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Perry & Wilson: The Accord and Strikes Australian Journal of Labour Economics, Vol. 7, No. 1, March 2004, pp 89 - 108

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The Role of The Unit of Analysis

in Tax Policy Reform Evaluations

of Inequality and Social Welfare

John Creedy and Rosanna Scutella

The University of Melbourne

Abstract

This paper examines the implications, for overall social welfare and inequality

comparisons, of using different definitions of the unit of analysis in computing

summary measures. The units considered are households, individuals and adult

equivalent persons. Comparisons are made of the effects of flattening the marginal

tax rate structure using the Melbourne Institute Tax and Transfer Simulator

(MITTS), a simulation model of the Australian direct tax and benefit system. The

reform was found to reduce inequality, no matter which unit of analysis was chosen.

However, it was not always judged to improve social welfare, depending on the

degree of inequality aversion and the unit of analysis chosen.

1. Introduction

A major advantage of tax microsimulation models is that they deal with

the considerable heterogeneity found in populations. They can be used to

examine the effects of a policy reform on a wide variety of types of person

(distinguished, say, by household type, location, education, occupation or

age). But this heterogeneity raises problems when making an overall

evaluation of a policy change in terms of inequality or social welfare, since

standard measures are designed for homogeneous populations. In making

decisions about the two fundamental concepts of income and the unit of

analysis, the difficulty is, as Ebert (1997, p.235) put it, that ‘an (artificial)

income distribution for a fictitious population has to be constructed’.1

Most studies regard the only relevant non-income difference as the

household size and its composition. The first stage, involving the artificial

income concept, is to convert total household income into a measure of the

‘living standard’ of each household member by dividing income by the

adult equivalent household size. This method of constructing the ‘money

metric welfare measure’ for individuals in a household has many well-

known problems. However, it is taken as given here, where emphasis is

placed instead on the choice of a fictitious population.

1 Cowell (1984) discussed nine alternatives, arising from a distinction between three

types of income recipient and three income measures. A third decision concerns the

time period of analysis, but this is not considered here: attention is restricted to

annual incomes.

Address for correspondence: John Creedy, Department of Economics, The University

of Melbourne, Parkville, Vic 3010. Email: jcreedy@unimelb.edu.au

We are grateful to Guyonne Kalb and two referees for helpful comments on an earlier

draft of this paper.

© The Centre for Labour Market Research, 2004

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The aim of this paper is thus to examine the implications for overall social

welfare and inequality comparisons of using different definitions of the

unit of analysis - the income recipient - in computing summary measures.

Comparisons are made using the Melbourne Institute Tax and Transfer

Simulator (MITTS). This is a simulation model of the Australian direct tax

and benefit system; see Creedy et al. (2002).2 The database used is the 1997/

98 Survey of Income and Housing Costs (SIHC), made available by the

ABS as a confidentialised unit record file (CURF). Net incomes can be

calculated for each individual for different tax and transfer systems,

allowing hypothetical and actual policy changes to be analysed.

The model is used to simulate the effects of flattening the marginal tax rate

structure. No suggestion is made that the comparisons reflect general

properties. However, it is useful to provide some indication of potential

orders of magnitude in a realistic context. The present study therefore

supplements that of Decoster and Ooghe (2002), who made extensive

comparisons for a policy reform in Belgium.

Three units of analysis are discussed and summary measures for alternative

definitions of the unit are described formally in section 2. The units

considered are households, individuals and adult equivalent persons.

Section 3 examines the relationship between inequality and adult

equivalence scales, for net incomes before the reform. The hypothetical

policy reform is described in section 4.

The tax and transfer systems examined are based on the March 1998 system

as the 1997/98 Survey of Income and Housing Costs was the latest data

base publicly available at the time of writing. The MITTS model consists of

a non-behavioural component, MITTS-A, and a behavioural component

estimating the effect of changes in labour supply behaviour, MITTS-B.

Numerical results using MITTS-A are reported in section 5. The effects of

allowing for labour supply responses to tax changes are examined in section

6. Conclusions are in section 7.

2. Alternative Concepts and Measures

This section introduces the notation and describes the alternative summary

measures of inequality and welfare used. First, equivalence scales are

defined in subsection 1. The three types of unit of analysis are discussed in

subsection 2, and the resulting welfare functions and inequality measures

are defined in subsection 3.

Adult Equivalence Scales

Let yi denote the total income of the ith household, for i = 1,...,N. The number

of individuals in the household is ni and the demographic structure is

denoted by di. Here di can be regarded as a vector indicating the number of

people of each of a number of types defined by age and gender. The adult

equivalent size of the household, mi, can be expressed as:

(1)

2 MITTS is joint intellectual property of The Melbourne Institute of Applied Economic

and Social Research and the Commonwealth Department of Family and Community

Services.

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Evaluations of Inequality and Social Welfare

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This is normalised such that m(n = 1,d = adult) = 1. Thus a household

consisting of one adult with an income of y is regarded as having the same

‘living standard’ as an n-person household with y multiplied by m(n,d).

The form of m(ni,di) needs to be specified. If there are nk,i individuals of

demographic type k = 1,...,K in the ith household, the adult equivalent size

may be written:

(2)

The term θ is regarded as a measure of economies of scale within the

household, with 0 ≤ θ ≤ 1. This formulation is an extension of the simple

form, , used by Buhmann et al. (1988) and Coulter et al. (1992) and

modified by, for example, Cutler and Katz (1992), Banks and Johnson (1994)

and Jenkins and Cowell (1994) who differentiated between adults and

children.

The scales examined below are a special case of (2) which distinguishes the

number of adults, na,i and children, nc,i,, such that:

(3)

This makes a distinction between the ‘head’ of the unit, for whom the weight

is unity, additional adults, and children. No distinction is made between

age and gender. In the following analyses φ1 = 0.56 and φ2 = 0.32. This choice,

with θ = 1, corresponds to the Whiteford (1985) scales. With θ < 1, this

implies a smaller adult equivalent household size than intended in the

Whiteford scales. This should be kept in mind when considering the results

below.

Three Units of Analysis

A number of empirical studies have taken the household itself as the basic

unit of analysis, usually with little discussion. This involves each household

being assigned the living standard, defined above as total income per adult

equivalent, and making no further allowance for the demographic structure

of the unit. One way to describe this is to say that the living standard of

each household is given a weight of 1/N in computing inequality and

welfare summary measures. While this approach appears to have little

rationale, it is included for comparative purposes here.

An alternative, which appears to treat the income concept and the unit of

analysis consistently, is to define the basic unit of analysis as the ‘adult

equivalent person’.3 The ith household contains mi adult equivalent persons,

each getting the living standard of yi/mi, where yi is the household’s total

income. This approach means that an individual’s contribution to inequality

and social welfare depends on the composition of the household of which

that person is a member. For example, an adult in a one-person household

‘counts for one’, but the same person in a household containing other adults

3 This approach is recommended by Ebert (1997) who suggested that of the

alternatives, it is the ‘most recommendable’ (p.243).

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Australian Journal of Labour Economics, March 2004

92

and several children counts for ‘less than one’. A feature of this approach is

that it satisfies the basic equity principle, associated with the principle of

transfers, that a transfer of income to those worse-off results in a reduction

in inequality (and an increase in social welfare). As a result, Lorenz and

Generalised Lorenz curve analyses can be conducted using the resulting

distribution.4

A third alternative is to treat the individual as the basic unit of analysis.5

Each individual is assigned the living standard of the relevant household.

Thus each individual effectively ‘counts for one’ irrespective both of the

household to which he or she belongs and the person’s age or gender. This

approach therefore has the property of anonymity, such that welfare or

inequality remain unchanged when one person (of whatever type) in the

population is replaced by another person having the same living standard

but belonging to any other type of household. This property was called the

‘compensation principle’ by Shorrocks (1997) and the ‘Pareto indifference

principle’ by Decoster and Ooghe (2002).

This approach does not in general satisfy the equity principle (of transfers).

As shown by Glewwe (1991), an income transfer from a poor to a richer

(and larger) household can reduce inequality and raise social welfare.6

Despite being based on individuals, the application of anonymity can lead

to a preference for inequality: with economies of scale, large households

are regarded as being ‘more efficient’ at generating welfare.

An important implication is that in this context of heterogeneous

populations, the basic equity principle inherent in the principle of transfers

and the concept of Lorenz dominance (whereby one Lorenz curve lies

unambiguously closer to the diagonal of equality) are no longer equivalent.

This equivalence is a fundamental component of welfare analysis for

homogeneous populations.

The choice between individuals and adult equivalents as the basic unit of

analysis in inequality and social welfare calculations therefore involves a

choice between two incompatible value judgements. They can in principle

lead to opposite conclusions about the effects of a tax policy change on

inequality.7 Before examining how they perform in a practical case, the

approaches are described more formally in the following subsection.

4 Despite explicitly not treating individuals as the unit, but instead using adult

equivalents, this actually leads to a recommendation for equal standards of living;

see Ebert (1997, p.242).

5 Jorgenson and Slesnick (1984) and Slesnick (1994) use this method, as does Glewwe

(1991), who dismisses the use of adult equivalents in a footnote (p.213). It is also

preferred by Shorrocks (1997), Danziger and Taussig (1979) and Ringen (1991).

6 Transfers of money do not correspond to transfers of ‘living standard’ units between

individuals. Glewwe (1991, p.213) used a simple numerical example with three

households. Decoster and Ooghe (2002, pp.3-4) also construct some illustrative

examples using three persons.

7 Shorrocks (1997) suggested that if concern is with equity, the use of adult equivalents

is recommended, whereas if concern is primarily with social welfare, individuals

should be the basic income unit. The disinterested economist is thus required to

report results using both approaches.

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Evaluations of Inequality and Social Welfare

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Social Welfare Functions and Inequality

Social welfare is regarded as an additive function of income per equivalent

adult, zi = yi/mi the living standard of each individual in the household. In

the case where the unit of analysis is the individual, that is where the

principle of anonymity (referred to alternatively in terms of compensation,

or Pareto indifference) is required, social welfare per individual is given by:

(4)

where V(z) is increasing and concave.

If the unit of analysis is the adult equivalent person, that is where the equity

principle (of transfers) applies, social welfare is:

(5)

Finally, if the household is treated as the unit of analysis, where each

household is assigned its income per equivalent adult, the welfare function

is simply:

(6)

Each of the three welfare measures is simply a weighted sum, over all N

households, of a function V of the income per equivalent adult z. The only

difference concerns the choice of the weights, which are respectively

, and 1/N. In practice, microsimulation models assign

a sample weight to each household so that appropriate population values

can be obtained. The weights are often those provided by the statistical

agency which collects the data, but they may also be modified for specific

purposes.8 The survey weights can easily be added to the above expressions:

for example if the survey weight for the ith household is wi, the weights for

treating the individual as the unit of analysis become

ws have been omitted from the above expressions for convenience only;

they are used in the numerical examples reported below.

. The

The type of additive welfare function discussed above is known to be

consistent with the Atkinson inequality measure, A, of income. In the

analysis below, concern is with net incomes, z. The Atkinson measure is

defined as the proportional difference between the equally-distributed-

equivalent (net) income, and the arithmetic mean income, . Hence, is

the net income per equivalent adult which, if received by every ‘unit of

analysis’, produces the same social welfare as the actual distribution, and:

(7)

8 See Creedy and Tuckwell (2004) for an example of survey reweighting for

microsimulation purposes.