The adjusted generalised entropy curve as a general descriptive and normative measure of income inequality, with a compression and estimation procedure

The adjusted generalised entropy curve as a general descriptive and normative measure of income

inequality, with a compression and estimation procedure

Kieran Latty

[email protected]

Work in progress, comments welcome

This version 19 August 2015

The generalised entropy index gives a summary measure of concentration of a

distribution, where the order of the index determines the sensitivity of the index to top

and bottom tail inequality respectively. This paper introduces an adjusted generalised

entropy index as a normative and descriptive measure of income inequality.

Disagreement over normative and positive issues impacting upon the degree of

warranted inequality aversion provides a case for generalising the information within a

sample distribution, so that any Atkinson-Kolm-Sen social welfare function which is

itself a function of one or many GE indexes of any order may be calculable. This may be

achieved by specifying or estimating an adjusted GE curve giving the adjusted GE index

for all orders. A method for estimating or compressing the GE curve as a two tailed

generalised logistic function of order is given. The estimation procedure is accurate;

errors are typically less than 1% over a wide range of order.

1 Introduction

[Standard summary of content to go here]

2 The generalised entropy index

The generalised entropy (GE) indexes take the following form;

𝑇𝑣 =1

𝑁𝑣(𝑣 − 1)∑[(

𝑥𝑖

�̅�)𝑣

− 1]

𝑁

𝑖=1

, 𝑣 ≠ 0,1

Where 𝑣 denotes the order of the index. Two special cases exist for 𝑣 = 0 and 𝑣 = 1. These

are Theil’s two commonly calculated inequality indexes;

𝑇0 =1

𝑁∑(

𝑥𝑖

�̅�ln

𝑥𝑖

�̅�)

𝑁

𝑖=1

𝑇1 =1

𝑁∑(ln

�̅�

𝑥𝑖)

𝑁

𝑖=1

Additionally, note that 𝑇2 is half the coefficient of variation;

𝑇2 =1

2

𝜎

�̅�

3 The 𝑻𝟎𝒗 index

We define an index 𝑇0𝑣 as a GE index or order 0 or 1, as derived from an index 𝑇𝑣 under the

assumption of lognormality. Solving obtains;

𝑇0𝑣 =1

(𝑣2 − 𝑣)ln[(𝑣2 − 𝑣)𝑇𝑣 + 1]

Under lognormality we also have the convenient result

𝑇0𝑣 =1

2𝜎Ln

2

where 𝜎Ln2 is the variance of logarithms.

For any finite sample of a nondegenerate distribution, we may then define the GE curve as

the function 𝑇0𝑣 = 𝑓(𝑣).

The 𝑇0𝑣 curve can be further normalised by dividing by 𝑇0𝑣 for a specified value of 𝑣, which

gives a measure of deviation from lognormality.

4 The 𝑻𝟎𝒗 index as a normative and descriptive measure of income

or consumption inequality

The index 𝑇0𝑣 is a good candidate for a normative and descriptive measure of income or

consumption inequality. This may be demonstrated by comparing 𝑇0𝑣 to the Atkinson index

and to the unadjusted GE index, 𝑇𝑣.

The Atkinson index (Atkinson, 1970) is an exemplar normative measure of income

inequality, with several desirable properties:

1. The index gives a measure of social welfare losses from inequality in equally

distributed income equivalents, where the marginal value of income is given by 𝛿𝑉

𝛿𝑌∝ 𝑌− . Social welfare under utilitarianism is then given by an Atkinson index

where the inequality aversion parameter 휀 is set to be equal to the elasticity of

marginal utility of income, η

2. More generally, variability of the sensitivity of the index to existence of relatively low

incomes is set by the inequality aversion parameter, by raising the inequality

aversion parameter the sensitivity of the index to existence of relatively low income

is raised. Ethical stances, besides utilitarianism which find extreme relative poverty

repugnant, for example the ‘priority’ view (Holtug, 2007; Nagel, 1979a, 1979b; Parfit,

2000, 1997) may be represented adequately by the use of a suitably high inequality

aversion parameter.

Despite these strengths, there are serious limitations to the Atkinson index as a normative

measure of inequality, which relate to restrictions on the form of inequality aversion.

Consider the marginal social value of income function associated with the Atkinson index

below:

𝛿𝑉

𝛿𝑌∝ 𝑌−

Under the utilitarian interpretation of the index value is equated with utility, and so the

function is simply the ubiquitous isoelastic utility function which is well supported

empirically, however other ethical stances are consistent with inequality aversion and

potentially with inequality aversion as modelled by the Atkinson index (Creedy, 2007),

however if the utility function is not isoelastic, or if the value function under non-utilitarian

ethical stance is not isoelastic then the Atkinson SWF may not appropriate.

4.1 Isoelastic value and top tail inequality aversion

Especially troubling is the inability for the Atkinson index to be averse to income

concentration within the top tail of the income distribution. For the Atkinson marginal value

function, for large 휀 marginal social value rapidly approaches zero for very large incomes;

marginal social value of income in absolute terms at high income is insensitive to income,

and social welfare losses are insensitive to top tail inequality. Reducing the inequality

aversion parameter increases the relative sensitivity of the index to top tail inequality, but at

the same time inequality aversion in general is lowered, such that inequality aversion

approaches zero well before the index is sensitive to top tail inequality. Consider social

welfare losses from a small transfer of income between an individual with income 𝑌1 to

another with higher income, 𝑌2, which is given by

−𝛿𝑉

𝛿𝑌2∝ (

𝛿𝑉

𝛿𝑌1−

𝛿𝑉

𝛿𝑌2)

−𝛿𝑉

𝛿𝑌1∝ 𝑌1

− − 𝑌2−

We may normalise by expressing losses in term of a reduction in income for someone on

some reference income, for example the median income, giving

−

𝛿𝑉𝛿𝑌1

𝛿𝑉

𝛿�̂�

=𝑌1

− − 𝑌2−

�̂�−

which can be rearranged to give losses as a product of two terms:

𝛿𝑉𝛿𝑌1

𝛿𝑉

𝛿�̂�

= [1 − (𝑌2

𝑌1)−

] (𝑌1

�̂�)−

where the first term, which we view as term an ‘inefficiency’ term, gives the total welfare

losses as a ratio of the welfare losses that would result from a loss of income at 𝑌 = 𝑌1. As

the ratio of 𝑌2 to 𝑌1 increases, then ‘inefficiency’ approaches 100%, with the rate of

convergence increasing in 휀. The second term may be viewed as a ‘salience’ term, i.e. it

gives the importance of a loss of income at 𝑌 = 𝑌1 as a ratio of losses due to a reduction of

income at 𝑌 = �̂�. When considering a transfer from an individual with relatively high

income to one with even higher income, ‘inefficiency’ is increasing in 휀, however salience is

decreasing in 휀. Figure XX to XX gives ‘inefficiency’; ‘Salience’ and adjusted losses for a range

of values of 휀; 𝑌2

𝑌1; and

𝑌1

�̂�.

Figure XX: ‘Inefficiency’ of a Pigou-Dalton inequality increasing transfer

0

1

2

34

0.00%

20.00%

40.00%

60.00%

80.00%

100.00%

1

2

4

8

Epsilon Y1/Y2

Figure XX: ‘Salience’ of a Pigou-Dalton inequality increasing transfer

Figure XX: Total normalised losses from a Pigou-Dalton inequality increasing transfer; 𝒀𝟐

𝒀𝟏= 𝟐

01

2

3

4

0.00%

20.00%

40.00%

60.00%

80.00%

100.00%

1

2

4

8

Epsilon Y1/Median Y

00.5

11.5

22.5

33.5

40.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

80.00%

90.00%

100.00%

12

4

8

16

Epsilon Y1/Y median

As shown above, the Atkinson index, and any index with an isoelastic value function, cannot

model aversion to concentration of income amongst the very top income earners, either

due to utilitarian or welfarist considerations in the context of political economy effects (for

example due to political economy or relative income effects) or due to alternative ethical

stances which otherwise induce aversion to top tail inequality beyond welfarist

considerations. However, slight extension of the isoelastic value function can allow for

aversion to income concentration. Consider the following value function:

𝛿𝑉

𝛿𝑌∝ 𝑌− 1 − 𝑘1 (

𝑌

�̅�)

2

Where the second term models welfare losses from external effects associated with top

incomes. Then, for sufficiently large 휀2 and 𝑘, the associated SWF is highly averse to income

concentration amongst top income earners. Derivation of the resulting SWF is given in

appendix A.

4.2 Modelling aversion to top and bottom tail inequality

Ideally, a normative measure of inequality should allow for independent adjustment of the

relative sensitivity of the index to low and high incomes respectively, and of the general

sensitivity of the index to inequality in general. One option would be to specify and an

Atkinson-Kolm-Sen social welfare function (Atkinson, 1970; Kolm, 1969; Sen, 1979) of the

form:

𝑊$ = (1 − 𝐼)�̅�

𝐼 = 𝑓(𝑇𝑣, 휀)

where 𝑊$ denotes welfare in equally distributed income equivalents, and I is the associated

Atkinson-Kolm-Sen measure of inequality. The index 𝐼 is monotonically increasing in 𝑇𝑣 and

of 휀, and where 휀 regulates the degree of inequality aversion. The Atkinson index is a special

case where we have:

𝐼 = 1 − [휀(휀 − 1)𝑇𝑣 + 1]1

1− ; 휀 ≠ 1; 0

𝐼 = 1 − 𝑒−𝑇𝑜; 휀 = 1

𝐼 = 0; 휀 = 0

휀 = 1 − 𝑣

Ideally, we would be able to generalise such a SWF by allowing independence of 𝑣 and of 휀.

In this case we also avoid the result that 𝐼 is positive only for negative values of 𝑣, which Sen

(1982, p. 419) has highlighted as a limitation of the Atkinson index, at least as a descriptive

measure of inequality.

One option would be to simply remove the restriction, 휀 = 1 − 𝑣. However, in this case I is

still sensitive to v independently of 휀, because for typical income distributions 𝑇𝑣 becomes

very large for absolutely large values of 𝑣. For example, consider 𝑇𝑣 as a function of v for the

lognormal distribution:

𝑇𝑣 =𝑒(𝑣2−𝑣)𝑇0;1 − 1

(𝑣2 − 𝑣)

𝑇0;1 =1

2𝜎Ln

2

where 𝜎Ln2 is the variance of logarithms. Figure XX gives 𝑇𝑣 as a function of v, showing the

very strong variation in 𝑇𝑣 as 𝑣 changes.

Figure XX: 𝑻𝒗 as a function of v

The problem of sensitivity of 𝐼 to 𝑣 may be solved by replacing 𝑇𝑣 with a transformation of

𝑇𝑣, 𝐼𝑣 , where 𝐼𝑣 is a constant for a particular reference distribution. For the lognormal

distribution this condition is satisfied by the adjusted GE index introduced in this paper, 𝑇0𝑣.

We may then posit the following generalised SWF:

0.1

1

10

100

-5 -4 -3 -2 -1 0 1 2 3 4 5

T0=0.125

TO=0.250

𝐼 = 1 − 𝑒− 𝑇0𝑣

Which is equivalent to the Atkinson index under the condition, 휀 = 1 − 𝑣, however this

condition may be relaxed to produce a normative index of inequality where 𝑣 may be set

independently of 휀. There are a series of reasons why such freedom may desirable:

1. The Atkinson SWF is utilitarian only under restricted conditions – namely that

population is fixed, utility is isoelastic and no relative income effects are present. If

relative income effects are important, then social welfare losses from inequality may

be better estimated utilising an index of higher order than 𝑣 = 1 − 휀, where 휀 is set

to be equal to the elasticity of marginal utility of income. In general, such effects

tend to raise 휀 above the personal elasticity of marginal utility of income, as relative

income effects raise the degree of warranted inequality aversion (Aronsson and

Johansson-Stenman, 2010, 2009; Aronsson and Johansson‐Stenman, 2010; Boskin

and Sheshinski, 1978; Johansson-Stenman et al., 2002; Kapteyn and Van

Herwaarden, 1980; Latty, 2015)

2. Normative measures of inequality should ideally be somewhat flexible in response to

ethical disagreements, for example within welfarism over the correct method for

aggregating utilities, and more broadly to deviations from welfarism such as

philosophical egalitarianism (Temkin, 2003). Some views within this broad spectrum

will find concentration of wealth amongst a minority of individuals repugnant,

perhaps as a consideration over and above any utilitarian justification for such

concern. In this case some additional sensitivity to top tail inequality is warranted.

Generally, we may take 𝐼 to be a good contender for a normative measure of income

inequality. Given that 𝑇0𝑣 enables the derivation of 𝐼 via a monotone transform, (alongside

휀 which is independent of the distribution), 𝑇0𝑣 contains all of the normatively relevant

information contained in the income distribution, if 𝐼 is a sufficient normative measure of

inequality. Thus 𝑇0𝑣 is also a good potential normative measure of inequality, and the GE

curve ten provides for all potentially relevant normative information regarding the income

distribution, even if v or 휀 are unknown.

Additionally, 𝑇0𝑣 is, unlike other normative measures a somewhat suitable descriptive

measure – if actual distributions are approximately lognormal, then 𝑇0𝑣 may be somewhat

insensitive to the order of the index. In this case, if 𝑇0𝑣 is used primarily as a normative

indicator, then some descriptive utility is retained - normative or other considerations which

may prompt disagreement over, or revision of, the index order may not lead to large

revisions in the level of inequality as reported by the 𝑇0𝑣 index, thus the description content

of the inequality index is not overly sensitive to normative considerations, 𝑇0𝑣 is also

completely independent of the inequality aversion parameter, 휀. In this case there is a

partial solution to the problem identified by Sen (1982, 1973) where normative measures of

inequality such as the Atkinson index tend to be poor descriptive measures, due to the

strong dependence of the degree of stated inequality on the degree of inequality aversion -

furthermore for non-positive values of 휀 the Atkinson index leads to a failure to respect the

Pigou-Dalton criteria of unambiguous inequality rankings.1 For the 𝑇0𝑣 index these

problems are partially mitigated - 𝑇0𝑣 is, as a monotone transform of the GE index,

unconditionally Pigou-Dalton respecting when order is fixed at any arbitrary value- however

if there are large deviation from lognormality then 𝑇0𝑣 may depend strongly on order,

however this is an unavoidable corollary of variable sensitivity to top and bottom tail

inequality, which is important to retain for a normative measure, and also for a descriptive

measure as it allows the GE curve to encode additional information regarding the

distribution, as opposed to the case where some summary measure (Gini, variance of

logarithms) alone gives no indication of the distribution shape. One solution to the problem

of sensitivity of the inequality level on 𝑣 would be to further normalise the GE curve by

adjusting for typical deviations from lognormality – if a ‘typical’ GE curve is estimated,

perhaps as some averaging procedure over a sample of distributions, and indexed by

division of its own 𝑇01;0 value, then dividing the GE curve by this index will produce a

normalised GE curve which will be approximately constant for distributions of

approximately ‘typical’ shape.

Furthermore, 𝑇0𝑣 may enable comparison of the level of inequality in the context of ethical

or other disagreements over the correct value for 𝑣. Consider for example two countries

with differing approached to inequality reduction – one has placed emphasis on reducing

concentration of income at the very top of the distribution, and thus reduction of 𝑇0𝑣 of

high order, whilst another has placed emphasis on raising the incomes of the very poor, and

hence of 𝑇0𝑣 of low order. In this case relative success in comparison to differing objectives

may be judged by some summary measure, for example the Gini index or a GE index of

some arbitrary order. However, in this case some unfairness is assessment may be alleged -

for example if 𝑇0 is utilised as the measure of inequality then the efforts of the second

country will tend to appear more impressive, as 𝑇0 is relatively more sensitive to bottom

than to top tail inequality. In this case, a fairer approach may be to compare inequality

reductions by utilising 𝑇0𝑣 but by allowing v to be set independently for each country, i.e.

by setting v to be high for the first, and low for the second country respectively.

Comparisons of inequality where Lorentz curves cross may then be undertaken on the basis

of GE curve dominance. In a pairwise comparison with crossing Lorentz curves there will be

some range over which one distribution will feature lower GE for any given 𝑣. Furthermore,

there will be some range of 𝑣 where 𝑣 may be set independently for each distribution such

that one distribution will always have lower 𝑇0𝑣. In this case clear normative rankings of

inequality may be obtained, even if it is accepted that the correct or ‘fair’ value for 𝑣 may

differ across the relevant distributions, either due to acceptance of the validity of different

policy objectives across jurisdictions, or variation in the positive determinants of the SWF,

1 The Pigou-Dalton criteria requires inequality to increase whenever a transfer

for example the parameter η, or the degree to which top or bottom tail inequality

respectively determines the degree of indirect negative effects form inequality on welfare

levels.

4.3 A further generalisation of a Generalised Entropy SWF

The discussion above suggests a case for independence of 𝑣 and 휀. However note also that

there may be a series of independent ethical or positive considerations which may make

social welfare losses from inequality a function of 𝑇0𝑣 of different orders. For example in

Latty (2015) an Atkinson like index is introduce which features relative income effects, and

social welfare losses from inequality may then be simultaneously sensitive to bottom tail

inequality due to a high elasticity of marginal utility of income, and also to top tail inequality

due to large negative externalities associated with very high incomes. In Latty (2015) the

following result is obtained for the Atkinson like index

𝐴𝐸 = 1 − 𝑒−T01−𝜂𝜂+T0𝑏𝜏(𝑏2−𝑏)

1−𝜏

Where 𝜏 regulates the strength of relative income concerns, and b determines the relative

weight on top incomes in determining reference income levels. For large b, reference

income is highly sensitive to top incomes, and hence social welfare losses are sensitive

to 𝑇0𝑣 of high order, as specified by the condition 𝑣 = 𝑏. Additional, when η is large, the

index is also sensitive to 𝑇0𝑣 of low order, given the condition 𝑣 = 1 − 𝜂.

A general form of SWF which allows for simultaneous sensitivity to 𝑇0𝑣 indexes of various

order is given below

𝐼 = 1 − 𝑒−𝜔

𝜔 = 휀1𝑇0𝑣1 + 휀2𝑇0𝑣2 . . . + 휀𝑛𝑇0𝑣𝑛

휀1 + 휀2 . . . + 휀𝑛 = 휀

or alternatively in continuous form:

𝜔 = ∫ 휀𝑣

𝑣=∞

𝑣=−∞

𝑇0𝑣; 𝑇0𝑣 = 𝑓(𝑣)

∫ 휀𝑣

𝑣=∞

𝑣=−∞

= 휀

which under the special case of lognormality gives

𝜔 = 휀1

2𝜎Ln

2

Not that we can rearrange the index in latty (2015) to obtain results in the form above,

𝜔 = 휀1𝑇01−𝜂 + 휀2𝑇0𝑏

휀1 =1

1 − 𝜏

휀2 =𝜏(𝑏2 − 𝑏)

1 − 𝜏

The generalised SWF above, given its linear specification, may be particularly useful for

econometric estimation of the effects of top, middle and bottom tail inequality on national

or regional summary measures of welfare, for example the mean of some subjective

wellbeing index, or for estimation of inequality or relative poverty aversion using surveys

(Amiel et al., 1999; Bernasconi, 2002; Carlsson et al., 2011; Johansson-Stenman et al., 2002;

Pirtilla and Uusitalo, 2007).

The income elasticity of marginal utility of income, 𝜂 when income is defined as subjective

wellbeing, is readily estimable - for example Layard et al (2008), utilising the happiness

measure of subjective wellbeing and 6 large datasets, altogether comprising 50 countries,

finds values of η ranging from 1.19 to 1.34. If inequality imposes indirect negative effects on

subjective wellbeing, for example via political economy or relative income effects, then

estimation of the above SWF will tend to produce estimates of 𝜔 that are larger than in the

absence of such effects , as given by 𝜔 = 𝜂𝑇0𝑣=1−𝜂.

Alternatively, we may directly estimate indirect effects utilising micro data. For the SWF

above we may utilise following equation

𝑈 = 𝑘(𝜌𝑤$)

1−𝜂 − 1

1 − 𝜂+ 𝑐 + 𝑋𝛽𝑋

𝑤$ = 𝑌𝑒−(𝜔−𝜂𝑇0𝑣=1−𝜂)

Where 𝑈 is the individual SWB score; 𝑤$ is individual welfare in monetary equivalents in

the absence of background inequality; 𝑋 is a vector of individual and country level

determinants of SWB; 𝛽𝑋 is the vector of coefficients for X; and 𝑘 and 𝜌 are (necessary)

scale parameters.

The equation above has a macro level analogue:

�̅� = 𝑘(𝜌𝑊$)

1−𝜂 − 1

1 − 𝜂+ 𝑐 + �̅��̅�

𝑊$ = �̅�𝑒−𝜔

One problem with the more generalised SWF above is that 𝑇0𝑣 for various orders of v must

be known or calculable. In section XX a method for stylised description of the function

𝑇0𝑣 = 𝑓(𝑣) is given that allows for data compression of the GE curve, and with sufficient

data points, estimation of the GE curve from selected GE indexes of variable order.

5 Sample T0 curves

This section gives sample 𝑇0 curves for the lognormal distribution, and a variety of realistic

deviations from lognormality - four sample distributions are given; lognormal (‘Lognormal’)

lognormal with a floor (‘Floor’); lognormal-Pareto; (‘Pareto’) and lognormal –Pareto with a floor

(‘Pareto Floor’).

The generalised form for these distributions is given below:

𝑥(𝑝) = Φln 𝑥−1(𝑝|𝜇𝑥; 𝜎) + 𝑚�̅�; 𝑝 < 𝑙

𝑥(𝑝) = 𝑥𝑙 (1 − 𝑙

1 − 𝑝)1/𝜑

+ 𝑚�̅�; 𝑝 > 𝑙

𝑌𝑙 = Φln𝑥−1(𝑝 = 𝑙|𝜇𝑥; 𝜎Ln)

Where Φln𝑥−1(𝑝|𝜇𝑥; 𝜎) denotes the inverse cumulative density function of the lognormal

distribution of 𝑥 with mean 𝜇𝑥 and standard deviation of logarithms of 𝜎Ln. The variable 𝑚

gives the ratio of minimum income to the mean.

The parameter 𝑙 gives the cumulative probability for the transition between the lognormal

and Pareto sections of the distribution, and 𝑌𝑙 gives the income level at the transition; 𝜑

denotes the Pareto parameter. For all sample distributions, we fix 𝑇0 at 0.125, and then

adjust 𝜎 accordingly.

We have four sample distributions; lognormal; lognormal with a floor; Pareto-lognormal;

and Pareto-lognormal with a floor. Summary statistics are given in table X. Figure X gives

𝑇0𝑣 for a range of v for all sample distributions, normalised by dividing by 𝑇0 and utilising a

sample size of 107. For all distributions we standardise by setting the mean to unity.

Table X: Summary statistics for the 4 sample distributions

Distribution 𝑇0 𝑇1 𝑚 𝜎 𝜑 𝑙

Lognormal 0.1250 0.1250 0 0.5000 - -

Lognormal floor 0.1250 0.1361 0.25 0.6907 - -

Lognormal-Pareto 0.1250 0.1562 0 0.4496 2 98.50%

Lognormal-Pareto with floor 0.1250 0.1672 0.25 0.6203 2 98.50%

Figure X: Normalised adjusted GE curves

6 Estimation method and results

We estimate the GE curve as a sum of a pair of simplified two-tailed generalised logistic

functions.

𝑇0𝑣 = [𝐴1 +𝐾1 − 𝐴1

(1 + 𝑄1𝑒−𝐵1𝑣)1/𝑉1] [

1 − 𝐴2

(1 + 𝑄2𝑒𝐵2𝑣)1/𝑉2]

+ [𝐴3 +𝐾3 − 𝐴3

(1 + 𝑄3𝑒−𝐵3𝑣)1/𝑉1] [

1 − 𝐴4

(1 + 𝑄3𝑒𝐵3𝑣)1/𝑉3]

Results were obtained by utilising the generalised reduced gradient algorithm to minimise

squared log errors, with 𝑣 ranging from -20 to 20.

Given the nonlinearity of the equation, it is possible for multiple local optima to provide

very good fits to the data. In order to limit the number of potential good solutions, all

parameter values were constrained to be nonnegative. For the ‘lognormal’ distribution the

second component could be dropped with little loss in accuracy. Parameter estimates are

given in table XX, and proportional errors, defined as 𝑒𝑠𝑡𝑖𝑚𝑎𝑡𝑒−𝑟𝑒𝑎𝑙

𝑟𝑒𝑎𝑙 are given in figure XX.

Note that results for 𝑇0𝑣 of very high or low order have a simple asymptotic solution. For

high order, we have

lim𝑣→∞

𝑇0𝑣 =ln

max (x)�̅�

𝑣 − 1

whereas for low order we have

0.25

0.5

1

2

4

8

-20 -15 -10 -5 0 5 10 15 20

v

Lognormal

Floor

Pareto

Pareto floor

lim𝑣→∞

𝑇0𝑣 =ln

min (x)�̅�

1 − 𝑣

In general, the larger the sample size, the lower is the accuracy of the above solution for any

absolute value of 𝑣 – as sample size increases the minimum or maximum of the sample

becomes a smaller share of the total sample, slowing convergence.

Table XX: Estimation results

Lognormal Floor Pareto Pareto floor

A1 0.225089 0.478194 1.271327 1.274207

K1

Q1 1273.318757 13.629385 616.082517 975.051957

B1 0.785857 0.515954 2.605140 2.772278

V1 25.338319 2.728398 0.947121 1.103590

A2

Q2 150.612168 117.857593 71.512212 83.636859

B2 0.520878 0.245957 40.802248 39.194021

V2 16.473030 6.915707 319.494452 307.555735

A3

0.028754 0.088375 0.035019

K3

0.082700 7.197261 2.031272

Q3

0.015498 1240.261972 815.529641

B3

0.053824 1.470661 1.488404

V3

0.026810 8.059648 7.193594

K4

0.029854 0.107787

Q4

0.146614 0.210576 4.942082

B4

0.149080 0.093368 0.183569

V4

0.265325 0.042719 0.577874

Ln(1-R^2) -8.90480811 -15.066482 -10.3427854 -10.804816

Root mean squared error 0.07% 0.02% 0.30% 0.32%

Figure XX: Errors

7 Conclusion

[standard summary to go here]

8 Appendix A: Derivation of a two tailed inequality averse SWF

We proceed from the following value function:

𝛿𝑉

𝛿𝑌∝ 𝑌− 1 − 𝑘1 (

𝑌

�̅�)

2

which implies the following individual value function

𝑉 = 𝑘

[ 𝑌1−휀1 − 1

1 − 휀1− 𝑘1

(𝑌�̅�)1+휀2

− 1

1 + 휀2+ 𝑐

]

Then mean value is given by

�̅� = 𝑘 [[(1 − 𝐴1)�̅�]

1−휀1 − 1

1 − 휀1− 𝑘1

[(1 − 𝐴2)]1+휀2 − 1

1 + 휀2+ 𝑐]

Which under the no inequality condition reduces to

-2.50%

-2.00%

-1.50%

-1.00%

-0.50%

0.00%

0.50%

1.00%

1.50%

2.00%

2.50%

-20 -15 -10 -5 0 5 10 15 20

Lognormal

Floor

Pareto

Pareto floor

�̅� = 𝑘 [�̅�

1−휀1 − 1

1 − 휀1+ 𝑐]

If we define welfare in Atkinson terms as 𝑊$ = �̅�(1 − 𝐴3) we have

�̅� = 𝑘 [[(1 − 𝐴3)�̅�]

1−휀1 − 1

1 − 휀1+ 𝑐]

then we have:

𝐴3 = 1 −[1 + [

�̅�𝑘

− 𝑐] (1 − 휀1)]1/(1− 1)

�̅�

Or in expanded form

𝐴3 = 1 −

[1 + [[(1 − 𝐴1)�̅�]

1−휀1 − 11 − 휀1

− 𝑘1[(1 − 𝐴2)]

1+휀2 − 11 + 휀2

] (1 − 휀1)]

1/(1− 1)

�̅�

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