empirical illustration of the application of LES with a cross-

section of about 10,000 Australian households. As a way of dealing with the functional form

issue mentioned above, we disaggregated households into quintiles and estimated a separate

LES for each quintile. This also goes some way in reducing aggregation problems. A

comparison of the quintile estimates with the one-consumer case (when the data are pooled) is

revealing. There is a rich diversity among households, leading to substantial differences in

22

demand responses that are masked when LES is constrained to have the same parameters across

the income distribution. More specifically, there are important implications for the income

elasticities of two of the most prominent goods in many budgets, food and housing. In the one-

consumer case, the food elasticity rises in the direction of unity as income increases, which

means food is less of a necessity (or more of a luxury) for more affluent households. Moreover,

this elasticity exceeds unity for the richest, which violates Engel’s law. This deeply counter-

intuitive result vanishes with the quintile estimates. The income elasticity of housing is about

unity when the data are pooled. But the situation changes noticeably when quintiles are

distinguished -- it is greater than unity for the poorest and richest households, possibly reflecting

the “special status” of housing in Australia where it is valued for both its residential services

and as an investment that attracts substantial tax incentives. We also established the further

advantage that the quintile estimates tend to fit the data considerably better than the single LES.

23

Table 1. SUMMARY OF MAJOR CONCEPTS

Symbol/Equation Meaning/Name Constraints

A. Notation

𝑞𝑖 Quantity consumed of good i, 𝑖 =1, … , 𝑛

𝑝𝑖 Price of good i

𝑀 =∑𝑝𝑖𝑞𝑖.

𝑛

𝑖=1

Total expenditure, “income” for short

𝑤𝑖 =𝑝𝑖𝑞𝑖𝑀

Budget share of i 𝑤𝑖 > 0, ∑ 𝑤𝑖 = 1𝑛𝑖=1

B. Expenditure System

𝑝𝑖𝑞𝑖 = 𝑝𝑖𝛾𝑖 + 𝛽𝑖 (𝑀 −∑𝑝𝑗𝛾𝑗

𝑛

𝑗=1

) LES, 𝑖 = 1, … , 𝑛

𝑝𝑖𝛾𝑖 Cost of subsistence of i 𝑝𝑖𝛾𝑖 < 𝑝𝑖𝑞𝑖

𝛽𝑖 =𝜕(𝑝𝑖𝑞𝑖)

𝜕𝑀 Marginal share of i 𝛽𝑖 > 0∑𝛽𝑖 = 1

𝑛

𝑖=1

𝑀 −∑𝑝𝑗𝛾𝑗

𝑛

𝑗=1

Supernumerary income 𝑀 −∑𝑝𝑗𝛾𝑗

𝑛

𝑗=1

> 0

C. Demand Elasticities

𝜂𝑖 =𝜕 log 𝑞𝑖𝜕 log𝑀

=𝛽𝑖𝑤𝑖

Income elasticity of i 𝜂𝑖 > 0,∑𝑤𝑖𝜂𝑖 = 1

𝑛

𝑖=1

𝜂𝑖𝑗∗ =

𝜕 log 𝑞𝑖𝜕 log 𝑝𝑗

= 𝛿𝑖𝑗 (𝑠𝑖𝑤𝑖− 1) −

𝛽𝑖𝑤𝑖𝑠𝑗

Marshallian elasticity of demand for i with respect to price of j, 𝑖, 𝑗 =

1, … , 𝑛, with 𝑠𝑖 =𝑝𝑖𝛾𝑖

𝑀 the

subsistence share of i. Money income constant

𝜂𝑖 +∑𝜂𝑖𝑗∗

𝑛

𝑗=1

= 0

(Homogeneity)

𝜂𝑖𝑗 =𝜕 log 𝑞𝑖𝜕 log 𝑝𝑗

= 𝛿𝑖𝑗 (𝑠𝑖𝑤𝑖− 1)

+𝛽𝑖𝑤𝑖(𝑤𝑗 − 𝑠𝑗)

(𝑖, 𝑗)𝑡ℎ Slutsky price elasticity. Real income constant

∑𝜂𝑖𝑗

𝑛

𝑗=1

= 0

(Homogeneity)

𝑤𝑖𝜂𝑖𝑗 = 𝑤𝑗𝜂𝑗𝑖

(Slutsky symmetry)

D. Marshallian Price Elasticities with Positive Gamma

𝜂𝑖𝑖∗ Marshallian own-price elasticity

𝜂𝑖𝑖∗ =

𝑝𝑖𝛾𝑖𝑝𝑖𝑞𝑖

(1 − 𝛽𝑖) − 1

−1 < 𝜂𝑖𝑖∗ < 0 , price

inelastic

𝜂𝑖𝑗∗ , 𝑖 ≠ 𝑗 Marshallian cross-price elasticity

𝜂𝑖𝑗∗ = −𝛽𝑖

𝑝𝑗𝛾𝑗𝑝𝑖𝑞𝑖

< 0

i, j gross complements

24

Table 1. SUMMARY OF MAJOR CONCEPTS (continued)

Symbol/Equation Meaning/Name Constraints

E. Marshallian Price Elasticities with Negative Gamma

𝜂𝑖𝑖∗

Marshallian own-price elasticity

𝜂𝑖𝑖∗ < −1, price elastic

𝜂𝑖𝑗∗ , 𝑖 ≠ 𝑗

Marshallian cross-price elasticity

𝜂𝑖𝑗∗ > 0

i, j gross substitutes

F. Slutsky Price Elasticities with Positive Gamma

𝜂𝑖𝑖 Slutsky own-price elasticity

𝜂𝑖𝑖 = 𝜂𝑖𝑖∗ + 𝛽𝑖

𝜂𝑖𝑖 > 𝜂𝑖𝑖∗ < −1

𝜂𝑖𝑖 not unambiguously < -1

𝜂𝑖𝑗 , 𝑖 ≠ 𝑗 Slutsky cross-price elasticity

𝜂𝑖𝑗 = 𝜂𝑖𝑗∗ +

𝛽𝑖𝑤𝑖𝑤𝑗

𝜂𝑖𝑗 > 𝜂𝑖𝑗∗ > 0

i, j net substitutes

G. Slutsky Price Elasticities with Negative Gamma

𝜂𝑖𝑖 Slutsky own-price elasticity

𝜂𝑖𝑖 = 𝜂𝑖𝑖∗ + 𝛽𝑖

𝜂𝑖𝑖 > 𝜂𝑖𝑖∗ < −1

𝜂𝑖𝑖 not unambiguously < -1, as for 𝛾𝑖 > 0

𝜂𝑖𝑗 , 𝑖 ≠ 𝑗 Slutsky cross-price elasticity

𝜂𝑖𝑗 = 𝜂𝑖𝑗∗ +

𝛽𝑖𝑤𝑖𝑤𝑗

𝜂𝑖𝑗 > 𝜂𝑖𝑗∗ > 0

i, j net substitutes, as for 𝛾𝑖 > 0

H. Utility

𝑢(𝑞1,⋯ , 𝑞𝑛) =∑𝛽𝑖 log(𝑞𝑖 − 𝛾𝑖)

𝑛

𝑖=1

Stone-Geary utility 0 < 𝛽𝑖 < 1,∑𝛽𝑖 = 1

𝑛

𝑖=1

, 𝛾𝑖 < 𝑞𝑖

𝜆 =1

𝑀 − ∑ 𝑝𝑖𝑛𝑖=1 𝛾𝑖

Marginal utility of income

𝜆 > 0

𝜔 =𝜕 log 𝜆

𝜕 log𝑀= −

𝑀

𝑀 − ∑ 𝑝𝑖𝑛𝑖=1 𝛾𝑖

Frisch parameter -- income elasticity of marginal utility of income

𝜔 < −1

25

Table 2. FRISCH-LES IMPLIED INCOME

Income

Group g (1)

Frisch

parameter 𝜔

(2)

Average 𝜔

1

2(𝜔𝑔+1+𝜔𝑔+1)

(3)

Difference in

Frisch parameter

log𝜔𝑔+1𝜔𝑔

× 100

(4)

Difference

log𝑀𝑔+1𝑀𝑔

× 100

(5)

Level

(Group 1 = 100)

(6)

1. Extremely poor -10 - 100 2. Slightly better off -4 -7 -92 15 116 3. Middle income -2 -3 -69 35 165 4. Better off -0.7 5. Rich -0.1

Notes:

1. The income difference between groups 𝑔 + 1 and 𝑔 of column 5 is defined as in equation (12), viz.,

log(𝑀𝑔+1 𝑀𝑔⁄ ) = (1 + �̅�)−1 log(𝜔𝑔+1 𝜔𝑔⁄ ), with �̅� = (1 2⁄ )(𝜔𝑔+1+𝜔𝑔).

2. The income level of group 𝑔 + 1 of column 6 is 𝑀𝑔+1 = 𝑀𝑔 exp{log(𝑀𝑔+1 𝑀𝑔⁄ )}, with 𝑀1 = 100.

3. As 𝜔 > −1 for groups 4 and 5, the LES condition (7) is violated. As the corresponding income differences

are no longer given by equation (12), the shaded cells are left blank.

26

Table 3. FRISCH PARAMETERS, ILLUSTRATIVE VALUES

Summary of income distribution/country

(1)

GDP per capita in $US at

Frisch parameter, −𝜔, with assumed US inflation

2018 prices 1970 prices, with assumed

US inflation

(2) 2% p. a.

(3) 5% p. a.

(4) 2% p. a.

(5) 5% p. a.

(6)

A. 196 Countries from World Development Indicators

Poorest (Burundi) 272 105 26 6.74 11.12

25th percentile (≈ Solomon Is) 2,142 828 206 3.20 5.29

Median (≈ Fiji) 6,290 2,431 605 2.17 3.59

75th percentile (≈ St Kitts) 19,214 7,427 1,847 1.45 2.40

Maximum (Monaco) 185,741 71,796 17,858 0.64 1.06

B. Selected High-Income Countries

Canada 46,233 17,871 4,445 1.06 1.75

Belgium 47,519 18,368 4,569 1.05 1.73

Germany 47,603 18,400 4,577 1.05 1.73

Hong Kong SAR, China 48,676 18,815 4,680 1.04 1.72

Finland 50,152 19,386 4,822 1.03 1.70

Austria 51,462 19,892 4,948 1.02 1.68

Netherlands 53,024 20,496 5,098 1.01 1.67

Sweden 54,608 21,108 5,250 1.00 1.65

Australia 57,374 22,177 5,516 0.98 1.62

Denmark 61,350 23,714 5,898 0.96 1.58

United States 62,795 24,272 6,037 0.95 1.57

Singapore 64,582 24,963 6,209 0.94 1.55

Notes: 5. Column 2 is from World Development Indicators,

https://databank.worldbank.org/reports.aspx?source=world-development-indicators

6. Column 3 (4) converts GDP from 2018 prices (in column 2) to those of 1970 under the assumption of average US inflation of 2% p. a. (5% p. a.). That is, as there are 48 years between 1970 and 2018, (GDP in 1970 prices) =(GDP in 2018 prices) × (1 + 𝜋)−48, 𝜋 = 0.02, 0.05.

7. Column 5 and 6 contain values of (the negative of) the Frisch parameter derived from the equation estimated by Lluch et al. (1977, p. 76), viz., −𝜔 = 36 × (GDP in 1970 prices)−0.36. Column 5 (6) assumes 2% (5%) inflation by using the GDP values of column 3 (4).

8. Since (GDP in 1970 prices) = (GDP in 2018 prices) × (1 + 𝜋)−48, −𝜔 = 36 × (GDP in 2018 prices)−0.36 ×

(1 + 𝜋)48×0.36. Thus, for 𝜋 = 0.02, 0.05, column 6 (for 5% inflation) is a multiple (1.05

1.02)48×0.36

= 1.65 of column 5 (2%

inflation). When inflation is assumed to be higher, GDP in 1970 prices is lower, leading to larger values of −𝜔.

27

Table 4. SUMMARY OF FRISCH PARAMETER

Percentile of income distribution

(1)

−𝜔 with assumed inflation Frisch conjecture

2% (2)

5% (3) (4)

25th percentile 3.20 5.29 4 (“the slightly better off”)

75th percentile 1.45 2.40 0.7 (“the better off”)

Median 2.17 3.59 2 (“the ‘median part’ of the population”)

Source: Columns 2 and 3 are from columns 5 and 6 of Table 3.

Table 5. RECIPROCAL FRISCH PARAMETERS, OECD COUNTRIES

1. Australia -0.414 (0.054) 21. Lithuania -0.394 (0.089) 2. Austria -0.463 (0.062) 22. Luxembourg -0.848 (0.143) 3. Belgium -0.050 (0.047) 23. Mexico -0.830 (0.000) 4. Canada -0.350 (0.048) 24. Netherlands -0.282 (0.066) 5. Colombia -0.082 (0.045) 25. New Zealand -0.624 (0.067) 6. Czech Rep. -0.438 (0.067) 26. Norway -0.906 (0.082) 7. Denmark -0.530 (0.087) 27. Poland -0.606 (0.071) 8. Estonia -0.321 (0.055) 28. Portugal -0.612 (0.083) 9. Finland -0.486 (0.059) 29. Slovak Rep. -0.477 (0.058) 10. France -0.444 (0.046) 30. Slovenia -0.336 (0.084) 11. Germany -0.300 (0.051) 31. South Africa -0.376 (0.046) 12. Greece -0.148 (0.134) 32. Spain -0.219 (0.062) 13. Hungary -0.524 (0.052) 33. Sweden -0.515 (0.068) 14. Iceland -0.806 (0.095) 34. Switzerland -1.706 (0.031) 15. Ireland -2.459 (0.053) 35. Turkey -0.183 (0.039) 16. Israel -0.017 (0.059) 36. UK -0.812 (0.078) 17. Italy -0.526 (0.048) 37. United States -0.601 (0.047) 18. Japan -0.605 (0.160) 19. Korea -0.209 (0.046) Mean -0.538 (0.067) 20. Latvia -0.423 (0.106)

Note: Standard errors are in parentheses

Table 6. MORE RECIPROCAL FRISCH PARAMETERS, 176 ICP COUNTRIES

Income group of countries

Income (US = 100)

Reciprocal Frisch parameter

Mean SD Mean SD

Top quartile 68.5 13.9 -0.217 0.375 Quartile 3 35.3 7.5 -0.709 0.512 Quartile 2 16.2 4.6 -0.685 0.389 Quartile 1 4.1 1.9 -0.463 0.830 All countries 31.0 25.7 -0.518 0.592

28

Table 7. DISTRIBUTION OF HOUSEHOLDS, 2015-16

Capital City Rest of State Total

New South Wales 1,778 479 2,257

Victoria 2,004 386 2,390

Queensland 1,050 439 1,489

South Australia 1,044 254 1,297

Western Australia 1,002 216 1,218

Tasmania 557 204 761

Northern territory 259 81 340

Australian Capital Territory 293 — 293

Australia 7,987 2,059 10,046

Source: ABS, 2015-16 Household Expenditure Survey (HES) https://www.abs.gov.au/ausstats/abs@.nsf/Lookup/by%20Subject/6503.0~2015-16~Main%20Features~Sampling~13

29

Table 8. THE AGGREGATED SECTORS

Aggregated Sector Input-Output Product Group

1. Food 1. Meat and Meat product Manufacturing, 2. Processed Seafood Manufacturing, 3. Dairy Product Manufacturing, 4. Fruit and Vegetable Product Manufacturing, 5. Oils and Fats Manufacturing, 6. Grain Mill and Cereal Product Manufacturing, 7. Bakery Product Manufacturing, 8. Sugar and Confectionery Manufacturing, 9. Other Food Product Manufacturing

2. Beverages and tobacco 1. Soft Drinks, Cordials and Syrup Manufacturing, 2. Beer Manufacturing, 3. Wine, Spirits and Other Alcoholic Beverage Manufacturing, 4. Cigarette and Tobacco Product Manufacturing

3. Clothing and Footwear 1. Textile Manufacturing, 2. Tanned Leather, Dressed Fur and Leather Product Manufacturing, 3. Textile Product

Manufacturing, 4. Knitted Product Manufacturing, 5.

Clothing Manufacturing, 6. Footwear Manufacturing,

4. Light Goods Manufactures 1. Sawmill Product Manufacturing, 2. Other Wood Product Manufacturing, 3. Pulp, Paper and Paperboard Manufacturing, 4. Paper Stationery and Other Converted Paper Product Manufacturing, 5. Printing (including the reproduction of recorded media), 6. Domestic Appliance Manufacturing, 7. Specialised and other Machinery and Equipment Manufacturing, 8. Furniture Manufacturing, 9. Other Manufactured Products

5. Heavy Goods Manufactures 1. Petroleum and Coal Product Manufacturing, 2. Human Pharmaceutical and Medicinal Product Manufacturing, 3. Veterinary Pharmaceutical and Medicinal Product Manufacturing, 4. Basic Chemical Manufacturing, 5. Cleaning Compounds and Toiletry Preparation Manufacturing, 6. Polymer Product Manufacturing, 7. Glass and Glass Product Manufacturing, 8. Ceramic Product Manufacturing, 9. Metal Containers and Other Sheet Metal Product manufacturing, 10. Other Fabricated Metal Product manufacturing, 11. Professional, Scientific, Computer and Electronic Equipment Manufacturing, 12. Electrical Equipment Manufacturing

6. Utilities 1. Electricity Generation, 2. Gas Supply, 3. Water Supply, Sewerage and Drainage Services, 4. Waste Collection, Treatment and Disposal Services

7. Housing 1. Residential Building Construction, 2. Rental and Hiring Services (except Real Estate), 3. Ownership of dwellings

8. Hotels and Restaurants 1. Accommodation, 2. Food and Beverage Services

9. Private Transportation 1. Motor Vehicles and Parts; 2. Other Transport Equipment manufacturing, 3. Ships and Boat Manufacturing, 4. Aircraft Manufacturing

10. Public Transportation 1. Road Transport, 2. Rail Transport, 3. Water, Pipeline and Other Transport, 4. Air and Space Transport, 5. Postal and Courier Pick-up and Delivery Service, 6. Transport Support services and storage

30

Table 8. THE AGGREGATED SECTORS (continued)

11. Communications 1. Publishing (except Internet and Music Publishing), 2. Motion Picture and Sound Recording, 3. Internet Publishing and Broadcasting and Services Providers, Web search Portals and Data Processing Services, 4. Telecommunication Services, 5. Library and Other Information Services, 6. Broadcasting (except Internet)

12. Finance 1. Finance

13. Insurance 1. Insurance and Superannuation Funds

14. Professional Services 1. Professional, Scientific and Technical Services

15. Administration and Support Services

1. Building Cleaning, Pest Control, Administrative and Other Support Services, 2. Employment, Travel Agency and Other Administrative Services

16. Public Administration 1. Public Administration and Regulatory Services

17. Education 1. Primary and Secondary Education Services (incl Pre-Schools and Special Schools), 2. Technical, Vocational and Tertiary Education Services (incl undergraduate and postgraduate), 3. Arts, Sports, Adult and Other Education Services (incl community education)

18. Health 1. Health Care Services, 2. Residential Care and Social Assistance Services

19. Recreation 1. Heritage, Creative and Performing Arts, 2. Sports and Recreation, 3. Gambling

20. Other Services 1. Automotive Repair and Maintenance, 2. Other Repair and Maintenance, 3. Personal Services, 4. Other Services

No HEC classification 1. Coal mining, 2. Oil and gas extraction, 3. Iron Ore Mining,

4. Non Ferrous Metal Ore Mining, 5. Non-metallic Mineral Mining, 6. Exploration and Mining Support Services, 7. Natural Rubber Product Manufacturing, 8. Cement, Lime and Ready-Mixed Concrete Manufacturing, 9. Plaster and Concrete Product Manufacturing, 10. Other Non-Metallic Mineral Product Manufacturing, 11. Iron and Steel Manufacturing, 12. Basic Non-Ferrous Metal Manufacturing, 13. Forged Iron and Steel Product Manufacturing, 14. Structural Metal Product Manufacturing, 15. Railway Rolling Stock Manufacturing, 16. Electricity Transmission, Distribution, On Selling and Electricity Market Operation, 17. Non-Residential Building Construction, 18. Heavy and Civil Engineering Construction, 19. Construction Services, 20. Wholesale Trade, 21. Retail Trade, 22. Non-Residential Property Operators and Real Estate Services, 23. Computer Systems Design and Related Services, 24. Defence, 25. Public Order and Safety

31

Table 9. SUMMARY OF EXPENDITURE DATA, AUSTRALIA 2015-16

All households

n = 10,046

Lowest quintile

n = 2,009

Second quintile

n = 2,009

Third quintile

n = 2,009

Fourth quintile

n = 2,009

Highest quintile

n = 2,010

Mean

($)

SD

($)

Share

(%)

Mean

($)

SD

($)

Share

(%)

Mean

($)

SD

($)

Share

(%)

Mean

($)

SD

($)

Share

(%)

Mean

($)

SD

($)

Share

(%)

Mean

($)

SD

($)

Share

(%)

1. Food 4,559 2,645 9.9 3,008 1,654 19.6 3,948 1,978 15.5 4,462 2,200 12.2 5,233 2,397 10.0 6,142 3,464 6.0

2. Beverages and tobacco 1,477 2,371 3.2 595 1,141 3.9 1,089 1,823 4.3 1,488 2,114 4.1 1,794 2,522 3.4 2,419 3,280 2.4

3. Clothing and Footwear 1,430 2,834 3.1 369 830 2.4 777 1,333 3.1 1,103 1,710 3.0 1,809 2,586 3.5 3,093 4,849 3.0

4. Light Goods Manufactures 3,448 6,283 7.5 900 1,176 5.9 1,812 1,968 7.1 2,751 2,947 7.5 3,967 3,875 7.6 7,811 11,810 7.7

5. Heavy Goods Manufactures 2,531 2,227 5.5 1,211 1,063 7.9 1,883 1,306 7.4 2,451 1,772 6.7 3,080 2,138 5.9 4,029 3,088 4.0

6. Utilities 1,867 1,118 4.0 1,529 857 10.0 1,669 950 6.6 1,810 994 5.0 1,995 1,060 3.8 2,335 1,453 2.3

7. Housing 7,902 27,017 17.1 2,432 3,493 15.8 4,393 4,888 17.3 5,676 5,793 15.6 7,789 7,106 15.0 19,217 57,915 18.9

8. Hotels and Restaurants 3,012 4,042 6.5 729 961 4.7 1,486 1,599 5.8 2,412 2,231 6.6 3,788 3,190 7.3 6,642 6,433 6.5

9. Private Transportation 1,738 7,422 3.8 55 409 0.4 371 1,631 1.5 941 3,030 2.6 1,938 4,879 3.7 5,385 14,866 5.3

10. Public Transportation 1,190 3,321 2.6 278 753 1.8 567 1,210 2.2 869 1,548 2.4 1,354 2,267 2.6 2,884 6,430 2.8

11. Communications 1,770 1,343 3.8 1,030 730 6.7 1,428 916 5.6 1,723 1,053 4.7 2,094 1,336 4.0 2,574 1,829 2.5

12. Finance 2,865 5,577 6.2 267 957 1.7 855 2,053 3.4 2,148 3,676 5.9 3,814 4,983 7.3 7,237 8,985 7.1

13. Insurance 4,895 13,499 10.6 1,445 1,391 9.4 2,058 1,912 8.1 3,156 3,098 8.7 4,740 4,727 9.1 13,073 27,994 12.8

14. Professional Services 561 3,944 1.2 136 574 0.9 282 1,051 1.1 403 1,426 1.1 623 2,173 1.2 1,359 8,287 1.3

15. Admin and Support Services 1,016 4,107 2.2 191 686 1.2 374 1,324 1.5 757 2,334 2.1 1,202 3,459 2.3 2,556 7,820 2.5

16. Public Administration 491 1,473 1.1 227 328 1.5 342 706 1.3 453 688 1.2 592 1,032 1.1 841 2,911 0.8

17. Education 1,107 5,679 2.4 95 420 0.6 289 913 1.1 633 1,849 1.7 1,323 3,352 2.5 3,194 11,802 3.1

18. Health 1,147 6,023 2.5 242 669 1.6 498 1,125 2.0 971 2,155 2.7 1,324 2,631 2.5 2,700 12,822 2.7

19. Recreation 719 1,991 1.6 194 553 1.3 369 830 1.5 615 1,213 1.7 834 1,564 1.6 1,595 3,706 1.6

20. Other Services 2,495 7,902 5.4 430 992 2.8 959 1,809 3.8 1,637 2,998 4.5 2,793 4,904 5.4 6,654 15,817 6.5

Note: Dollar values are average annual equivalent household expenditures at 2015-16 prices.

Source: See text.

32

Table 10. LES ESTIMATES, CONSUMPTION SHARES AND ELASTICITIES, POOLED DATA

Commodity

group

Coefficients

(Standard errors) Budget

share

Marginal share

Subsistence budget share Income

elasticity

Own-price elasticities

Intercept αi

Slope βi

Proportion of income

Proportion of total

subsistence Marshallian Slutsky

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

1. Food 1,476 (29.24) 7.95 (0.13) 9.86 7.95 7.17 14.34 0.806 -0.408 -0.328

2. Beverages and tobacco 67 (23.03) 3.45 (0.10) 3.20 3.44 1.86 3.73 1.077 -0.498 -0.463

3. Clothing and Footwear -179 (19.12) 3.63 (0.09) 3.09 3.62 1.43 2.85 1.171 -0.576 -0.539

4. Light Goods Manufactures -349 (30.38) 8.36 (0.13) 7.46 8.36 3.43 6.85 1.120 -0.587 -0.504

5. Heavy Goods Manufactures 295 (19.62) 5.48 (0.09) 5.48 5.47 3.37 6.74 0.999 -0.478 -0.423

6. Utilities 1,252 (14.71) 1.51 (0.07) 4.04 1.51 3.46 6.92 0.373 -0.191 -0.176

7. Housing -132 (67.93) 16.31 (0.29) 17.10 16.30 7.87 15.73 0.954 -0.589 -0.426

8. Hotels and Restaurants -352 (23.3) 7.47 (0.10) 6.52 7.47 2.97 5.94 1.146 -0.590 -0.515

9. Private Transportation -542 (31.14) 4.51 (0.14) 3.76 4.51 1.08 2.16 1.198 -0.690 -0.645

10. Public Transportation -152 (17.73) 2.89 (0.08) 2.58 2.89 1.12 2.23 1.122 -0.576 -0.548

11. Communications 531 (13.14) 3.05 (0.06) 3.83 3.04 2.67 5.34 0.795 -0.383 -0.352

12. Finance -795 (33.31) 7.85 (0.15) 6.20 7.84 2.20 4.41 1.266 -0.668 -0.590

13. Insurance -39 (38.83) 9.62 (0.17) 10.59 9.61 4.72 9.45 0.908 -0.552 -0.456

14. Professional Services -43 (15.6) 1.26 (0.07) 1.21 1.25 0.53 1.07 1.032 -0.545 -0.533

15. Admin and Support Services -166 (22.64) 2.48 (0.10) 2.20 2.47 0.88 1.76 1.124 -0.595 -0.570

16. Public Administration 90 (8.9) 0.93 (0.04) 1.06 0.93 0.66 1.31 0.871 -0.419 -0.410

17. Education -275 (21.23) 2.72 (0.10) 2.39 2.71 0.76 1.52 1.133 -0.650 -0.623

18. Health -154 (19.9) 2.77 (0.09) 2.48 2.76 1.05 2.10 1.114 -0.580 -0.553

19. Recreation -79 (12.91) 1.81 (0.06) 1.56 1.80 0.73 1.46 1.154 -0.560 -0.542

20. Other Services -457 (31.99) 6.06 (0.14) 5.40 6.06 2.04 4.08 1.122 -0.622 -0.561

Sum 0 100 100 100 50 100

Note: Columns 3 to 7 are X100

33

Table 11. DEMAND ELASTICITIES, QUINTILES

Commodity group

Income elasticity for quintile Marshallian own-price elasticity for quintile Slutsky own-price elasticity for quintile

1 2 3 4 5 1 2 3 4 5 1 2 3 4 5

1. Food 0.970 0.451 0.100 0.284 0.228 -0.578 -0.28 -0.062 -0.167 -0.125 -0.388 -0.21 -0.049 -0.138 -0.111

2. Beverages and tobacco 1.190 1.215 0.549 0.607 0.245 -0.612 -0.628 -0.291 -0.318 -0.127 -0.566 -0.576 -0.268 -0.297 -0.121

3. Clothing and Footwear 1.406 1.967 1.287 1.288 0.701 -0.723 -0.984 -0.658 -0.66 -0.351 -0.689 -0.924 -0.619 -0.615 -0.33

4. Light Goods Manufactures 1.277 1.232 1.042 0.937 0.899 -0.674 -0.65 -0.559 -0.507 -0.478 -0.599 -0.562 -0.48 -0.435 -0.409

5. Heavy Goods Manufactures 1.120 0.804 0.546 0.609 0.407 -0.598 -0.438 -0.3 -0.329 -0.212 -0.51 -0.378 -0.263 -0.293 -0.196

6. Utilities 0.322 0.400 0.287 0.214 0.369 -0.186 -0.221 -0.156 -0.114 -0.189 -0.154 -0.195 -0.141 -0.106 -0.18

7. Housing 1.174 0.688 0.831 0.922 1.402 -0.666 -0.422 -0.491 -0.535 -0.824 -0.48 -0.303 -0.362 -0.397 -0.559

8. Hotels and Restaurants 1.187 1.559 1.475 1.241 0.681 -0.621 -0.799 -0.763 -0.655 -0.36 -0.565 -0.708 -0.665 -0.564 -0.316

9. Private Transport 1.577 3.921 2.340 1.625 1.177 -0.817 -1.907 -1.16 -0.824 -0.595 -0.812 -1.85 -1.099 -0.764 -0.533

10. Public Transport 1.295 1.196 1.305 0.989 1.036 -0.661 -0.609 -0.663 -0.508 -0.515 -0.637 -0.582 -0.632 -0.482 -0.485

11. Communications 0.723 0.560 0.217 0.395 0.330 -0.391 -0.302 -0.118 -0.21 -0.169 -0.343 -0.271 -0.107 -0.194 -0.161

12. Finance 1.597 2.106 2.178 1.690 0.730 -0.843 -1.049 -1.078 -0.864 -0.384 -0.815 -0.979 -0.949 -0.74 -0.332

13. Insurance 0.758 0.760 1.026 1.161 1.451 -0.422 -0.418 -0.556 -0.625 -0.777 -0.351 -0.357 -0.467 -0.519 -0.591

14. Professional Services 1.021 1.200 2.331 1.139 1.332 -0.511 -0.606 -1.162 -0.576 -0.673 -0.502 -0.592 -1.136 -0.563 -0.655

15. Admin Support 1.214 1.886 1.486 1.365 1.052 -0.612 -0.945 -0.751 -0.692 -0.535 -0.597 -0.917 -0.72 -0.66 -0.509

16. Public Admin 0.661 0.952 1.223 0.886 0.800 -0.336 -0.483 -0.617 -0.449 -0.39 -0.326 -0.47 -0.602 -0.439 -0.384

17. Education 1.072 2.296 3.382 1.704 1.445 -0.557 -1.145 -1.651 -0.858 -0.723 -0.55 -1.119 -1.593 -0.815 -0.678

18. Health 1.260 1.175 1.578 1.082 1.110 -0.644 -0.597 -0.798 -0.554 -0.578 -0.624 -0.574 -0.756 -0.526 -0.549

19. Recreation 1.385 0.948 0.825 0.764 0.596 -0.7 -0.481 -0.421 -0.39 -0.297 -0.683 -0.467 -0.407 -0.377 -0.288

20. Other Services 1.377 1.877 1.437 1.724 1.267 -0.71 -0.943 -0.737 -0.875 -0.655 -0.671 -0.872 -0.672 -0.783 -0.572

34

Table 12. AGGREGATION AND GOODNESS OF FIT

Commodities Pooled Quintile 1 Quintile 2 Quintile 3 Quintile 4 Quintile 5

(1) (2) (3) (4) (5) (6) (7)

1. Food 0.0313 0.0476 0.0325 0.0236 0.0159 0.0118

2. Beverages and tobacco 0.0260 0.0341 0.0331 0.0231 0.0204 0.0113

3. Clothing and Footwear 0.0203 0.0209 0.0216 0.0184 0.0185 0.0219

4. Light Goods Manufactures 0.0367 0.0294 0.0342 0.0360 0.0329 0.0482

5. Heavy Goods Manufactures 0.0181 0.0255 0.0169 0.0184 0.0143 0.0091

6. Utilities 0.0130 0.0244 0.0120 0.0077 0.0052 0.0034

7. Housing 0.0982 0.1259 0.1068 0.0825 0.0673 0.0898

8. Hotels and Restaurants 0.0225 0.0215 0.0237 0.0222 0.0223 0.0228

9. Private Transportation 0.0424 0.0102 0.0334 0.0436 0.0476 0.0582

10. Public Transportation 0.0196 0.0189 0.0195 0.0162 0.0170 0.0252

11. Communications 0.0104 0.0157 0.0113 0.0081 0.0081 0.0044

12. Finance 0.0393 0.0256 0.0393 0.0469 0.0409 0.0393

13. Insurance 0.0545 0.0372 0.0319 0.0416 0.0455 0.0871

14. Professional Services 0.0198 0.0153 0.0178 0.0177 0.0208 0.0256

15. Admin and Support Services 0.0279 0.0182 0.0236 0.0312 0.0334 0.0303

16. Public Administration 0.0107 0.0077 0.0147 0.0059 0.0072 0.0146

17. Education 0.0303 0.0108 0.0149 0.0232 0.0328 0.0505

18. Health 0.0252 0.0173 0.0181 0.0281 0.0213 0.0359

19. Recreation 0.0127 0.0121 0.0125 0.0120 0.0105 0.0156

20. Other Services 0.0435 0.0262 0.0321 0.0413 0.0488 0.0595

TOTAL 0.6024 0.5444 0.5501 0.5476 0.5307 0.6643

Note: Root-mean squared errors × 100. Highlighted cells indicate lower RMSE of quintiles relative to the pooled.

35

Figure 1. FRISCH PARAMETER AND INCOME

Source: Lluch et al. (1977, p. 77).

36

Figure 2. BUDGET SHARES AND TOTAL EXPENDITURE

0.00%

5.00%

10.00%

15.00%

20.00%

25.00%

0 20 40 60 80 100 120 140

Avg

. Bu

dge

t Sh

are

Average Expenditure ($'000 p.a.)

Food

0.00%

0.50%

1.00%

1.50%

2.00%

2.50%

3.00%

3.50%

4.00%

4.50%

5.00%

0 20 40 60 80 100 120 140

Avg

. Bu

dge

t Sh

are

Average Expenditure ($'000 p.a.)

Beverages and tobacco

0.00%

0.50%

1.00%

1.50%

2.00%

2.50%

3.00%

3.50%

4.00%

0 20 40 60 80 100 120 140

Avg

. Bu

dge

t Sh

are

Average Expenditure ($'000 p.a.)

Cloth and Footwear

0.00%

1.00%

2.00%

3.00%

4.00%

5.00%

6.00%

7.00%

8.00%

9.00%

0 20 40 60 80 100 120 140

Avg

. Bu

dge

t Sh

are

Average Expenditure ($'000 p.a.)

Light Goods Manufactures

0.00%

1.00%

2.00%

3.00%

4.00%

5.00%

6.00%

7.00%

8.00%

9.00%

0 20 40 60 80 100 120 140

Avg

. Bu

dge

t Sh

are

Average Expenditure ($'000 p.a.)

Heavy Goods Manufactures

0.00%

2.00%

4.00%

6.00%

8.00%

10.00%

12.00%

14.00%

0 20 40 60 80 100 120 140

Avg

. Bu

dge

t Sh

are

Average Expenditure ($'000 p.a.)

Utilities

0.00%

5.00%

10.00%

15.00%

20.00%

25.00%

0 20 40 60 80 100 120 140

Avg

. Bu

dge

t Sh

are

Average Expenditure ($'000 p.a.)

Housing

0.00%

1.00%

2.00%

3.00%

4.00%

5.00%

6.00%

7.00%

8.00%

0 20 40 60 80 100 120 140

Avg

. Bu

dge

t Sh

are

Average Expenditure ($'000 p.a.)

Hotel and Restaurant

0.00%

1.00%

2.00%

3.00%

4.00%

5.00%

6.00%

0 20 40 60 80 100 120 140

Avg

. Bu

dge

t Sh

are

Average Expenditure ($'000 p.a.)

Private Transportation

0.00%

0.50%

1.00%

1.50%

2.00%

2.50%

3.00%

0 20 40 60 80 100 120 140

Avg

. Bu

dge

t Sh

are

Average Expenditure ($'000 p.a.)

Public Transportation

37

Figure 2. BUDGET SHARES AND TOTAL EXPENDITURE (continuation)

0.00%

1.00%

2.00%

3.00%

4.00%

5.00%

6.00%

7.00%

8.00%

0 20 40 60 80 100 120 140

Avg

. Bu

dge

t Sh

are

Average Expenditure ($'000 p.a.)

Communications

0.00%

1.00%

2.00%

3.00%

4.00%

5.00%

6.00%

7.00%

8.00%

9.00%

0 20 40 60 80 100 120 140

Avg

. Bu

dge

t Sh

are

Average Expenditure ($'000 p.a.)

Finance

0.00%

2.00%

4.00%

6.00%

8.00%

10.00%

12.00%

14.00%

16.00%

0 20 40 60 80 100 120 140

Avg

. Bu

dge

t Sh

are

Average Expenditure ($'000 p.a.)

Insurance

0.00%

0.20%

0.40%

0.60%

0.80%

1.00%

1.20%

1.40%

1.60%

0 20 40 60 80 100 120 140

Avg

. Bu

dge

t Sh

are

Average Expenditure ($'000 p.a.)

Professional Services

0.00%

0.50%

1.00%

1.50%

2.00%

2.50%

3.00%

0 20 40 60 80 100 120 140

Avg

. Bu

dge

t Sh

are

Average Expenditure ($'000 p.a.)

Admin and Support Services

0.00%

0.20%

0.40%

0.60%

0.80%

1.00%

1.20%

1.40%

1.60%

1.80%

0 20 40 60 80 100 120 140

Avg

. Bu

dge

t Sh

are

Average Expenditure ($'000 p.a.)

Public Administration

0.00%

0.50%

1.00%

1.50%

2.00%

2.50%

3.00%

3.50%

4.00%

0 20 40 60 80 100 120 140

Avg

. Bu

dge

t Sh

are

Average Expenditure ($'000 p.a.)

Education

0.00%

0.50%

1.00%

1.50%

2.00%

2.50%

3.00%

0 20 40 60 80 100 120 140

Avg

. Bu

dge

t Sh

are

Average Expenditure ($'000 p.a.)

Health

0.00%

0.20%

0.40%

0.60%

0.80%

1.00%

1.20%

1.40%

1.60%

1.80%

2.00%

0 20 40 60 80 100 120 140

Avg

. Bu

dge

t Sh

are

Average Expenditure ($'000 p.a.)

Recreation

0.00%

1.00%

2.00%

3.00%

4.00%

5.00%

6.00%

7.00%

8.00%

0 20 40 60 80 100 120 140

Avg

. Bu

dge

t Sh

are

Average Expenditure ($'000 p.a.)

Other Services

38

Figure 3. INCOME ELASTICITIES, POOLED DATA

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

39

Figure 4. ESTIMATED SHARES AND ELASTICITIES, QUINTILES

A. SUBSISTENCE SHARES

B. MARGINAL SHARES

C. INCOME ELASTICITIES

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

Foo

d

Bev

gTob

aco

Clo

thFo

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Ligh

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Hea

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anu

f

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Hou

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Hot

elR

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Pri

vate

Tran

s

Pub

licTr

ans

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

Co

mm

un

icat

ion

Fin

ance

Insu

ran

ce

Pro

fSci

Tech

S

Ad

min

Sup

Srv

Pub

licA

dm

in

Edu

cati

on

Hea

lth

Rec

reat

ion

Oth

erS

erv

Quintile 1 Quintile 2 Quintile 3 Quintile 4 Quintile 5

0.00

0.05

0.10

0.15

0.20

0.25

0.30

Foo

d

Bev

gTob

aco

Clo

thFo

otw

r

Ligh

tDu

rab

les

Hea

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anu

f

Uti

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s

Hou

sin

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Hot

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Pri

vate

Tran

s

Pub

licTr

ans

0.00

0.05

0.10

0.15

0.20

0.25

0.30

Co

mm

un

icat

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Fin

ance

Insu

ran

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Pro

fSci

Tech

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Ad

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Sup

Srv

Pub

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dm

in

Edu

cati

on

Hea

lth

Rec

reat

ion

Oth

erS

erv

Quintile 1 Quintile 2 Quintile 3 Quintile 4 Quintile 5

-0.20

0.50

1.20

1.90

2.60

3.30

4.00

Foo

d

Bev

gTob

aco

Clo

thFo

otw

r

Ligh

tDu

rab

les

Hea

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anu

f

Uti

litie

s

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sin

g

Hot

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est

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Pri

vate

Tran

s

Pub

licTr

ans -0.20

0.50

1.20

1.90

2.60

3.30

4.00

Co

mm

un

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ion

Fin

ance

Insu

ran

ce

Pro

fSci

Tech

S

Ad

min

Sup

Srv

Pub

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Edu

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Oth

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Quintile 1 Quintile 2 Quintile 3 Quintile 4 Quintile 5

40

Figure 5. INCOME AND FOOD

Figure 6. STANDARD DEVIATIONS OF INCOME ELASTICITIES

DATA AVAILABILITY STATEMENT

The data that support the findings of this study are available from the corresponding author upon reasonable

request.

0.00

0.05

0.10

0.15

0.20

0.25

1 2 3 4 5

Income

Observed

A. BUDGET SHARES

0.00

0.05

0.10

0.15

0.20

0.25

1 2 3 4 5

IncomeDisaggregated Aggregated

B. MARGINAL SHARES

0.00

0.30

0.60

0.90

1.20

1.50

1 2 3 4 5Income

Disaggregated Aggregated

C. INCOME ELASTICITIES D. ENGEL CURVE

2,000

2,500

3,000

3,500

4,000

4,500

5,000

5,500

6,000

0 20,000 40,000 60,000 80,000 100,000

Exp

end

itu

re

Income

Slope = β1

Slope = β2

Slope = β3

Slope = β4

Slope = β5

Disaggregated Aggregated

0.201 0.318

0.824

0.820

0.463

0.414

0.073

0.146

0.171

0.199

0.201

0.209

0.254

0.276

0.284

0.292

0.298

0.317

0.339

0.343

0.426

0.450

0.530

0.578

0.902 1.086

A. Std dev of income elasticities, Pooled and Quintiles

B. Std dev of quintile income elasticities for each good

41

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Lluch, C. (1971). “Consumer Demand Functions, Spain, 1958-1964.” European Economic

Review 2: 277-302.

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Lluch, C. (1973). “The Extended Linear Expenditure System.” European Economic Review 4:

21-32.

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Household Budget Data: The Linear and Quadratic Expenditure Systems.” American

Economic Review 68: 348-59.

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Econometrica 49:1533-51.

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44

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http://siteresources.worldbank.org/ICPEXT/Resources/ICP_2011.html.

45

APPENDIX

The Utility Value of Income

The Frisch parameter is not constant in LES, but if we step outside of the LES framework,

specifying 𝜔 to take a constant value has some interesting implications for the form of the

utility function.

Following Theil and Suhm (1981, Chap. 7), suppose prices are held constant so the indirect

utility function depends on income only, that is, 𝑢I(𝑀), with marginal utility 𝜆 =𝜕𝑢

𝜕𝑀 and

elasticity 𝜕 𝜆

𝜕𝑀

𝑀

𝜆= 𝜔. Then, with 𝜔 constant, 𝜆 = 𝑘𝑀𝜔 . The term k is a positive proportionality

coefficient that transforms dollars (the units of M) into utils. It is convenient to measure utility

in monetary terms by setting 𝑘 = 1, which is an innocuous normalisation. Integrating, the

utility function takes the form

(A1) 𝑢I(𝑀) = 𝑘′ +

𝑀𝜔+1

𝜔 + 1,

where 𝑘′ is a constant of integration. If country c is richer than d (𝑀𝑐 > 𝑀𝑑) and both possess

utility function (A1), the utility-value of the income difference is 𝑢I(𝑀𝑐) − 𝑢I(𝑀𝑑) =1

𝜔+1(𝑀𝑐

𝜔+1 −𝑀𝑑𝜔+1). When 𝜔 = −2, for example,

𝑢I(𝑀𝑐) − 𝑢I(𝑀𝑑) =1

𝑀𝑑−1

𝑀𝑐> 0.

In words, the difference in utility is the difference in the reciprocals of income. The plausibility

of this simple result could be interpreted as a further attraction of setting 𝜔 = −2.

When 𝜔 < −1, the term 𝑀𝜔+1

𝜔+1 in equation (A1) is negative and goes to zero as M rises

indefinitely. In this case, 𝑢I(𝑀) → 𝑘′, the “bliss” value of utility. When −1 < 𝜔 < 0, 𝑀𝜔+1

𝜔+1 is

positive and unbounded in M (which is perfectly acceptable) and 𝑘′ can no longer be

interpreted as “bliss”; but the difference in utility continues to be 𝑢I(𝑀𝑐) − 𝑢I(𝑀𝑑) =1

𝜔+1(𝑀𝑐

𝜔+1 −𝑀𝑑𝜔+1). When 𝜔 = −

1

2 for example, 𝑢I(𝑀𝑐) − 𝑢I(𝑀𝑑) = 2(√𝑀𝑐 − √𝑀𝑑) >

0. Cross-Sectional Variation in Wages

It cannot be true that all consumers pay the same prices. As a stylised fact, the poor tend to pay

higher relative prices, for example. Nevertheless, the assumption of constant prices is more

likely to be approximately true for some goods, less so for others. In the case of leisure, the

constant-price assumption is most unlikely to be true as the opportunity cost is labour earnings,

“wages” for short, which surely differ substantially from one consumer to another. Thus,

another identification strategy, introduced by Betancourt (1971), is to incorporate leisure in

LES and exploit the cross-sectional variation in wages. The account of this approach that

follows is largely based on Betancourt (1971).19

Write LES as

(A2) 𝑝𝑖𝑞𝑖 = 𝑝𝑖𝛾𝑖 + 𝛽𝑖 (𝑀 −∑𝑝𝑗𝛾𝑗

𝑛−1

𝑗=1

− 𝑝𝑛𝛾𝑛) , 𝑖 = 1,⋯ , 𝑛.

19 Verikios et al. (2019, pp. 38-39) use essentially this approach with TELES (the twice extended LES, once for savings and once

of leisure) plus Howe’s (1975) short-cut way of deriving ELES (LES plus saving) of treating savings as another commodity in

LES with its 𝛾𝑖 = 0, as discussed in Section 2.3 of the text. On TELES, see Betancourt (1973), Tulpule (1980) and Tulpule and Powell (1978). Verikios et al. (2019, p. 42) additionally set the Frisch parameter 𝜔 to -1.5. Gharibnavaz and Verikios (2018, pp.

7-8) also exogenously set 𝜔.

46

Add both sides over 𝑘 = 1,⋯ , 𝑛 − 1:

𝑉 =∑𝑝𝑘𝛾𝑘

𝑛−1

𝑘=1

+ 𝐵(𝑀 −∑𝑝𝑗𝛾𝑗

𝑛−1

𝑗=1

− 𝑝𝑛𝛾𝑛) , 𝑉 = ∑𝑝𝑘𝑞𝑘

𝑛−1

𝑘=1

, 𝐵 = ∑𝛽𝑘

𝑛−1

𝑘=1

.

Thus,

(𝑀 −∑𝑝𝑗𝛾𝑗

𝑛−1

𝑗=1

− 𝑝𝑛𝛾𝑛) =1

𝐵(𝑉 −∑𝑝𝑘𝛾𝑘

𝑛−1

𝑘=1

),

which we substitute back into equation (A2) for 𝑖 = 1,⋯ , 𝑛 − 1 to give

(A3) 𝑝𝑖𝑞𝑖 = 𝑝𝑖𝛾𝑖 +𝛽𝑖𝐵(𝑉 −∑𝑝𝑘𝛾𝑘

𝑛−1

𝑘=1

) = 𝑝𝑖𝛾𝑖 + 𝛽𝑖′ (𝑉 −∑𝑝𝑗𝛾𝑗

𝑛−1

𝑗=1

) , 𝑖 = 1,⋯ , 𝑛 − 1,

where 𝛽𝑖′ =

𝛽𝑖

𝐵 is the marginal share of i in expenditure on the first 𝑛 − 1 goods, with

∑ 𝛽𝑖′ = 1.𝑛−1

𝑖=1

Next, suppose the prices of the first 𝑛 − 1 goods do not vary over consumers, but that of good

n does so. Then, apply the 𝑛𝑡ℎ equation of (A2) to cross-sectional data (indexed by c):

𝑝𝑛𝑐𝑞𝑛𝑐 = 𝑝𝑛𝑐𝛾𝑛 + 𝛽𝑛 (𝑀𝑐 −∑𝑝𝑗𝛾𝑗

𝑛−1

𝑗=1

− 𝑝𝑛𝑐𝛾𝑛) , 𝑐 = 1,⋯ , 𝐶.

Write the above equation as

(A4) 𝑝𝑛𝑐𝑞𝑛𝑐 = −𝛽𝑛∑𝑝𝑗𝛾𝑗

𝑛−1

𝑗=1

+ 𝑝𝑛𝑐𝛾𝑛(1 − 𝛽𝑛) + 𝛽𝑛𝑀𝑐 = 𝛼0+𝛼𝑛𝑝𝑛𝑐 + 𝛽𝑛𝑀𝑐,

where 𝛼0 = −𝛽𝑛 ∑ 𝑝𝑗𝛾𝑗𝑛−1𝑗=1 and 𝛼𝑛 = 𝛾𝑛(1 − 𝛽𝑛). Accordingly,

(A5) ∑𝑝𝑗𝛾𝑗

𝑛−1

𝑗=1

= −𝛼0𝛽𝑛

and 𝛾𝑛 =𝛼𝑛

1−𝛽𝑛.

Next, interpret good n as leisure, so the “quantity demanded” is 𝑞𝑛𝑐 = 365 × 24 − ℎ𝑐, where

ℎ𝑐 is the number of hours worked in a year by consumer c. Here, any time not worked

constitutes “leisure”. What if not all this non-work time is not in fact leisure? The time spent

sleeping, performing housework, etc. can be allowed by a corresponding adjustment of 𝛾𝑛, the

subsistence parameter for leisure. The “price of leisure” paid by c, 𝑝𝑛𝑐 , is the wage rate received

by c. Estimating equation (A4) as a single linear equation gives estimates of the intercept 𝛼0

and slope coefficient 𝛽𝑛, with ∑ 𝑝𝑗𝛾𝑗𝑛−1𝑗=1 defined by their ratio with the sign changed,

according to equation (A5).

System (A3) refers to the first n-1 goods, the prices of which are assumed constant. Thus, the

cross-sectional version is

𝑝𝑖𝑞𝑖𝑐 = 𝛼𝑖 + 𝛽𝑖′𝑉𝑐, 𝛼𝑖 =∑(𝛿𝑖𝑗 − 𝛽𝑖

′)

𝑛−1

𝑗=1

𝑝𝑗𝛾𝑗 , 𝑖 = 1,⋯ , 𝑛 − 1; 𝑐 = 1,⋯ , 𝐶.

Using the estimate of ∑ 𝑝𝑗𝛾𝑗𝑛−1𝑗=1 given by (A5), −

𝛼0

𝛽𝑛, each of the 𝑛 − 1 values of the 𝑝𝑗𝛾𝑗 ′𝑠 can

then be identified as

47

𝑝𝑖𝛾𝑖 = 𝛼𝑖 − 𝛽𝑖′ 𝛼0𝛽𝑛, 𝑖 = 1,⋯ , 𝑛 − 1.

This is just a modified version of equation (10). By construction, the sum of these values equals

the “extraneous” sum.

As 𝑀𝑐 = ∑ 𝑝𝑖𝑐𝑞𝑖𝑐𝑛𝑖=1 = 𝑝𝑛𝑐𝑞𝑛𝑐 + 𝑉𝑐, “full income” 𝑀𝑐 could be treated as endogenous. In such

a case, one might prefer to estimate the reduced form corresponding to equation (A4) which

simply involves replacing 𝑀𝑐 with 𝑉𝑐 and a reinterpretation of the coefficients:

𝑝𝑛𝑐𝑞𝑛𝑐 = −𝛽𝑛

1 − 𝛽𝑛∑𝑝𝑗𝛾𝑗

𝑛−1

𝑗=1

+ 𝑝𝑛𝑐𝛾𝑛 +𝛽𝑛

1 − 𝛽𝑛𝑉𝑐 = 𝛼0

′ + 𝛾𝑛𝑝𝑛𝑐 + 𝜃𝑉𝑐,

with 𝛼0′ = −

𝛽𝑛

1−𝛽𝑛∑ 𝑝𝑗𝛾𝑗 ,𝑛−1𝑗=1 𝜃 =

𝛽𝑛

1−𝛽𝑛, ∑ 𝑝𝑗𝛾𝑗 = −

𝛼0′

𝜃 𝑛−1

𝑗=1 .

There can also be situations in which the cross-sectional price variation extends to more than

just wages. For example, Lluch (1971) estimated demand systems (but not LES) from

household data that contained regional price variation, while Kravis et al. (1982, Chap. 9)

applied LES to data from a number of countries at a given point in time using PPP prices

(different in different countries). Along similar lines, Pollak and Wales (1978) estimate LES

with two cross sections of data.

48

Other Determinants of Consumption

Data at the individual household level frequently exhibits considerable variability due

to the heterogeneity of households. This can be accounted for within a single framework in

LES by making the subsistence parameters depend on household characteristics. Pollak and

Wales (1978, 1981) call this approach “demographic translating”. A brief account of this

procedure follows.

Suppose household characteristics are binary (with or without children, for example)

and let there be d characteristics so that households can be classified into the sets 𝑺1, ⋯ , 𝑺𝑑 . Define the indicator variable 𝐼𝑐𝑘(𝑐 ∈ 𝑺𝑘) that takes the value 1 if household c possesses

characteristic k, 0 otherwise. Under demographic translation, subsistence expenditures are

taken to be linear in characteristics:

𝑝𝑖𝛾𝑖𝑐 = 𝑝𝑖𝛾𝑖0 +∑(𝑝𝑖𝛾𝑖𝑘)𝐼𝑐𝑘(𝑐 ∈ 𝑺𝑘)

𝑑

𝑘=1

, 𝑖 = 1,⋯ , 𝑛.

The intercept, 𝑝𝑖𝛾𝑖0, refers to subsistence of the good for the base group of households that do

not possess any of the d characteristic, while the coefficients of the indicator variables,

𝑝𝑖𝛾𝑖𝑘 , 𝑘 = 1,⋯ , 𝑑, are intercept shifters, of which there are 𝑛 × 𝑑. The cross-sectional version

of LES in then 𝑝𝑖𝑞𝑖𝑐 = 𝛼𝑖𝑐 + 𝛽𝑖𝑀𝑐 , 𝑖 = 1,⋯ , 𝑛; 𝑐 = 1,⋯ , 𝐶,

where

𝛼𝑖𝑐 =∑(𝛿𝑖𝑗−𝛽𝑖)

𝑛

𝑗=1

{𝑝𝑗𝛾𝑗0 +∑(𝑝𝑗𝛾𝑗𝑘)𝐼𝑐𝑘(𝑐 ∈ 𝑺𝑘)

𝑑

𝑘=1

} , 𝑖 = 1,⋯ , 𝑛.

As ∑ (𝛿𝑖𝑗−𝛽𝑖) = 0,𝑛𝑖=1 it follows that ∑ 𝛼𝑖𝑐 = 0,

𝑛𝑖=1 as in the absence of demographic effects.

The comments can be made regarding this approach:

1. Demographic translating does not aid with the identification subsistence expenditures,

it just serves to model some of the non-price-non-income determinants of

consumption at the individual level. After appropriate modifications, the procedures

discussed in earlier sections of the main text can be used for identification.

2. A similar procedure can be used when in a time-series context: For example,

subsistence expenditures can be made dependent on a time trend or functions of

previous expenditures (Pollak and Wales, 1969, Stone, 1954, Stone et al., 1964).

These treatments eliminate much of the serial correlation frequently encountered with

LES time-series residuals.

3. As mentioned, the variable 𝛾𝑖 ′𝑠 generate shifting intercepts. This is a type of

functional form adjustment that can make LES more flexible.

Appendix References

Betancourt, R. R. (1971). “The Estimation of Price Elasticities from Cross-Section Data under

Additive Preferences.” International Economic Review 12: 283-92.

Betancourt, R. R. (1973). “Household Behaviour in a Less Developed Country: An

Econometric Analysis of Cross-Sectional Data.” University of Maryland, working

paper.

Gharibnavaz, R. and G. Verikios (2018). “Estimating LES Parameters with Heterogeneous

Households for a CGE Model.” 21st Annual Conference on Global Economic Analysis,

Cartagena, Colombia, June 13-15.

49

Howe, H. (1975). “Development of the Extended Linear Expenditure System from Simple

Saving Assumptions.” European Economic Review 6: 305-10.

Kravis, I. B., A. Heston, R. Summers (1982). World Product and Income: International

Comparisons of Real Gross Product. Baltimore, MD: Johns Hopkins University Press.

Lluch, C. (1971). “Consumer Demand Functions, Spain, 1958-1964.” European Economic

Review 2: 277-302.

Pollak, R. A., and T. J. Wales (1969). “Estimation of the Linear Expenditure System.”

Econometrica 37: 611-28.

Pollak, R. A., and T. J. Wales (1978). “Estimation of Complete Demand Systems from

Household Budget Data: The Linear and Quadratic Expenditure Systems.” American

Economic Review 68: 348-59.

Pollak, R. A., and T. J. Wales (1981). “Demographic Variables in Demand Analysis.”

Econometrica 49:1533-51.

Stone, R. (1954). “Linear Expenditure Systems and Demand Analysis: An Application to the

Pattern of British Demand.” Economic Journal 64: 511-27.

Stone, R., A. Brown and D. A. Rowe (1964). “Demand Analysis and Projections for Britain,

1900—1970: A Study in Method.” In J. Sandee (ed) Europe’s Future Consumption.

Vol. 2, Amsterdam: North-Holland.

Theil, H., and F. E. Suhm (1981). International Consumption Comparisons: A System-Wide

Approach. Amsterdam: North Holland.

Tulpule, A., and A. A. Powell (1978). “Estimates of Household Demand Parameters for the

Orani Model.” Impact Preliminary Working Paper OP-22, Industries Assistance

Commission, Melbourne.

Tulpule, A. (1980). “Revised Estimates of Labour Supply Elasticities.” Working Paper B-12,

Impact Project Research Centre, University of Melbourne, Melbourne.

Verikios, G., K. Hanslow, D. Bahyl and R. Gharibnavaz (2019). “UK-CGE, a Computable

General Equilibrium Model of the Four Countries of the United Kingdom: Model

Documentation.” May. KPMG Economics, Australia.

50

APPENDIX T1. MARSHALLIAN PRICE ELASTICITIES

Commodity group i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

1. Food -0.408 -0.013 -0.010 -0.024 -0.024 -0.025 -0.056 -0.021 -0.008 -0.008 -0.019 -0.016 -0.034 -0.004 -0.006 -0.005 -0.005 -0.007 -0.005 -0.015

2. Beverages & tobacco -0.069 -0.498 -0.014 -0.033 -0.032 -0.033 -0.076 -0.029 -0.010 -0.011 -0.026 -0.021 -0.045 -0.005 -0.008 -0.006 -0.007 -0.010 -0.007 -0.020

3. Cloth and Footwear -0.080 -0.021 -0.576 -0.038 -0.038 -0.039 -0.088 -0.033 -0.012 -0.013 -0.030 -0.025 -0.053 -0.006 -0.010 -0.007 -0.009 -0.012 -0.008 -0.023

4. Light Goods Manuf. -0.079 -0.020 -0.016 -0.587 -0.037 -0.038 -0.086 -0.033 -0.012 -0.012 -0.029 -0.024 -0.052 -0.006 -0.010 -0.007 -0.008 -0.012 -0.008 -0.022

5. Heavy Goods Manuf. -0.064 -0.017 -0.013 -0.031 -0.478 -0.031 -0.070 -0.027 -0.010 -0.010 -0.024 -0.020 -0.042 -0.005 -0.008 -0.006 -0.007 -0.009 -0.007 -0.018

6. Utilities -0.026 -0.007 -0.005 -0.012 -0.012 -0.191 -0.028 -0.011 -0.004 -0.004 -0.010 -0.008 -0.017 -0.002 -0.003 -0.002 -0.003 -0.004 -0.003 -0.007

7. Housing -0.073 -0.019 -0.015 -0.035 -0.034 -0.035 -0.589 -0.030 -0.011 -0.011 -0.027 -0.022 -0.048 -0.005 -0.009 -0.007 -0.008 -0.011 -0.007 -0.021

8. Hotel & Restaurant -0.080 -0.021 -0.016 -0.038 -0.038 -0.039 -0.088 -0.590 -0.012 -0.012 -0.030 -0.025 -0.053 -0.006 -0.010 -0.007 -0.008 -0.012 -0.008 -0.023

9. Private Transport -0.097 -0.025 -0.019 -0.046 -0.046 -0.047 -0.106 -0.040 -0.690 -0.015 -0.036 -0.030 -0.064 -0.007 -0.012 -0.009 -0.010 -0.014 -0.010 -0.028

10. Public Transport -0.081 -0.021 -0.016 -0.039 -0.038 -0.039 -0.089 -0.034 -0.012 -0.576 -0.030 -0.025 -0.053 -0.006 -0.010 -0.007 -0.009 -0.012 -0.008 -0.023

11. Communications -0.052 -0.014 -0.010 -0.025 -0.024 -0.025 -0.057 -0.022 -0.008 -0.008 -0.383 -0.016 -0.034 -0.004 -0.006 -0.005 -0.006 -0.008 -0.005 -0.015

12. Finance -0.092 -0.024 -0.018 -0.044 -0.043 -0.044 -0.101 -0.038 -0.014 -0.014 -0.034 -0.668 -0.060 -0.007 -0.011 -0.008 -0.010 -0.013 -0.009 -0.026

13. Insurance -0.072 -0.019 -0.014 -0.035 -0.034 -0.035 -0.079 -0.030 -0.011 -0.011 -0.027 -0.022 -0.552 -0.005 -0.009 -0.007 -0.008 -0.011 -0.007 -0.021

14. Professional Services -0.077 -0.020 -0.015 -0.037 -0.036 -0.037 -0.085 -0.032 -0.012 -0.012 -0.029 -0.024 -0.051 -0.545 -0.009 -0.007 -0.008 -0.011 -0.008 -0.022

15. Admin & Support Serv. -0.084 -0.022 -0.017 -0.040 -0.039 -0.040 -0.092 -0.035 -0.013 -0.013 -0.031 -0.026 -0.055 -0.006 -0.595 -0.008 -0.009 -0.012 -0.009 -0.024

16. Public Administration -0.059 -0.015 -0.012 -0.028 -0.028 -0.029 -0.065 -0.025 -0.009 -0.009 -0.022 -0.018 -0.039 -0.004 -0.007 -0.419 -0.006 -0.009 -0.006 -0.017

17. Education -0.092 -0.024 -0.018 -0.044 -0.043 -0.044 -0.101 -0.038 -0.014 -0.014 -0.034 -0.028 -0.060 -0.007 -0.011 -0.008 -0.650 -0.013 -0.009 -0.026

18. Health -0.081 -0.021 -0.016 -0.039 -0.038 -0.039 -0.089 -0.034 -0.012 -0.013 -0.030 -0.025 -0.054 -0.006 -0.010 -0.007 -0.009 -0.580 -0.008 -0.023

19. Recreation -0.079 -0.021 -0.016 -0.038 -0.037 -0.038 -0.087 -0.033 -0.012 -0.012 -0.029 -0.024 -0.052 -0.006 -0.010 -0.007 -0.008 -0.012 -0.560 -0.023

20. Other Services -0.086 -0.022 -0.017 -0.041 -0.040 -0.041 -0.094 -0.036 -0.013 -0.013 -0.032 -0.026 -0.056 -0.006 -0.010 -0.008 -0.009 -0.013 -0.009 -0.622

51

APPENDIX T2. SLUTSKY PRICE ELASTICITIES

Commodity group i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

1. Food -0.328 0.012 0.013 0.030 0.020 0.005 0.058 0.027 0.016 0.010 0.011 0.028 0.034 0.004 0.009 0.003 0.010 0.010 0.006 0.022

2. Beverages & tobacco 0.038 -0.463 0.017 0.040 0.026 0.007 0.078 0.036 0.022 0.014 0.015 0.038 0.046 0.006 0.012 0.004 0.013 0.013 0.009 0.029

3. Cloth and Footwear 0.044 0.019 -0.539 0.047 0.031 0.008 0.091 0.042 0.025 0.016 0.017 0.044 0.054 0.007 0.014 0.005 0.015 0.015 0.010 0.034

4. Light Goods Manuf. 0.044 0.019 0.020 -0.504 0.030 0.008 0.090 0.041 0.025 0.016 0.017 0.043 0.053 0.007 0.014 0.005 0.015 0.015 0.010 0.033

5. Heavy Goods Manuf. 0.036 0.015 0.016 0.037 -0.423 0.007 0.073 0.033 0.020 0.013 0.014 0.035 0.043 0.006 0.011 0.004 0.012 0.012 0.008 0.027

6. Utilities 0.014 0.006 0.006 0.015 0.010 -0.176 0.029 0.013 0.008 0.005 0.005 0.014 0.017 0.002 0.004 0.002 0.005 0.005 0.003 0.011

7. Housing 0.040 0.018 0.018 0.043 0.028 0.008 -0.426 0.038 0.023 0.015 0.015 0.040 0.049 0.006 0.013 0.005 0.014 0.014 0.009 0.031

8. Hotel & Restaurant 0.044 0.019 0.020 0.047 0.030 0.008 0.091 -0.515 0.025 0.016 0.017 0.044 0.054 0.007 0.014 0.005 0.015 0.015 0.010 0.034

9. Private Transport 0.054 0.023 0.024 0.056 0.037 0.010 0.110 0.050 -0.645 0.020 0.021 0.053 0.065 0.008 0.017 0.006 0.018 0.019 0.012 0.041

10. Public Transport 0.045 0.019 0.020 0.047 0.031 0.008 0.092 0.042 0.025 -0.548 0.017 0.044 0.054 0.007 0.014 0.005 0.015 0.016 0.010 0.034

11. Communications 0.029 0.012 0.013 0.030 0.020 0.005 0.059 0.027 0.016 0.010 -0.352 0.028 0.035 0.005 0.009 0.003 0.010 0.010 0.007 0.022

12. Finance 0.051 0.022 0.023 0.054 0.035 0.010 0.104 0.048 0.029 0.019 0.019 -0.590 0.062 0.008 0.016 0.006 0.017 0.018 0.012 0.039

13. Insurance 0.040 0.017 0.018 0.042 0.028 0.008 0.082 0.038 0.023 0.015 0.015 0.040 -0.456 0.006 0.012 0.005 0.014 0.014 0.009 0.031

14. Professional Services 0.043 0.019 0.020 0.045 0.030 0.008 0.088 0.040 0.024 0.016 0.016 0.042 0.052 -0.533 0.013 0.005 0.015 0.015 0.010 0.033

15. Admin & Support Serv. 0.046 0.020 0.021 0.049 0.032 0.009 0.095 0.044 0.026 0.017 0.018 0.046 0.056 0.007 -0.570 0.005 0.016 0.016 0.011 0.035

16. Public Administration 0.033 0.014 0.015 0.035 0.023 0.006 0.067 0.031 0.019 0.012 0.013 0.032 0.040 0.005 0.010 -0.410 0.011 0.011 0.007 0.025

17. Education 0.051 0.022 0.023 0.054 0.035 0.010 0.104 0.048 0.029 0.018 0.019 0.050 0.062 0.008 0.016 0.006 -0.623 0.018 0.012 0.039

18. Health 0.045 0.020 0.021 0.048 0.031 0.009 0.093 0.042 0.026 0.016 0.017 0.045 0.055 0.007 0.014 0.005 0.015 -0.553 0.010 0.034

19. Recreation 0.044 0.019 0.020 0.046 0.030 0.008 0.090 0.041 0.025 0.016 0.017 0.043 0.053 0.007 0.014 0.005 0.015 0.015 -0.542 0.033

20. Other Services 0.047 0.021 0.022 0.050 0.033 0.009 0.097 0.045 0.027 0.017 0.018 0.047 0.057 0.007 0.015 0.006 0.016 0.017 0.011 -0.561

52

APPENDIX T3. ESTIMATES OF LES BY QUINTILE

Commodity

Quintile 1 Quintile 2 Quintile 3

Intercept

𝛼𝑖

Slope

𝛽𝑖

Intercept

𝛼𝑖

Slope

𝛽𝑖

Intercept

𝛼𝑖

Slope

𝛽𝑖

1. Food 126 (74.92) 0.1899 (0.0059) 2168 (396.21) 0.07 (0.0159) 4015 (483.28) 0.0123 (0.0135)

2. Beverages and tobacco -112 (50.25) 0.0462 (0.004) -234 (362.29) 0.052 (0.0145) 672 (462.21) 0.0224 (0.0129)

3. Clothing and Footwear -156 (34.21) 0.0338 (0.0027) -751 (260.43) 0.0601 (0.0105) -318 (371.16) 0.039 (0.0104)

4. Light Goods Manufacturers -263 (47.69) 0.0748 (0.0038) -420 (385.6) 0.0877 (0.0155) -117 (635.03) 0.0787 (0.0177)

5. Heavy Goods Manufacturers -143 (46.69) 0.0883 (0.0037) 370 (260.4) 0.0596 (0.0105) 1113 (389) 0.0368 (0.0109)

6. Utilities 1055 (45.25) 0.0321 (0.0036) 1001 (191.95) 0.0263 (0.0077) 1291 (219.88) 0.0143 (0.0062)

7. Housing -436 (150.54) 0.1859 (0.0119) 1370 (978.32) 0.1189 (0.0392) 961 (1267.8) 0.1294 (0.0354)

8. Hotels and Restaurants -143 (40.62) 0.0564 (0.0032) -830 (311.36) 0.0911 (0.0125) -1145 (474.5) 0.0976 (0.0133)

9. Private Transportation -34 (16.14) 0.0057 (0.0013) -1084 (311.14) 0.0572 (0.0125) -1262 (644.44) 0.0604 (0.018)

10. Public Transportation -85 (31.35) 0.0235 (0.0025) -111 (241.12) 0.0267 (0.0097) -265 (336.71) 0.0311 (0.0094)

11. Communications 290 (33.76) 0.0485 (0.0027) 629 (184.38) 0.0315 (0.0074) 1350 (231.78) 0.0103 (0.0065)

12. Finance -172 (36.92) 0.0278 (0.003) -946 (404.66) 0.0708 (0.0162) -2531 (782.12) 0.1284 (0.0218)

13. Insurance 354 (64.31) 0.0714 (0.0051) 494 (380.97) 0.0615 (0.0153) -82 (661.33) 0.0889 (0.0185)

14. Professional Services -2 (25.61) 0.0091 (0.0021) -57 (207.71) 0.0133 (0.0084) -537 (299.39) 0.0258 (0.0084)

15. Admin & Support Services -41 (30.78) 0.0151 (0.0025) -332 (260.45) 0.0277 (0.0105) -368 (507.85) 0.0309 (0.0142)

16. Public Administration 78 (16.12) 0.0098 (0.0013) 17 (132.42) 0.0128 (0.0053) -101 (144.15) 0.0152 (0.0041)

17. Education -10 (18.51) 0.0067 (0.0015) -375 (177.82) 0.0261 (0.0072) -1509 (384.74) 0.0588 (0.0108)

18. Health -66 (29.13) 0.0199 (0.0023) -88 (224.86) 0.0231 (0.009) -561 (455.87) 0.0421 (0.0127)

19. Recreation -76 (23.37) 0.0175 (0.0019) 20 (165.72) 0.0138 (0.0067) 108 (264.66) 0.014 (0.0074)

20. Other Services -168 (40.69) 0.0385 (0.0032) -842 (353.44) 0.0708 (0.0142) -716 (640.93) 0.0646 (0.0179)

Note: Standard errors are in parentheses.

53

APPENDIX T3. ESTIMATES OF LES BY QUINTILE (continued)

Commodity

Quintile 4 Quintile 5

Intercept

𝛼𝑖

Slope

𝛽𝑖

Intercept

𝛼𝑖

Slope

𝛽𝑖

1. Food 3747 (476.25) 0.0286 (0.0094) 4792 (256.06) 0.0138 (0.0031)

2. Beverages and tobacco 707 (507.32) 0.0209 (0.01) 1839 (248.06) 0.0059 (0.003)

3. Clothing and Footwear -521 (496.79) 0.0448 (0.0098) 1047 (355.75) 0.0214 (0.0043)

4. Light Goods Manufacturers 248 (761.99) 0.0714 (0.015) 975 (642.34) 0.069 (0.0077)

5. Heavy Goods Manufacturers 1206 (424.77) 0.036 (0.0084) 2478 (220.95) 0.0162 (0.0027)

6. Utilities 1569 (211.28) 0.0082 (0.0042) 1510 (107.37) 0.0085 (0.0013)

7. Housing 608 (1403) 0.1379 (0.0275) -9229 (1083.46) 0.2648 (0.0129)

8. Hotels and Restaurants -912 (614.26) 0.0903 (0.0121) 2318 (420.76) 0.0445 (0.005)

9. Private Transportation -1213 (954.3) 0.0605 (0.0187) -763 (779.95) 0.0624 (0.0093)

10. Public Transportation 15 (442.26) 0.0258 (0.0087) -1 (352.62) 0.0294 (0.0042)

11. Communications 1269 (264.72) 0.0159 (0.0052) 1763 (130.47) 0.0084 (0.0016)

12. Finance -2632 (965.27) 0.1238 (0.0189) 2256 (641.89) 0.052 (0.0077)

13. Insurance -763 (908.5) 0.1057 (0.0178) -5908 (997.05) 0.1865 (0.0119)

14. Professional Services -88 (413.9) 0.0137 (0.0081) -454 (316.23) 0.0178 (0.0038)

15. Administration and Support Services -438 (681.48) 0.0316 (0.0134) -116 (442.28) 0.0265 (0.0053)

16. Public Administration 68 (199.66) 0.0101 (0.004) 199 (187.14) 0.0067 (0.0023)

17. Education -931 (646.7) 0.0433 (0.0127) -1367 (574.26) 0.0454 (0.0069)

18. Health -109 (520.36) 0.0275 (0.0102) -348 (421.52) 0.0295 (0.0051)

19. Recreation 197 (308.49) 0.0123 (0.0061) 684 (256.66) 0.0094 (0.0031)

20. Other Services -2024 (936.38) 0.0925 (0.0184) -1675 (731.86) 0.0829 (0.0087)

Note: Standard errors are in parentheses.

54

APPENDIX T4. CONSUMPTION SHARES, QUINTILES

Commodity group

Budget shares for quintile Marginal shares by quintile Subsistence shares by quintile

1 2 3 4 5 1 2 3 4 5 1 2 3 4 5

1. Food 0.196 0.155 0.122 0.100 0.060 0.190 0.070 0.012 0.029 0.014 0.206 0.240 0.232 0.172 0.108

2. Beverages and tobacco 0.039 0.043 0.041 0.034 0.024 0.046 0.052 0.022 0.021 0.006 0.032 0.034 0.059 0.048 0.042

3. Clothing and Footwear 0.024 0.031 0.030 0.035 0.030 0.034 0.060 0.039 0.045 0.021 0.014 0.001 0.022 0.025 0.042

4. Light Goods Manufacturers 0.059 0.071 0.075 0.076 0.077 0.075 0.088 0.079 0.071 0.069 0.041 0.055 0.072 0.081 0.088

5. Heavy Goods Manufacturers 0.079 0.074 0.067 0.059 0.040 0.088 0.060 0.037 0.036 0.016 0.070 0.089 0.098 0.082 0.065

6. Utilities 0.100 0.066 0.050 0.038 0.023 0.032 0.026 0.014 0.008 0.008 0.169 0.105 0.085 0.068 0.038

7. Housing 0.158 0.173 0.156 0.150 0.189 0.186 0.119 0.129 0.138 0.265 0.129 0.226 0.182 0.161 0.083

8. Hotels and Restaurants 0.047 0.058 0.066 0.073 0.065 0.056 0.091 0.098 0.090 0.044 0.038 0.026 0.035 0.055 0.090

9. Private Transportation 0.004 0.015 0.026 0.037 0.053 0.006 0.057 0.060 0.060 0.062 0.001 -0.028 -0.009 0.014 0.047

10. Public Transportation 0.018 0.022 0.024 0.026 0.028 0.023 0.027 0.031 0.026 0.029 0.012 0.018 0.017 0.026 0.029

11. Communications 0.067 0.056 0.047 0.040 0.025 0.048 0.031 0.010 0.016 0.008 0.086 0.081 0.084 0.065 0.043

12. Finance 0.017 0.034 0.059 0.073 0.071 0.028 0.071 0.128 0.124 0.052 0.005 -0.004 -0.011 0.023 0.096

13. Insurance 0.094 0.081 0.087 0.091 0.128 0.071 0.061 0.089 0.106 0.186 0.117 0.100 0.084 0.076 0.070

14. Professional Services 0.009 0.011 0.011 0.012 0.013 0.009 0.013 0.026 0.014 0.018 0.009 0.009 -0.004 0.010 0.009

15. Admin & Support Services 0.012 0.015 0.021 0.023 0.025 0.015 0.028 0.031 0.032 0.026 0.010 0.002 0.011 0.015 0.024

16. Public Administration 0.015 0.013 0.012 0.011 0.008 0.010 0.013 0.015 0.010 0.007 0.020 0.014 0.010 0.013 0.011

17. Education 0.006 0.011 0.017 0.025 0.031 0.007 0.026 0.059 0.043 0.045 0.005 -0.003 -0.024 0.008 0.019

18. Health 0.016 0.020 0.027 0.025 0.027 0.020 0.023 0.042 0.027 0.029 0.011 0.016 0.011 0.023 0.023

19. Recreation 0.013 0.015 0.017 0.016 0.016 0.017 0.014 0.014 0.012 0.009 0.008 0.015 0.020 0.020 0.023

20. Other Services 0.028 0.038 0.045 0.054 0.065 0.038 0.071 0.065 0.092 0.083 0.017 0.005 0.025 0.015 0.050

Sum 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

1