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\begin{document}
\date{}
\author{John Muellbauer\thanks{%
Nuffield College, Oxford University. \texttt{%
john.muellbauer@nuffield.oxford.ac.uk}} \ and Justin van de Ven\thanks{%
National Institute of Economic and Social Research. \texttt{%
jvandeven@niesr.ac.uk}}}
\title{Estimating Equivalence Scales for Tax and Benefits Systems\thanks{%
We should like to thank Tony Atkinson, Frank Cowell, Peter Lambert, Sig
Prais and John Creedy for useful comments and suggestions. Any omissions or
errors remain our own.}}
\maketitle
\baselineskip20pt
\section{Introduction}
When comparing, say, the welfare derived from income by a family that is
comprised of two adults and three children to that of a single adult, it is
necessary to take into consideration the relative needs of the respective
households. The most common means by which applied studies in economics
currently relate the needs of heterogeneous income units is through the use
of equivalence scales. Despite a considerable research effort, however,
almost every aspect of equivalence scale specification remains
controversial. What characteristics should equivalence scales take into
account? Should the scales apply an additive or multiplicative adjustment to
income? Is the assumption of base independence valid?\footnote{%
Base independence (Lewbel, 1989, and Blundell and Lewbel, 1991) requires the
equivalence scale to be unaffected by the level of utility (or income). This
requirement is referred to as `equivalence scale exactness' by Blackorby and
Donaldson (1993).} How should a reference unit be selected? Is it reasonable
to assume that there is no inequality within an income unit? What criteria
are most sensible for selecting a functional form? And, arguably most
important, do the cardinal relations implied by equivalence scales permit
income units to be compared in terms of underlying welfare? All of these
questions remain largely unresolved.
This paper is concerned with estimating the relativities that are implicit
in tax and benefits policy. Using observed tax and benefits payments to
estimate equivalence scales may mitigate some of the criticisms to which
alternative scale specification criteria have been subject. For example,
most econometric estimates of equivalence scales used for distributional
analysis are based on consumer demand behaviour. Pollak and Wales (1979, p.
216) have notably criticised this methodology on the basis that
\textquotedblleft the equivalence scales required for welfare comparisons
are logically distinct from those which arise in demand
analysis\textquotedblright .\footnote{%
Deaton and Muellbauer (1980, chapters 7, 8 and 9) provide a comprehensive
discussion of the theoretical underpinnings of the demand based approach for
estimating equivalence scales. Muellbauer (1975) pointed out the
difficulties of making welfare comparisons, given taste differences of the
type used in equivalence scale models.} The central difficulty is that
demand analysis fails to provide a basis for making cardinal comparisons of
welfare between households, and so equivalence scales that are estimated
from expenditure data necessarily depend upon exogenously imposed value
judgements. In contrast, part of the intuitive appeal of equivalence scales
based on a country's transfer system is the perception that such
relativities embody a social consensus; that the tax and benefits system,
being an observable instrument of government, can be used to infer the value
judgements made by government when acting in its role as administrative
agent for society.\footnote{%
See, for example, Atkinson \& Stiglitz (1980, pp. 404-405), and Foley
(1967). This argument in favour of equivalence scales based on tax and
benefits systems has been questioned by Coulter \textit{et al.} (1992, p.
100) who, referring to the social security reforms that were made in the UK
in April 1988, state that the \textquotedblleft controversy that accompanied
the social security changes gives little support to the idea that the
reforms represented a social consensus opinion.\textquotedblright\
Alternatively, Atkinson \& Stiglitz (1980, p. 9) warn that \textquotedblleft
Tax and expenditure policy may be designed more with a view to electoral
success, or the goals of an established bureaucracy, than to social welfare
maximisation.\textquotedblright}
Comparison of the relativities that underlie taxation with equivalence
scales based on the costs borne by heterogeneous income units could also
provide a useful means for evaluating the adequacy of associated provisions
made by transfer systems. Alternatively, the equivalence scales implicit in
transfer systems could be used to compare the provisions made through time
and/or between countries. Such equivalence scales could also play a role in
the tax design process itself. In view of the fact that the redistributive
systems of many countries are comprised of numerous different tax and
benefit schemes, it is difficult to ensure that the overall system does in
fact achieve the desired redistribution. Information about implicit scales
-- which may on reflection be found to differ from the values held by policy
makers -- can be useful in suggesting that certain features of the tax
system need adjusting.
The prevailing uncertainty regarding how an equivalence scale should be
specified has particular relevance when considering the redistributive
effects of taxation. This is because measures of progressivity and
horizontal inequity calculated from equivalised income are often observed to
depend upon the equivalence scale assumed. Hence, distributional analyses of
equivalent income are typically subject to the criticism that horizontal
inequity is `imposed from the outside'.\footnote{%
See referee comments cited by Lambert (2003, footnote 2).} The analysis
presented in this paper essentially reverses the prevailing methodology by
assuming that \textit{horizontal equity} is, on average, satisfied by a
transfer system. This assumption enables the relativities implicit in
transfer policy to be inferred from observed tax and benefits payments.
The reversal of methodology suggested here is analogous to the analysis
presented in a recent paper by Bourguignon and Spadaro (2002), which
considers the social preferences that are consistent with observed marginal
tax rates. The traditional approach adopted to evaluate a transfer system
using optimal tax theory involves assuming a `reasonable' social welfare
function, and then comparing the implied `optimal tax schedule' with the
schedule that is observed in practice.\footnote{%
This methodology is attributable to Mirrlees (1971, 1986). Recent literature
in this field includes Diamond (1998), Saez (1998), Salani\'{e} (1998),
d'Autume (1999), and Bourguignon and Spadaro (2000), cited by Bourguignon
and Spadaro (2002).} In contrast, Bourguignon and Spadaro (2002)
\textquotedblleft consider the effective marginal tax rates schedule that
corresponds to an actual redistribution system and...look for the social
welfare function according to which that schedule would be
optimal\textquotedblright\ (p. 2). The analysis advocated by Bourguignon and
Spadaro (2002) thus enables a practitioner to determine whether the social
welfare function implied by the observed marginal tax rate schedule is
consistent with \textit{a priori} expectations. Similarly, the framework of
analysis presented in this paper enables the relativities implicit in
transfer policy to be compared against the consistency and monotonicity
properties that could be expected of a `sensible' equivalence scale.%
\footnote{%
If the scales implied by transfer policy are found to be inconsistent with
\textit{a priori} expectations (are found, for example, to be decreasing in
household size), then this may reveal that existing transfer policy is
inconsistent with the concept of horizontal equity - which is itself an
interesting finding.} Whereas the approach suggested by Bourguignon and
Spadaro (2002) interprets observed marginal tax rates in terms of social
welfare, the approach presented here interprets the relativities implicit in
a transfer system in terms of the equivalence scale methodology.\footnote{%
The applied analysis presented by Bourguignon and Spadaro (2002) considers
two extremes with regard to the relationship assumed between heterogeneous
households. In the first instance household differences are ignored. In the
second, households are considered exclusively within homogenous demographic
subgroupings, from which qualitative comparisons are made.}
Very few studies have attempted to estimate the equivalence scales implicit
in tax policy, and those that do have focused on a subset of the transfer
system.\footnote{%
See, for example, the Royal Commission on the Distribution of Income and
Wealth (1978), which calculates equivalence scales based on the short-term
scale rates of supplementary benefits.} This apparent lack of interest can
be attributed to the perception that \textquotedblleft income taxes are not
typically coherent with equivalence scales\textquotedblright\ (Lambert,
1993, p. 364). In the absence of a generally agreed optimal solution%
\footnote{%
See, for example, the impossibility result of Moyes and Shorrocks (1998).},
it is unsurprising that different countries have adopted transfer systems
that take a range of different forms. The system that is perhaps the most
transparently consistent with the equivalence scale methodology is the
\textit{quotient familial} applied in France, as described by Atkinson
\textit{et al.} (1988). Others, such as the system of exemptions and credits
applied in the UK, bear less resemblance to the equivalence scale framework.%
\footnote{%
See Lambert and Yitzhaki (1997)} The perception that many tax and benefits
systems are inconsistent with the equivalence scale methodology is
strengthened by the observation that the individual tax and benefit schemes
from which transfer systems are comprised often imply different
relativities. How should an analyst decide which scheme, and consequently
which relativities, to use? Furthermore, given that many benefits provide
safety-net incomes, how can an analyst be sure that the implied equivalence
scales are relevant for the entire distribution of income, rather than just
the bottom tail?
An equivalence scale that embodies only part of a tax and benefits system is
evidently of limited interest.\footnote{%
See Ebert and Lambert (1999).} Section \ref{theory} introduces a formal
framework that is consistent with the equivalence scale methodology, and
demonstrates how the framework relates to the concept of horizontal equity.
It is frequently assumed that the adjustments made by fiscal policy for
household heterogeneity describe a range of equivalence scales, rather than
a single set of relativities.\footnote{%
See, for example, Coulter \textit{et al.} (1992).} In Section \ref{general}
it is shown that the suggested framework, based on a single equivalence
scale, is sufficiently flexible to reflect \textit{any} transfer system. The
generality of the model is a product of relaxing all \textit{a priori }%
restrictions, including the assumption of base independence, upon the
equivalence scale. In Section \ref{est}, a non-parametric procedure for
estimating equivalence scales implicit in tax and benefits systems is
introduced. In Section \ref{estn} estimates are reported for the
relativities implicit in the Australian transfer system.\footnote{%
The Australian transfer system is considered here in response to the prior
expertise of the authors. Estimating the equivalence scales implicit in the
UK transfer system is planned to be the subject of a subsequent study.} The
equivalence scale estimates derived from tax and benefits data are also
compared with a range of alternative scales, including those used by
government agencies in Australia, the scale proposed by the OECD, and scales
estimated from household expenditure data. It is found that the scales
implicit in the Australian transfer system compare well with the official
and demand based scales, and the relationship with income is both
interesting and intuitive. Conclusions from the study are summarised in
Section \ref{conc}.
\section{Equivalence Scales and Taxation\label{theory}}
Assume that the redistributive objectives of tax policy designers (the
government) are framed in terms of income per equivalent adult. Values in
adult equivalent terms, as measured by the government, are denoted by a *
superscript. Let $a_{i}^{\ast }$ denote the number of equivalent adults in
tax unit $i$. Each $a_{i}^{\ast }$ can be regarded as a function of a set of
$m$ observable variables (which may include income) of the tax unit $%
v_{i}=\left( v_{1i},...,v_{mi}\right) $, so that:\footnote{%
Seneca and Taussig (1971) recognised the importance of allowing the scales
implicit in transfer policy to be income dependent. See, also, the empirical
results presented by Banks and Brewer (2002).}
\begin{equation}
a_{i}^{\ast }=a^{\ast }\left( v_{i}\right)
\end{equation}
Let $x_{i}$ and $y_{i}$ denote the pre-tax and post-tax income of tax unit $%
i $.\footnote{%
Throughout this paper, pre-tax and post-tax income refer, respectively, to
income gross and net of associated tax and benefit payments.} Hence, $%
x_{i}^{\ast }=x_{i}/a_{i}^{\ast }$ and $y_{i}^{\ast }=y_{i}/a_{i}^{\ast }$
are respectively pre-tax and post-tax equivalent income.
Given the distribution of $x_{i}^{\ast }$ for a population, the government
is considered to impose a tax structure that is capable (subject to a budget
constraint) of achieving its distributional aims. This involves a tax
function, $T^{\ast }\left( x_{i}^{\ast }\right) $, so that the net adult
equivalent income of unit $i$ is:
\begin{equation}
y_{i}^{\ast }=x_{i}^{\ast }-T^{\ast }\left( x_{i}^{\ast }\right)
\label{etax}
\end{equation}
In equation (\ref{etax}), the tax function depends only upon pre-tax
equivalent income, which ensures that the requirement of horizontal equity
is satisfied.\footnote{%
The \textquotedblleft principle of equity, or \textit{horizontal equity}, is
fundamental to the ability-to-pay approach, which requires equal taxation of
people with equal ability and unequal taxation of people with unequal
ability\textquotedblright\ (Musgrave, 1959, p. 160). This requirement is
discussed further in the following subsection.} This assumption can be
relaxed by adding an individual specific term, $\varepsilon _{i}^{\prime }$,
to the right hand side of equation (\ref{etax}).
Assuming that the same equivalence scale is applied to both pre-tax and
post-tax income, equation (\ref{etax}) translates to:
\begin{equation}
y_{i}=x_{i}-a_{i}^{\ast }T^{\ast }\left( \frac{x_{i}}{a_{i}^{\ast }}\right)
\label{etax2}
\end{equation}
This specification was alluded to by Vickrey (1947, pp. 295-296) who wrote;
\textquotedblleft A more thoroughgoing and equitable procedure [than
exemptions and credits] would be to set up some factor indicative of the
needs of the entire family, divide the total income by this factor, compute
a per capita tax on this `per capita income', and multiply the tax so
computed by the family size factor to obtain the total tax for the
family\textquotedblright .\footnote{%
Quoted by Lambert and Yitzhaki (1997, p. 346). See also Pyatt (1990), and
Ebert (1997).}
\subsection{Family size, horizontal equity, and taxation\label{opt}}
Horizontal equity is the command that equals be treated equally by a
transfer system. Although there is widespread support for this concept,
there is an ongoing debate regarding who should be defined as equals and
what constitutes `equal tax treatment'.\footnote{%
See Lambert (2003) and Ebert and Lambert (2002) for alternative
interpretations of horizontal equity and their relation to equivalence
scales.} Consequently, it is important to provide a clear definition of what
horizontal equity means here, and how it may be used to motivate the formal
framework considered by this paper.
Horizontal equity is defined here as the requirement that: \textquotedblleft
\textit{If two individuals would be equally well off (have the same utility)
in the absence of taxation, they should also be equally well off if there is
a tax}\textquotedblright\ (Feldstein, 1976, p. 83, emphasis in the
original). Consider, for example, a population that is comprised of one and
two adult households (singles and couples), where households are further
differentiated by their respective ability levels (wage rates). Household
size is the only non-income characteristic that is relevant for tax
purposes, and is defined exogenously - hence behavioural effects are not
considered in this dimension.\footnote{%
See, for example, Nerlove\textbf{\ }\textit{et al.} (1984), and Barro and
Becker (1989) for models of endogenous fertility.} Define $s$ subscripts for
single adult households and $c$ subscripts for couples. Pre-tax and post-tax
income are defined by $x$ and $y$ respectively (as in the previous
subsection), and it is assumed that pre-tax income is equal to earnings
derived through labour, $l$, via the (unobserved) wage rate, $w$, $x=wl$.
The variable, $l$, is defined as the proportion of a household's total time
spent on labour, which takes a value between zero and one for both singles
and couples. The government is considered to impose a tax schedule, $%
T_{k}\left( x\right) $, which may differ between couples and singles (hence
the subscript $k$), such that household pre-tax and post-tax income are
related by:
\begin{equation}
y=wl-T_{k}\left( wl\right) \label{bchh}
\end{equation}
The household utility function, $u\left( y,l\right) $, is assumed to be
strictly quasi-concave, (strictly) increasing in $y$, (strictly) decreasing
in $l$, and continuously differentiable. Furthermore, it is assumed that $%
u\rightarrow -\infty $ as $l\rightarrow 1$ from below. An assumption of
equal sharing is made, such that every member of a household enjoys the same
level of utility, indicated by the function $u\left( .\right) $.\footnote{%
This assumption is made in view of the scarcity of data regarding
intra-household distributions. See, for example, Kaplow (1996) for an
explicit consideration of alternatives to the assumption of equal sharing.}
An equivalence scale, $a_{c}=a\left( u_{c}\right) $, relates the utility of
each member of a couple to a single adult household by:\footnote{%
Equation (\ref{etax2}) implies that $a_{c}$ can be stated as a function of
(observable) income, whereas it is fundamentally a function of
(unobservable) utility. For the government to be able to impose a
horizontally equitable tax, it must be able to infer a household's ability
by observing its type and income. Hence, it is necessary for there to exist
a one-to-one mapping of pre-tax income to ability for any household type.
That is, we require $\frac{\partial l}{\partial w}>-\frac{l}{w}$ for all
conceivable abilities (wage rates).}
\begin{equation}
u_{c}\left( y_{c},l_{c}\right) =u_{s}\left( \frac{y_{c}}{a_{c}},l_{c}\right)
\label{equivs}
\end{equation}
Equation (\ref{equivs}) reflects the fact that a couple are likely to
convert post-tax income into utility differently to single adults because of
the need to share. Labour, in contrast, is not similarly affected. No
attempt is made to estimate the utility functions that are considered here -
rather the functions $u\left( .\right) $ can be interpreted as the utility
that the government attributes to households when formulating transfer
policy.\footnote{%
Rosen (1978) considers a similar framework to the one defined here, but
takes a different view by estimating utility functions from household income
and labour data to consider the issue of horizontal equity.}
Households are assumed to select their labour supply (given perfectly
competitive markets, no profits, and a clearing labour market) to maximise
their utility, subject to the budget constraint defined by equation (\ref%
{bchh}).\footnote{%
The model considered here assumes that individuals are free to choose how
much labour they supply. Involuntary unemployment could be included in the
model, by recognising it as an individual characteristic (in much the same
way as health or invalidity might be included as a characteristic in the
equivalence scale specification).} For single adult households the solution
is obtained by maximising the Lagrangian:
\begin{equation}
\tciLaplace =u_{s}\left( y,l\right) +\lambda \left( wl-T_{s}\left( wl\right)
-y\right) \label{ls}
\end{equation}
Defining equivalent post-tax income for couples by $y^{\ast }=y/a_{c}$, and
the equivalent wage rate by $w^{\ast }=w/a_{c}$, the decision for couples
can be identified by maximising:
\begin{equation}
\tciLaplace =u_{s}\left( y^{\ast },l\right) +\lambda \left( w^{\ast }l-\frac{%
T_{c}\left( a_{c}w^{\ast }l\right) }{a_{c}}-y^{\ast }\right) \label{lc}
\end{equation}
The first order conditions of equations (\ref{ls}) and (\ref{lc}) indicate
that there is a critical wage rate for singles $(w_{s0})$, and for couples $%
\left( w_{c0}^{\ast }\right) $, such that a wage rate in excess of the
critical rate is sufficient to induce some labour supply, $l>0$. To simplify
the analysis, the wage rate of any household is assumed to be sufficient to
reach an interior solution throughout the remainder of this section.
Substituting $T_{s}\left( wl\right) =T_{c}\left( wl\right) =0$ into
equations (\ref{ls}) and (\ref{lc}) reveals that, when single adult and two
adult households enjoy the same equivalised ability ($w^{\ast }$ in equation
(\ref{lc}) $=w$ in equation (\ref{ls})), they are subject to the same
maximising problem in the absence of taxation and hence, given the
assumptions made, will enjoy the same utility. We therefore define
households as pre-tax equals if they possess the same equivalised ability.%
\footnote{%
This finding implies that, if the equivalence scale $a_{c}$ embodies
economies of scale, then couples will require a lower (average) ability than
single adults to enjoy the same utility. Consider, for example, the case
when $a_{c}=1.6$, and each member of a couple supplies the same number of
labour hours and earns the same hourly wage rate. Given these conditions, if
each member of a couple are to enjoy the same utility as a single with
ability $w_{s}$, then $w_{c}=a_{c}w_{s}=1.6w_{s}$. Since $w_{c}$ is shared
equally between the members of the couple, each must consequently earn 80
per cent of the single's wage.} The specification of pre-tax equals as
households with the same equivalised ability (and the same pre-tax
equivalent income\footnote{%
Pre-tax equals will share the same pre-tax equivalent income if there exists
a unique solution to the utility maximising problem, which (given the
assumptions made regarding $u_{s}\left( .\right) $) is ensured if $%
T_{s}\left( x\right) =T_{c}\left( x\right) =0$.}) can be contrasted with the
framework of Balcer and Sadka (1986), and Banks and Brewer (2002), who
define pre-tax equals as households with the same \textit{pre-tax} \textit{%
unequivalised} incomes, and to Seneca and Taussig (1971) who define pre-tax
equals as households with the same \textit{post-tax unequivalised }incomes.
A horizontally equitable tax in the current context must consequently imply
that the same level of post-tax utility will be enjoyed by households with
the same equivalised ability. Comparison of equation (\ref{ls}) with (\ref%
{lc}) reveals that single adult and couple households with the same
equivalised ability will enjoy the same utility if their respective tax
functions are related by:
\begin{equation}
T_{c}\left( x\right) =a_{c}T_{s}\left( \frac{x}{a_{c}}\right) \label{hit}
\end{equation}
which restates equation (\ref{etax2}).\footnote{%
No attempt is made to generalise this discussion, which remains an issue for
subsequent research.}
Given the strong support that horizontal equity has received from a diverse
range of views regarding redistributive justice\footnote{%
See, for example, Musgrave (1990) for a brief review.}, it seems reasonable
to suppose that it does play an important role in the design of transfer
policy. This is the premise upon which the remainder of the current paper is
based. Specifically, we assume that the government makes value judgements
regarding the relative needs of different households, and that transfer
policy is designed with reference to a population whose preferences are
consistent with the value judgements made. It is important to note, however,
that the value judgements (implicitly) made by a government may (and
arguably \textit{will}) bear little resemblance to the preferences of the
actual population. Furthermore, policy objectives that run orthogonal to
horizontal equity are likely to complicate the relativities implicit in tax
and benefits systems.\footnote{%
For example, unemployment benefits may be designed to encourage labour
market participation, or family benefits specified to affect fertility rates
(compare the different fertility objectives of state policy in France and
China).} Hence, although equivalence scales that reflect the value
judgements made by government are an important descriptive tool, they have
limited appeal as a means of analysing social welfare. Interpretation of the
welfare implications of equivalence scales implicit in transfer policy will
necessarily depend upon how such scales relate to \textit{a priori}
expectations regarding, for example, monotonicity and uniformity.
\section{Generality of the equivalence scale framework\label{general}}
Consider a transfer system that is comprised of numerous tax and benefit
schemes, where each scheme depends upon household characteristics (such as
the number and age of household members, their health status, pre-tax
income, and so on), $v,$ and may embody a different set of relativities to
those of other schemes. It is clear that there will exist a tax function, $%
T\left( x,v\right) $, which relates pre-tax and benefit and post-tax and
benefit income, such that:
\begin{equation}
y_{i}=x_{i}-T\left( x_{i},v_{i}\right) \label{plyr}
\end{equation}
for each household $i$.\footnote{%
As in the case of equation (\ref{etax}), equation (\ref{plyr}) omits any
horizontal inequity, consistent with the official specifications commonly
implied by tax and benefit policy. Horizontal inequity can be included by
adding an error term to the right hand side of equation (\ref{plyr}).} The
generality of equation (\ref{plyr}) is sufficient to capture a broad range
of redistributive systems; from the quotient familial applied in France, to
the system of exemptions and credits applied in the UK. How might the system
described by equation (\ref{plyr}) be related to the equivalence scale
framework, using a single set of relativities denoted by $a^{\ast }$? While
the equivalence scale framework is, in principle, quite general, a few
restrictions, plausible in themselves, can result in significant
restrictions on the tax functions. \ We first illustrate this proposition,
and then consider some implications for the properties of the equivalence
scale function $\ a^{\ast }$ of assumptions about the tax function $T\left(
x,v\right) $ ,\ such as being progressive and diminishing in household size,
given x.
\bigskip
To illustrate restrictions on the tax functions, suppose that the equivalent
tax function $T^{\ast }\left( x^{\ast }\right) $ takes the quadratic form:
\begin{equation}
T^{\ast }\left( x^{\ast }\right) =\beta _{0}+\beta _{1}x^{\ast }+\beta
_{2}x^{\ast 2} \label{hmm}
\end{equation}
If $a^{\ast }$ is a (potentially base dependent) equivalence scale, then
from equation (\ref{etax2}) we require:
\begin{eqnarray}
a^{\ast }T^{\ast }\left( x^{\ast }\right) &=&T\left( x,v\right) \notag \\
&\therefore &\beta _{0}a^{\ast 2}+\beta _{1}xa^{\ast }+\beta
_{2}x^{2}=a^{\ast }T\left( x,v\right) \notag \\
&\therefore &a^{\ast }=\frac{T\left( x,v\right) -\beta _{1}x\pm \sqrt{\left(
\beta _{1}x-T\left( x,v\right) \right) ^{2}-4\beta _{0}\beta _{2}x^{2}}}{%
2\beta _{0}} \label{quad}
\end{eqnarray}
Supposing that $T\left( 0,v\right) <0$ for all $v$ (consistent with the view
that the tax system provides a net benefit to households with zero pre-tax
income), then the negative square root of equation (\ref{quad}) must be
selected to ensure that $a^{\ast }\neq 0$ when $x=0$. When pre-tax income is
equal to zero, equation (\ref{quad}) implies that:
\begin{equation}
a^{\ast }=\frac{T\left( 0,v\right) }{\beta _{0}} \label{b0}
\end{equation}
Hence, if the government provides a transfer benefit for all tax units with
zero pre-tax income, such that $T\left( 0,v\right) <0$ for all $v$, then $%
\beta _{0}$ must be less than zero to obtain $a^{\ast }\left( 0,v\right) $
greater than zero for all $v$. If $\beta _{0}<0$, then we require:
\begin{equation}
T\left( x,v\right) -\beta _{1}x-\sqrt{\left( \beta _{1}x-T\left( x,v\right)
\right) ^{2}-4\beta _{0}\beta _{2}x^{2}}<0
\end{equation}
to observe $a^{\ast }\left( x,v\right) >0$ for any $x$ and $v$. Therefore:
\begin{equation}
T\left( x,v\right) -\beta _{1}x<\sqrt{\left( T\left( x,v\right) -\beta
_{1}x\right) ^{2}-4\beta _{0}\beta _{2}x^{2}} \label{b1}
\end{equation}
It is clear that
\begin{equation*}
T\left( x,v\right) -\beta _{1}x\leq \left| T\left( x,v\right) -\beta
_{1}x\right|
\end{equation*}
and that
\begin{equation*}
\left| T\left( x,v\right) -\beta _{1}x\right| <\sqrt{\left( T\left(
x,v\right) -\beta _{1}x\right) ^{2}-4\beta _{0}\beta _{2}x^{2}}
\end{equation*}
for any $x>0$ when $\beta _{0}<0$ and $\beta _{2}>0$. Hence, assuming that $%
T\left( 0,v\right) <0$ for all $v$, we require $\beta _{0}<0$ and $\beta
_{2}>0$ for $a^{\ast }\left( x,v\right) $ to be positive for any tax unit -
the equivalent tax function must be strictly progressive in the sense that
the average tax rate must be an increasing function of pre-tax income. This
restriction on the tax function is the outcome of the three assumptions: a
quadratic tax function, $T\left( 0,v\right) <0$ for all $v,$and $a^{\ast
}\left( x,v\right) $ to be positive for any tax unit.
The restriction, $T\left( 0,v\right) <0$, considered above is consistent
with most practical cases, since transfer systems generally provide a net
benefit to households with no pre-tax income. There are, however, exceptions
to this rule, particularly for households with large wealth stocks. If $%
T\left( 0,v_{i}\right) <0$ for some $v_{i}$ and $T\left( 0,v_{j}\right)
\nless 0$ for some $v_{j}$, then equation (\ref{b0}) indicates that there
are no equivalent tax parameters which will ensure that $a^{\ast }\left(
0,v\right) >0$ for all conceivable specifications of the tax unit. In the
case of wealthy households, however, the equivalence scales obtained by the
above framework do retain a sensible interpretation. When, for example, the
transfer system provides no net benefit to a tax unit, $v_{j}$, with zero
pre-tax income, $T\left( 0,v_{j}\right) =0$, then it is clear from equation (%
\ref{b0}) that $a^{\ast }\left( 0,v_{j}\right) =0$ for any $\beta _{0}\neq 0$%
. This is consistent with the interpretation that the tax unit $v_{j}$ has
no tax relevant needs when it has no pre-tax income, a value judgement that
is possibly justified for households with large stocks of wealth.
Furthermore, it is arguably preferable to consider an income concept that
includes all pecuniary accruals of wealth measured over a long period of
time for households with large wealth stocks, in which case it is less
likely that $T\left( 0,v_{j}\right) \nless 0$ will be observed for some $%
v_{j}$.
\subsection{Properties of the implicit equivalence scale}
It is useful to consider how the properties of the equivalence scale $%
a^{\ast }$ defined by equation (\ref{etax2}) relate to properties of an
observed tax and benefits system. Assume that the observed tax burden of
household $i$ depends upon pre-tax income, $x_{i}$, and household
characteristics including the number, age, and health status of household
members, $v_{i}$, such that post-tax income is given by:
\begin{equation}
y_{i}=x_{i}-T\left( x_{i},v_{i}\right) \label{sst}
\end{equation}
For equations (\ref{etax2}) and (\ref{sst}) to be equivalent over the
relevant domain of $x$ and $v$, we need:
\begin{equation}
a^{\ast }T^{\ast }\left( x^{\ast }\right) \equiv T\left( x,v\right)
\label{a}
\end{equation}
and:
\begin{equation}
T^{\ast }\left( x\right) \equiv T\left( x,v_{0}\right) \label{b}
\end{equation}
where $a^{\ast }=1$ for the reference household type with characteristics $%
v_{0}$. Equations (\ref{a}) and (\ref{b}) can be combined into (\ref{c}):
\begin{equation}
a^{\ast }T\left( \frac{x}{a^{\ast }},v_{0}\right) \equiv T\left( x,v\right)
\label{c}
\end{equation}
Equation (\ref{c}) defines an implicit function for $a^{\ast }$ in terms of
pre-tax income, household characteristics and reference characteristics:
\begin{equation}
a^{\ast }=a^{\ast }\left( x,v,v_{0}\right) \label{d}
\end{equation}
To analyse the properties of the function $a^{\ast }$, consider the
following assumptions about $T\left( .\right) $. Assume that the tax
function $T\left( .\right) $ is differentiable in pre-tax income $x$ and in
size $s$, a component of $v$.\footnote{%
Differentiability with respect to size $s$, which is an integer, is easily
relaxed by considering unit changes $\triangle s=1$ in what follows.}
Furthermore, we assume for the moment that $T\left( .\right) $ is
progressive in $x$ $\left( \partial \log T/\partial \log x>1\right) $ given $%
v$\footnote{%
Empirically, this is often untrue. See, for example, Brewer and Clark (2002)
who show that marginal effective tax rates are frequently higher at lower
incomes in the UK.}, and decreasing in household size, given $x$. Taking
logs of equation (\ref{c}) and differentiating with respect to $s$ yields:
\begin{equation}
\frac{\partial \log a^{\ast }}{\partial s}=\frac{t_{s}}{\left( 1-t_{x}^{\ast
}\right) }
\end{equation}
where $t_{s}=\partial \log T\left( x,v\right) /\partial s$ and $t_{x}^{\ast
}=\partial \log T\left( x/a^{\ast },v_{0}\right) /\partial \log \left(
x/a^{\ast }\right) $. The assumption of progressivity implies that $%
t_{x}^{\ast }>1$. If, as we have assumed, $t_{s}<0$ where $s$ is a household
size characteristic, then $a^{\ast }$ will be increasing in $s$, as we
should expect. If, in contrast, $T\left( .\right) $ is locally regressive,
then the opposite conclusion may apply to specific pre-tax income domains.
The relationship between the equivalence scale $a^{\ast }$ and pre-tax
income $x$ is now investigated. Differentiation in logs of equation (\ref{c}%
) with respect to $x$ implies:
\begin{equation}
\frac{\partial \log a^{\ast }}{\partial \log x}=\frac{t_{x}-t_{x}^{\ast }}{%
1-t_{x}^{\ast }}
\end{equation}
where $t_{x}=\partial \log T\left( x,v\right) /\partial \log x$. Since $%
1-t_{x}^{\ast }<0$, $a^{\ast }$ will be diminishing in $x$ if $%
t_{x}>t_{x}^{\ast }$ and increasing in $x$ if $t_{x}0$. For $a^{\ast }>1$, it is possible to show that $%
t_{x}>t_{x}^{\ast }$ in this case, so that $a^{\ast }$ is decreasing in $x$.
An iso-elastic progressive system with exemptions has the same property.
Here:
\begin{equation}
T\left( x,v\right) =\alpha x^{1+\beta }-c\left( v\right)
\end{equation}
where $\beta >0$. Hence the relationship between $a^{\ast }$ and $x$ is
dependent upon the specification of the tax function. Furthermore, the
examples considered here suggest that, where the tax function describes a
progressive system with (income independent) exemptions, the equivalence
scale will be decreasing in pre-tax income.
\section{Estimating the Equivalence Scales Implicit in Transfer Systems\label%
{est}}
Given that the equivalence scales discussed in this paper are implicit, and
hence unobservable, non-parametric estimation methods that impose limited
\textit{a priori} restrictions on the specification of the equivalence scale
are particularly useful. The remainder of this paper is consequently
concerned with non-parametric estimates. Two alternative procedures that use
standard econometric regression techniques to estimate assumed tax and
equivalence scale functions are described in Appendix \ref{app-regn}.%
\footnote{%
Regression techniques are complicated by the highly non-linear nature of the
model described by equation (\ref{etax2}). See Muellbauer and van de Ven
(2003) for a detailed discussion of econometric estimation issues.}
\subsection{Non-Parametric Estimation\label{iter}}
A number of alternative non-parametric procedures can be devised for
estimating the equivalence scales implicit in tax and benefits policy. The
approach that is described here is one that we found useful when deriving
the estimates that are reported in the following section. The approach
involves three principal stages:
\begin{enumerate}
\item \textit{Population Division}. The sample population used to estimate
the equivalence scales is divided into subgroups, within which tax units are
considered to be homogenous in all tax relevant respects other than pre-tax
and post-tax income.
\item \textit{Tax Function Estimation}. Standard non-parametric methods are
used to estimate the tax functions of each individual subgroup identified in
(1) above.
\item \textit{Equivalence Scale Inference}. The tax functions estimated in
(2) are used to obtain functions of the average tax rate versus pre-tax
income, $AVt_{i}\left( x\right) $, for each of the subgroups, $i$,
identified in (1). Let the subscript $i=0$ denote the reference unit used to
define the equivalence scale. The equivalence scale of subgroup $i$ with
pre-tax income $x_{i}$ is then obtained by:
\begin{equation}
a\left( x_{i}\right) =\frac{x_{i}}{x_{0}},\text{ where }AVt_{i}\left(
x_{i}\right) =AVt_{0}\left( x_{0}\right)
\end{equation}
\end{enumerate}
This last stage of the estimation procedure warrants some comment. If it is
assumed that the same equivalence scale is applicable for pre-tax and
post-tax income then:
\begin{equation}
\frac{y^{\ast }}{x^{\ast }}=\frac{\left( y/a\right) }{\left( x/a\right) }=%
\frac{y}{x} \label{chk}
\end{equation}
Equation (\ref{chk}) indicates that the equivalence scale does not affect
the ratio of pre-tax to post-tax income, or equivalently the average tax
rate, $AVt\left( x\right) =T\left( x\right) /x=\left( x-y\right) /x$. The
equivalence scale discussed here consequently applies a proportional
adjustment to pre-tax and to post-tax income, so that the tax function of
any household is mapped onto the tax function of the reference household by
way of the average tax rate. This is described graphically in Figure \ref%
{avtf}.
\FRAME{ftbpFU}{319.75pt}{246.8125pt}{0pt}{\Qcb{Relating the Average Tax
Rates of Alternative Population Subgroups}}{\Qlb{avtf}}{Figure }{\special%
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"0";croptop "1";cropright "1";cropbottom "0";tempfilename
'HT11810D.wmf';tempfile-properties "XPR";}}Figure \ref{avtf} displays tax
functions for two household types, $T_{0}$ (singles) and $T_{1}$ (couples).
Consider an equivalence scale that specifies the tax adjustment made for
couples relative to the associated adjustment that is made for singles. When
a couple earn pre-tax income $x_{1}$, their average tax rate is given by $%
t_{1}/x_{1}$, and the figure indicates that the equivalence scale is equal
to $a\left( x_{1}\right) =x_{1}/x_{0}=t_{1}/t_{0},$ $t_{1}/x_{1}=t_{0}/x_{0}$%
.
It is evident from the above discussion that, if the tax function of the
reference unit is not strictly progressive in the sense that the average tax
rate is strictly increasing with pre-tax income (or, more accurately, that
the average tax rate does not vary strictly monotonically with pre-tax
income), then the equivalence scale implicit in tax policy may not be unique.%
\footnote{%
This point is first alluded to by the analysis that is reported in Section %
\ref{general}. Indeed, it is possible to devise examples where the
equivalence scale will not be defined at all.} This observation serves to
highlight the care that must be exercised when selecting a reference unit,
and when calculating the equivalence scales that are implicit in tax and
benefits policy. Furthermore, the above discussion does not address the
issue of calculating associated standard errors, which remains an issue for
further research.\footnote{%
Given that the non-parametric estimates discussed here are sample
statistics, it is presumably possible to use resampling methods to obtain
bootstrap estimates for associated standard errors.}
\section{Equivalence Scale Estimates for Australia\label{estn}}
This section presents equivalence scale estimates calculated using simulated
and survey data for Australia. The section begins with a description of the
data used, before discussing the non-parametric equivalence scale estimates,
and then the parametric estimates obtained. The section concludes by
comparing the tax implicit equivalence scales with scales that have received
some official recognition, and with scales estimated from household
expenditure data.
\subsection{Data\label{data}}
\subsubsection*{Simulated Data}
A simulation model is used to generate a synthetic population of households
that are differentiated by their number of adults, number of children, and
their pre-tax and post-tax incomes. The synthetic population is comprised of
single adults with up to two dependant children and couples with up to three
dependant children (7 different demographic combinations), for 51 measures
of pre-tax annual income ranging between \$0 and \$100,000. Given the number
of adults, the number of children and the pre-tax income of a household, the
simulation model generates measures of post-tax income based upon the rates
and thresholds of the following 10 tax and benefit schemes, as they were
defined in 1997/98:\footnote{%
Post-tax income is generated assuming that; all couples are married; all
household income is earned by one individual; no household is eligible for
an `over 60s bonus' for the basic NS rate; there are no part year
recipients; there are no wealth tests; if there are children in a household,
then at least one child is under 5 years of age (for the Medicare Levy).}
\begin{itemize}
\item \textit{Newstart }$\left( NS\right) $: An unemployment benefit payable
to individuals who are available for, capable of, and actively seeking work
between the ages of 18 and 65.
\item \textit{Family Payment }$\left( FP\right) $: A benefit that is
structured to support low income families with dependant children.
\item \textit{Parenting Payment }$\left( PP\right) $: A benefit paid instead
of \textit{Newstart} to one member of a married couple with at least one
dependant child. It has a more generous income test than \textit{Newstart}
and does not require the recipient to be seeking employment.
\item \textit{Sole Parent Payment }$\left( SPP\right) $: The sole parent
equivalent of the \textit{Parenting Payment.}
\item \textit{Income Tax}: Income taxation is levied on individual rather
than joint incomes, and takes a standard multi-step form with $5$
progressive marginal rates.
\item \textit{Medicare Levy} $\left( ML\right) $: Charged in addition to
\textit{Income Tax}, to fund the costs of a universal health care system.
\item \textit{Family Tax Initiative }$\left( FTI\right) $: A scheme designed
to support households with dependant children (in addition to the $FP$). It
is comprised of two parts; Family Tax Assistance for households that earn a
sufficiently high taxable income, and Family Tax Payment for low income
families.
\item \textit{Dependent Spouse Rebate }$\left( DSR\right) $: A tax rebate
that can be claimed by individuals who have a spouse who earns a
sufficiently low income.
\item \textit{Sole Parent Rebate }$\left( SPR\right) $: A tax rebate that
can be claimed by single parents with dependant children.
\item \textit{Low Income Earner Rebate }$\left( LIR\right) $: A tax rebate
that can be claimed by individuals who earn a sufficiently low income.
\end{itemize}
The procedures that are used to simulate these transfer schemes are based
upon a study by Creedy and van de Ven (1999). In 1997/98, the transfer
schemes listed above accounted for 73.0 per cent of all social security
expenditure excluding benefits for the elderly, and 82.0 per cent of
individual taxation liability.
\subsubsection*{Survey Data}
The survey data are derived from the Confidentialised Unit Record File
(CURF) of the 1997-1998 Survey of Income and Housing Costs (SIHC) for
Australia. This survey provides income and demographic data for individuals
who are aggregated into households.\footnote{%
Refer to the Australian Bureau of Statistics for detailed information
regarding the SIHC.} The SIHC records annual household income measured in
1997 Australian dollars, and attempts to account for all direct pecuniary
flows. Importantly, for the analysis that is undertaken here, no attempt is
made to impute indirect taxes or transfer benefits. The results presented
here must consequently be interpreted bearing this limitation in mind.
\subsection{Non-Parametric Estimates\label{npmmm}}
Figures \ref{anps} to \ref{anpcc} display non-parametric estimates of the
equivalence scales implicit in the Australian tax and benefits system. Also
included in Figures \ref{anps} to \ref{anpcc} are equivalence scale
estimates based on household expenditure data, the scales recommended by the
OECD, and Henderson equivalence scales.\footnote{%
See Appendix \ref{app-scls} for a detailed description of the demand based,
OECD and Henderson scales referred to here.} The alternatives to the tax
implicit equivalence scales considered, are the scales that are most
commonly applied in the existing literature, and consequently allow a
comparison with the status quo. The comparative analysis undertaken here is
preliminary and incomplete insofar as it omits reference to associated
standard errors. A more thorough interpretative analysis remains an issue
for further research.
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\caption{Australian Non-Parametric Equivalence Scale Estimates - Single
Adults with Children\label{anps}}$%
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\caption{Australian Non-Parametric Equivalence Scale Estimates - Couples
without Children\label{anpc}}$%
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\caption{Australian Non-Parametric Equivalence Scale Estimates - Couples
with Children\label{anpcc}}$%
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Four data series are displayed in each of the panels of Figures \ref{anps}
to \ref{anpcc}, three of which are derived from simulated data, and one from
survey data. The series denoted `smoothed - bandwidth = 0.2' describes the
non-parametric estimates of equivalence scales derived from simulated data,
using tax functions estimated by the `lowess' procedure in STATA (with a
bandwidth equal to 0.2). Similarly, `smoothed - bandwidth = 0.8' refers to
non-parametric estimates obtained from simulated data using lowess with a
bandwidth of 0.8.\footnote{%
In STATA, lowess calls a non-parametric regression procedure that obtains a
separate estimate, $\hat{y}_{i}$, for each observation in a data set, $%
\left( x_{i},y_{i}\right) $, by regressing $y$ on $x$ for a limited
proportion of the data set (defined by the bandwidth). The regression
weights the data such that points further from the central point $\left(
x_{i},y_{i}\right) $ receive less weight. See Cleveland (1993) for further
details.} Equivalence scale estimates obtained without smoothing the
simulated tax functions are denoted as `no smoothing' in each of the
figures. Finally, `survey data (bandwidth = 0.8)' refer to non-parametric
estimates obtained from survey data using a bandwidth of 0.8.\footnote{%
A bandwidth of 0.8 was selected for the survey based estimates displayed
here, after testing a number of alternatives. The authors may be contacted
for associated results.} The alternatives to the tax implicit equivalence
scales all take base independent specifications, and are consequently
displayed as points on the vertical axis to avoid excessive clutter.
The tax implicit equivalence scale estimates obtained from simulated data
are easier to analyse than those derived from survey data because of the
limited sources of household heterogeneity involved. These are consequently
discussed, before making comparisons with the estimates obtained from survey
data, and the alternative equivalence scales displayed in each graph. In all
cases, the equivalence scale estimates are specified with reference to
single adults without dependant children (for whom the equivalence scale
takes a value of one). Muellbauer and van de Ven (2003) note that tax
implicit equivalence scales may be sensitive to the reference unit adopted.
Single adults are adopted for the analysis presented in this paper because
of their prevalence in the survey data used, and the relatively wide range
of their average tax rates. A relatively high prevalence in the survey data
improves the accuracy of the estimate obtained for the tax function of
reference households, and a wide range of average tax rates is useful
because of the comparisons that are made to infer tax implicit scales as
discussed in Section \ref{est}.
Figure \ref{anps} indicates that the Australian tax and benefit schemes
included in the simulation analysis tend to treat single parents more
generously than they do single adults without dependant children, where the
equivalence scales for single parents are greater than one for all measures
of pre-tax income considered. Focussing upon the estimates obtained from
simulated data that were not smoothed (no smoothing), the scales for single
parents with one and two children are highly base dependant, increasing at
approximately the same rate from \$0 to peak at \$24,000, before falling
away at higher pre-tax incomes. Furthermore, the scale estimates obtained
for single parents with one child are strictly less than the estimates for
single parents with two children when pre-tax income is less than \$74,000,
where the two are equal for higher incomes.
The trends observed for the equivalence scales relate to elements of the
simulated tax and benefit schemes. \textit{SPP }provides a higher benefit
rate and less punitive means testing than \textit{NS}. This drives a wedge
between the post-tax incomes of single people with and without dependant
children, which increases in size until pre-tax income reaches \$14,000,
before falling away at higher incomes. The effects of \textit{FP}, \textit{%
FTI}, and $SPR$ also increase the post-tax incomes of single parents
relative to otherwise similar single adults without children. It is not
until the means testing of $FP$ takes effect that the equivalence scales of
single adults tend to fall with higher pre-tax income. The fact that $SPR$
is not means tested implies that the wedge between single parents and
otherwise similar single adults without children will not go to zero as
pre-tax income is increased, and hence the equivalence scales of single
parents remain greater than one for higher incomes.
The scale estimates derived from unsmoothed simulated data displayed for
single adults in Figure \ref{anps} suggest that the government takes greater
consideration of the needs of single parents (relative to single adults
without dependant children) as incomes rise up to a threshold, but that this
consideration decreases thereafter. This is consistent with the view that
single parents face a greater burden associated with working than those
without children due, for example, to the financial costs associated with
childcare and the emotional strain of parental responsibility. The fact that
the equivalence scales for single parents fall away after a threshold income
level is consistent with the view that, at higher incomes, parenthood
becomes a consumption decision that should be borne by the individual rather
than by society.
The amount of smoothing increases with the bandwidth used for the lowess
estimation procedure - this is reflected by the three series derived from
simulated data displayed in Figure \ref{anps} (for reference, the `no
smoothing' condition is equivalent to assuming a bandwidth of 0). It is
interesting to note that the equivalence scale estimates obtained from
survey data, which were derived using a bandwidth of 0.8, bear a closer
relationship to the estimates obtained from simulated data using a bandwidth
of 0.2 than to those obtained using a bandwidth of 0.8. This result is
observed because the distribution of the survey population is concentrated
between \$15,000 and \$30,000 (AUD), as opposed to the uniform distribution
that is generated by the simulation model. Furthermore, and perhaps more
importantly, the survey data include more noise than the simulated data do.
With regard to the alternatives to the tax implicit equivalence scales
displayed in Figure \ref{anps}, it can be seen that the demand system,
Henderson, and OECD scales take values that are less than the Engel scales
and greater than the Rothbarth scales. The values of the demand system,
Henderson, and OECD scales are consequently consistent with biasses that
have been associated with the Engel and Rothbarth methods, which tend
respectively to overstate and understate the actual child costs borne by
households.\footnote{%
On biasses associated with the Engel and Rothbarth scales see, for example,
Deaton and Muellbauer (1986, p. 732) and van de Ven (2003).} The tax
implicit equivalence scale estimates of single parents with no pre-tax
income and one dependant child are higher than the Engel scales, and rise
(except for the scales with a band-width of 0.8) to a maximum at
approximately \$20,000 per annum, or 65 per cent of mean pre-tax income
(MPTI, \$31,450), before falling to approach the Rothbarth scales at incomes
in excess of 200 per cent of MPTI.
To assign some normative interpretation to these observations, consider the
implications of assuming that the demand system estimates provide an
accurate reflection of the relative costs borne by single parents, and that
the non-parametric estimates obtained from unsmoothed simulated data provide
an accurate reflection of the relativities implicit in transfer policy. The
top panel of Figure \ref{anps} consequently suggests that the Australian
transfer system makes a greater proportional adjustment for household need
than the proportional increase in the costs incurred due to the addition of
a dependant child for single parents on very low pre-tax incomes, and that
this disparity increases up to 60 per cent of MPTI. This variation is
consistent with the view that the children of single parents should be
adequately provided for, either because they are not responsible for the
decisions of their parents or because there are social benefits to be
enjoyed from such provision. In contrast policy makers may take the view
that the onus of adequate provision for single adults without dependant
children should rest with the individual (and hence under-provide for their
associated costs). Above 60 per cent of MPTI the excess proportional
adjustment for household need made by the transfer system falls, and drops
below the proportional increase in costs incurred due to the addition of a
dependant child when income exceeds 150 per cent of MPTI.
A similar profile is observed for the equivalence scales of single parents
with two children, displayed in the lower panel of Figure \ref{anps}. The
tax implicit equivalence scales for single parents with two children are,
however, lower relative to the associated demand based and official scales
than for single parents with one child. This suggests that Australian policy
makers may take into consideration the incentive effects associated with
making larger provisions for the needs of single parents relative to the
costs that parents actually incur due to their dependant children.
Figures \ref{anpc} and \ref{anpcc} suggest that the Australian tax and
benefits system is more generous to couples than to single adults without
dependant children, and tends to make a larger adjustment for household need
as the number of children increases. Focussing upon the series associated
with the unsmoothed simulated data, the equivalence scale estimates obtained
for couples without children take a value of approximately 1.7 until pre-tax
income reaches \$20,000, after which the estimates fall away sharply to
restabilise at approximately 1.2. These observations are principally driven
by two schemes; at low pre-tax incomes, the $NS$ benefit is more generous
and imposes a less severe means test for couples than for singles; and at
high incomes the $DSR$ (which is not means tested) drives a wedge between
the post-tax incomes of couples and singles. Comparison of Figure \ref{anpc}
with \ref{anpcc} indicates that the base dependence of the equivalence
scales derived for couples with children is more smooth than for couples
without dependant children. This is attributable to the gradual way that
means testing is applied for $NS$, $PA$, $FP$, and $FTI$.
The equivalence scale estimates displayed in Figures \ref{anps} to \ref%
{anpcc} suggest that, in general, the government implicitly assumes some
economies of scale when adjusting for household need. This is indicated by
the fact that smaller vertical shifts are observed as household size
increases. Furthermore, the equivalence scales for couples without dependant
children are consistent with the value judgement that adults should be
supported up to a pre-tax income threshold (of approximately \$22,000),
after which they are largely responsible for taking care of their own needs.
The 20 per cent adjustment made for adults with high incomes and a dependant
spouse relative to otherwise similar single individuals is likely to be less
than the costs that are actually incurred by the addition of a spouse (a
proposition that is supported by the demand based equivalence scale
estimates discussed below). This is consistent with the view that having one
member of a childless couple take on a `home-maker' role becomes a
consumption decision at higher household incomes. It also suggests that the
adjustment for household need takes into consideration the value of home
production for couples.
Turning attention to the alternative equivalence scales displayed in Figures %
\ref{anpc} and \ref{anpcc} for two adult Australian households, it can be
seen that both the Engel and Rothbarth scales for couples without dependant
children exceed the OECD, demand system, and Henderson scales. van de Ven
(2003) suggests that this observation is attributable to biasses that are
associated with the Engel and Rothbarth methods, which cause both methods to
over-estimate the costs to households of adult members. Given that biasses
associated with the Rothbarth method tend to underestimate the costs to
households of children, the observation that the OECD, demand system and
Henderson scales tend to increase relative to the Rothbarth scales as the
number of children in a household increases is expected.
In general, the tax implicit equivalence scale estimates for two adult
households exhibit a negative relationship with income. In all three panels
displayed in Figures \ref{anpc} and \ref{anpcc}, the non-parametrically
estimated equivalence scales for tax and benefits policy cross through the
demand system estimates at approximately average pre-tax income. Hence, if
the demand system equivalence scale estimates accurately reflect the costs
of heterogeneous households relative to a single adult without dependents,
and if the tax implicit estimates obtained by the non-parametric procedure
embody the relativities implicit in the Australian transfer system, then
couples appear to be treated preferentially at low incomes, and relatively
harshly at higher incomes. The equivalence scales at low incomes
consequently suggest that the government may perceive a larger social
benefit to be gained by supporting couples than singles (potentially for
social reasons). In contrast, the equivalence scale estimates observed at
high incomes are consistent with the view that couples enjoy benefits beyond
economies of consumption due, for example, to their ability to share
household responsibilities.
\bigskip
Comparing the estimates obtained from survey data with those derived using
simulated data indicates the extent to which the relativities described by
the survey data are captured by the simulation model. The estimates
displayed in Figure \ref{anpc}, for example, indicate that couples are
treated more generously on average relative to single adults in the survey
data than by the simulation model. This can be attributed to some form of
income splitting that is undertaken by couples when calculating their tax
burden, which is not taken into consideration by the simulation model.%
\footnote{%
The simulation model assumes that all household income is earned by a single
individual, which is not the case for survey data.} Nevertheless, despite
the simplicity of the simulated tax and benefits system, most of the
estimates derived from simulated data displayed in the five panels of
Figures \ref{anps} to \ref{anpcc} bear a close relationship to the
associated estimates derived from survey data, suggesting that the
simulation model does a (surprisingly) good job at mimicking the
relativities implicit in the actual Australian tax and benefits system.
\section{Conclusions\label{conc}}
Redistributive policy that makes explicit reference to non-income
characteristics of households - such as their size and demographic
composition - embodies an implicit set of relativities. Tax and benefits
policy consequently provides an important, and as yet under-utilised source
of information for identifying the value judgements (implicitly) made by
policy makers regarding the needs of heterogeneous households. In this paper
it is suggested that the equivalence scale framework can be used to make the
relativities embedded in transfer policy explicit.
A formal model for considering observed tax and benefits systems within the
equivalence scale framework is introduced, and its relationship with the
concept of horizontal equity is described. Furthermore, it is shown that the
ability of the model to reflect transfer systems that are commonly observed
depends crucially upon the assumptions made regarding the equivalence scale
specification. Our equivalence scale specification is general enough to
capture transfer systems observed in practice.
A number of methods can be used to estimate the equivalence scale implicit
in tax and benefits policy. We consider a non-parametric approach here that
infers the relativities implicit in transfer policy using data that are
described by common household surveys, such as the Survey of Income and
Housing Costs in Australia. The fact that the estimation procedure
considered here is based upon survey data, rather than relying upon
microsimulation models, is of particular importance because simulation
models can provide only a limited reflection of transfer systems; the
effects of imperfect take-up rates, tax avoidance and various tax
minimisation strategies imply that the real world impact of a transfer
system may be quite different from the impact that is implied by official
rates and thresholds.
The estimation method is used to explore the relativities implicit in the
Australian transfer system, based upon simulated data for a subset of the
system, and survey data from the SIHC. The estimates derived using tax and
benefits data are compared with a range of alternative scales, including
estimates obtained from expenditure data, scales applied by government
statistical agencies in the two countries, and the scales suggested by the
OECD. The implicit equivalence scale estimates obtained suggest that the
adjustments made by the Australian transfer system for differences in
household need describe an interesting and intuitive set of value judgements.
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\pagebreak
\appendix
\section{Regression Estimation Methods\label{app-regn}}
The regression methods that are discussed here are of interest because they
enable functional forms to be estimated directly from survey data, and
because they allow for an explicit treatment of associated error terms. In
practice, however, practical application of the regression methods can prove
difficult because of the highly non-linear nature of the regression models.
A study that is principally concerned with practical application of these
models is currently a work in progress - see Muellbauer and van de Ven
(2003) for some preliminary analysis.
\subsection{Joint Econometric Estimation\label{econo}}
The term `joint estimation' is used to refer to the following procedure
because it involves estimating the equivalence scale and equivalent tax
functions jointly.\footnote{%
This is distinct from the common use of the term `joint estimation' in the
econometric literature, where it refers to a procedure that takes into
consideration hypothesised correlations between the error terms of two
regression models. In the analysis undertaken here, the correlation between
the error terms of the equivalent tax function and the equivalence scale is
unidentifiable, and is consequently assumed to be zero. See below for
further discussion.} Restating equation (\ref{etax2}):
\begin{equation}
y_{i}=x_{i}-a_{i}^{\ast }T^{\ast }\left( \frac{x_{i}}{a_{i}^{\ast }}\right)
+a_{i}^{\ast }\varepsilon _{i}^{\prime } \label{estt}
\end{equation}
where $\varepsilon _{i}^{\prime }$ allows for the possibility of horizontal
inequity and measurement error. Assuming:
\begin{eqnarray*}
T^{\ast }\left( x_{i}^{\ast }\right) &=&T\left( x_{i}^{\ast }\right) +\omega
_{i} \\
a_{i}^{\ast } &=&a\left( v_{i}\right) +\lambda _{i}
\end{eqnarray*}
and substituting into equation (\ref{estt}) obtains:
\begin{equation}
y_{i}=x_{i}-\left( a\left( v_{i}\right) +\lambda _{i}\right) T\left( \frac{%
x_{i}}{a\left( v_{i}\right) +\lambda _{i}}\right) +\left( a\left(
v_{i}\right) +\lambda _{i}\right) \left( \varepsilon _{i}^{\prime }+\omega
_{i}\right) \label{esta}
\end{equation}
which can be estimated using standard non-linear regression techniques. It
is evident from the specification of equation (\ref{esta}), however, that
the error term will be heteroscedastic. Consider, for example, the case in
which the function $T\left( .\right) $ takes the form of an $N$th order
polynomial. Then:
\begin{equation}
y_{i}=x_{i}-\left( a_{i}+\lambda _{i}\right) \left\{ \sum_{j=0}^{N}\beta
_{j}\left( \frac{x_{i}}{a_{i}+\lambda _{i}}\right) ^{j}\right\} +\left(
a_{i}+\lambda _{i}\right) \varepsilon _{i} \label{estb}
\end{equation}
where the $\beta _{j}$ terms are tax function coefficients, $a_{i}=a\left(
v_{i}\right) $, and $\varepsilon _{i}=\varepsilon _{i}^{\prime }+\omega _{i}$%
. For the $j$th term of the tax function:
\begin{eqnarray}
\left( a_{i}+\lambda _{i}\right) \beta _{j}\left( \frac{x_{i}}{a_{i}+\lambda
_{i}}\right) ^{j} &=&\beta _{j}\frac{x_{i}^{j}}{\left( a_{i}+\lambda
_{i}\right) ^{j-1}} \notag \\
&\simeq &\beta _{j}\frac{x_{i}^{j}}{a_{i}{}^{j-1}\left( 1+\left( j-1\right)
\frac{\lambda _{i}}{a_{i}}\right) } \notag \\
&\simeq &\beta _{j}\frac{x_{i}^{j}}{a_{i}^{j-1}}\left( 1-\left( j-1\right)
\frac{\lambda _{i}}{a_{i}}\right) \notag \\
&=&\beta _{j}\frac{x_{i}^{j}}{a_{i}^{j-1}}-\beta _{j}\left( j-1\right)
\left( \frac{x_{i}}{a_{i}}\right) ^{j}\lambda _{i} \label{error1}
\end{eqnarray}
where the approximations assume small $\lambda _{i}$. Substituting for the
tax function terms in equation (\ref{estb}), we obtain the following:
\begin{eqnarray}
y_{i} &=&x_{i}-\sum_{j=0}^{N}\beta _{j}\frac{x_{i}^{j}}{a_{i}^{j-1}} \notag
\\
&&+a_{i}\varepsilon _{i}+\lambda _{i}\varepsilon _{i} \notag \\
&&+\sum_{k=0}^{N}\beta _{k}\left( k-1\right) \left( \frac{x_{i}}{a_{i}}%
\right) ^{k}\lambda _{i} \notag \\
&&+\psi _{i} \label{estc}
\end{eqnarray}
where $\psi _{i}$ accounts for the approximations made to obtain equation (%
\ref{error1}). The error term of this equation is defined by:
\begin{equation}
\epsilon _{i}=a_{i}\varepsilon _{i}+\lambda _{i}\varepsilon
_{i}+\sum_{k=0}^{N}\beta _{k}\left( k-1\right) \left( \frac{x_{i}}{a_{i}}%
\right) ^{k}\lambda _{i}+\psi _{i} \label{error}
\end{equation}
We assume that the error terms, $\varepsilon _{i}$, $\lambda _{i}$, and $%
\psi _{i}$ have an expectation of zero, and a constant variance for all $i$.
Furthermore, we assume that the error terms are all independent of the
exogenous variables of the model (that the model is correctly specified),
and that the error terms are independent of one another. This last
assumption warrants some discussion. Specifically, it is clear from equation
(\ref{esta}) that the tax function and the equivalence scale are related,
which implies that the associated error terms, $\varepsilon $ and $\lambda $%
, might also be correlated. Equation (\ref{error}), however, indicates that
it is not possible to derive explicit estimates for the individual error
terms; following regression of equation (\ref{estc}) we have only one
equation - (\ref{error}) - and three unknowns - $\varepsilon $, $\lambda $,
and $\psi $. Hence the assumption made here regarding the independence of $%
\varepsilon $ and $\lambda $ cannot be tested using the model. It is
possible, for example, to assume that no error is associated with the
equivalence scale $\left( \lambda =0\right) $, in which case (assuming $\psi
_{i}=0$ for all $i$) all of the error associated with an estimate of
equation (\ref{estc}) will be attributed to the tax function ($\varepsilon $%
). The assumption of zero correlation between $\varepsilon $ and $\lambda $
implies that there is some separability between the process by which the
transfer system identifies a household's `type' (and hence its needs), and
the process by which it determines the household's net transfer payment
given it's allocated type. In our view, this assumption appears reasonable.
Given the following assumptions:
\begin{equation*}
\begin{array}{l}
E\left( \varepsilon _{i}\right) =E\left( \lambda _{i}\right) =E\left( \psi
_{i}\right) =0 \\
var\left( \varepsilon _{i}\right) =\sigma _{\varepsilon }^{2} \\
var\left( \lambda _{i}\right) =\sigma _{\lambda }^{2} \\
var\left( \psi _{i}\right) =\sigma _{\psi }^{2} \\
cov\left( \varepsilon _{i},\lambda _{i}\right) =cov\left( \varepsilon
_{i},\psi _{i}\right) =cov\left( \lambda _{i},\psi _{i}\right) =0 \\
cov\left( \varepsilon _{i},z_{i}\right) =cov\left( \lambda _{i},z_{i}\right)
=cov\left( \psi _{i},z_{i}\right) =0%
\end{array}%
\end{equation*}
where $z_{i}$ defines all exogenous variables of equation (\ref{estc}), $%
E\left( .\right) $ defines the expectation, $var\left( .\right) $ the
variance, and $cov\left( .\right) $ the covariance (which is also assumed to
apply to all higher moments), the properties of $\epsilon _{i}$ are defined
by:
\begin{eqnarray}
E\left( \epsilon _{i}\right) &=&0 \\
var\left( \epsilon _{i}\right) &=&a_{i}^{2}\sigma _{\varepsilon }^{2}+\sigma
_{\lambda }^{2}\sigma _{\varepsilon }^{2} \notag \\
&&+\left[ \sum_{k=0}^{N}\beta _{k}\left( k-1\right) \left( \frac{x_{i}}{a_{i}%
}\right) ^{k}\right] ^{2}\sigma _{\lambda }^{2}+\sigma _{\psi }^{2}
\label{errorv} \\
cov\left( \epsilon _{i},z_{i}\right) &=&0
\end{eqnarray}
Since the form of the heteroscedasticity is known, but the associated
parameter estimates are not, this analysis suggests that it is appropriate
to estimate equation (\ref{estc}) by either Weighted Least Squares or
Generalised Methods of Moments.
\subsection{Two-stage or mixed estimation\label{2stg}}
In practice, using the joint estimation procedure described above to
estimate the equivalence scale and equivalent tax functions is complicated
by a number of factors. Most important of these is the income dependence of
the equivalence scale, which can take a highly non-linear form. This, and
the fact that such a relationship is unlikely to be known \textit{ex ante},
implies that the function adopted for the equivalence scale may not be
sufficiently flexible to accurately reflect the relativities implicit in the
transfer system, which will result in omitted variable bias.
When an equivalence scale is desired for distributional, rather than
interpretive purposes, some forms of omitted variable bias may actually help
to improve the estimates obtained. Omitting relevant health related
characteristics may, for example, lead to an upward bias of coefficients on
old age identifiers, which could help to correct the estimated equivalence
scale for distributional analysis. However, omitting pertinent variables
from the equivalence scale specification, and income related variables in
particular, is likely to bias the coefficients of the tax function as well
as those of the equivalence scale when derived from a joint estimation.
Estimating the equivalence scale and equivalent tax functions jointly means
that the biasses of one function can produce biases in the other function,
which complicates interpretation of the parameter estimates obtained.
Specifically, interpretation of the equivalence scale as a proportional
adjustment to income which ensures that the same tax function, primarily
applicable for reference households, is applicable for the entire
population, does not hold when biassed estimates are obtained for the
equivalent tax function of reference households.
One useful test for determining the adequacy of the equivalence scale and
tax functions used is to check whether the tax function coefficient
estimates derived from a non-linear regression of the entire population are
significantly different from the tax function coefficient estimates obtained
for the reference population when taken in isolation. In practice, it may be
difficult to find a specification for which the tax function parameter
estimates are stable when calculated using the restricted and unrestricted
populations. When this is the case an alternative method of estimation is
required. The test described above suggests one possibility; a two-stage
estimation procedure in which the tax function and equivalence scale are
estimated separately.\footnote{%
Again, the use of the term `two-stage estimation' that is adopted here
should not be confused with its common use in the econometric literature. In
the current context, it refers to separate estimation of the equivalent tax
function and equivalence scale in two distinct stages, as opposed to a
procedure that is designed explicitly to adjust for heteroscedasticity of
the associated error terms.}
Two-stage estimation is available in two alternative forms. \ The iterative
method set out in Section \ref{iter} can be used to generate values of $%
a_{i}^{\ast }$ for observed households with an income $x_{i}$ and
characteristics $v_{i}$. \ A parametric function for $a_{i}^{\ast
}(x,v,v_{0})$ can then be estimated. \ Note that this method is consistent
with parametric or non-parametric estimates of net tax functions.
The alternative two-stage method takes the parametric form (\ref{esta}) and
estimates the $\beta $'s for the reference household type. Given these
estimated values, now substitute the equivalence scale function (\ref{d})
into (\ref{esta}) and estimate the parameters of (\ref{d}) through weighted
non-linear least squares.
Two-stage estimation prevents the biasses of the equivalence scale feeding
into the tax function (and vice versa). As such, where the tax function used
provides a close approximation to the observed tax function of reference
households, the biasses that remain with regard to the equivalence scale
will be largely confined to the type that tend to improve the estimates for
distributional purposes, as discussed above.
The indeterminacy that is associated with the error structure of equation (%
\ref{error}), will continue to apply to the second stage of the two-stage
regression procedure. Specifically, we might assume that reference
households are only those for which the tax function estimate derived in the
first stage of the procedure provides an accurate description of the
relationship between pre-tax and post-tax income, in which case $\varepsilon
_{i}=0$ for all $i$, and hence all of the error observed for the second
stage of the estimation is associated with the equivalence scale where:%
\footnote{%
Assuming $\psi _{i}=0$ for all $i$.}
\begin{equation}
\epsilon _{i}=\sum_{k=0}^{N}\beta _{j}\left( j-1\right) \left( \frac{x_{i}}{%
a_{i}}\right) ^{j}\lambda _{i}
\end{equation}
Alternatively, it is possible to assume that both the tax function and the
equivalence scale are subject to error, in which case the error structure of
the second stage of the procedure will be described by equation (\ref{error}%
). These observations regarding the error structure highlight the importance
of the precise definition that is adopted for the reference household when
estimating an equivalence scale that is implicit in transfer policy.
\section{Alternative Equivalence Scales\label{app-scls}}
\subsubsection{Demand based equivalence scales}
Equivalence scale estimates based on household expenditure data are
considered by van de Ven (2003), using the base independent specification
defined by:
\begin{equation}
a_{i}=\exp \left\{ \phi \frac{c_{i}}{n_{i}}+\gamma \ln \left( n_{i}\right)
\right\} \label{demes}
\end{equation}
where $c_{i}$ defines the number of children (under the age of 18) in a
household, and $n_{i}$ denotes the total number of household members. Three
demand based estimation methods are considered by van de Ven (2003); the
Engel method, the Rothbarth method, and a method based on a demand system.
All of the scales considered by van de Ven (2003) are assumed to be base
independent (as indicated by the specification of equation (\ref{demes})),
which is consistent with the demand based literature. Equivalence scale
estimates derived using each of the three approaches are displayed in Table %
\ref{appta}, which are used to calculate the associated equivalence scales
displayed in Figures \ref{anpc} to \ref{anpcc}.
%TCIMACRO{\TeXButton{B}{\begin{table}[tbp] \centering}}%
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\caption{Demand Based Equivalence Scale Estimates\label{appta}}$%
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$
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\subsubsection{OECD scales}
The OECD scale is recommended by the OECD for use when there exists no other
preferred scale, or where international comparisons are to be made. The
scale considered here assigns a value of one to the reference person of a
household, 0.7 to their partner, and 0.5 to any dependant children
regardless of household income. Hence a household that is comprised of a
couple with two dependant children is allocated a scale of 2.7.\footnote{%
See \textit{The OECD List of Social Indicators}, OECD, 1982.}
\subsubsection{Henderson scales}
The Henderson equivalence scale is commonly used for analysis of Australian
income data - see, for example, the Australian Bureau of Statistics
publication, \textit{Income Distribution, Australia 1997-1998}.\footnote{%
ABS Catalog Number 6523.0. See, also, Henderson, \textit{et. al }(1970), and
Appendix F of the Australian Government Commission of Inquiry into Poverty
(1975).} This scale allocates points depending upon household member
characteristics, and the number of people in a household. The relevant
points, which are independent of household income, are defined in Table \ref%
{hend}.
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\caption{Henderson Scale\label{hend}}$%
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$
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\end{document}