Lecture 1 – Inequality and Principles of Measurement
CES Lectures, Fall 2004
Michael Hoy
Preliminaries
Useful Surveys:
• Stephen Jenkins (1991) – short and accessible
• Frank Cowell (2000) – more comprehensive
• Peter J. Lambert (2001) – comprehensive book with
several chapters on equity and taxation
Free Software:
• DAD – go to www.pep-net.org/
Structure of the Lectures
• Basic principles I + applications
• Basic principles II + applications
• Comparing distributional impact of alternative tax
schedules
1
Intended goals of this lecture series:
• Provide a background to underlying principles of
inequality measurement in a way that is useful for the
“nonspecialist” practitioner
• Perhaps entice someone to look more deeply into
principles of inequality and how this relates to
measurement in order to further theory of inequality
measurement
• Provide illustrative examples on applications (focus on
impact of government policies)
• Final lecture will be devoted to comparisons of
income tax schedules with a special emphasis on
recent reforms of tax flattening
2
Why study distributional impacts of economic/policy
changes?
• Poverty of efficiency analysis (growth issues
excepted)
• People care about distributional issues
• Make economic analysis more relevant to policy
making
Why should “nonspecialists” become more involved in
distributional analysis?
• Improve link between behavioural and normative
economics
• Thinking about distributional issues widens one’s
conceptual framework
• Help specialists do more relevant work
3
Lorenz Curves and Quantile Shares
Simplistically speaking, the goal of inequality analysis is to
rank income distributions according to the inequality or
distributive inequity they display.
Discrete Case:
Let x = (x1, x2, …, xn) and y = (y1, y2, …, yn) be two
(ordered-nonincreasing) income vectors with the same
mean income.
Continuous Case:
Compare two income distributions with the same mean,
x~F(x), x~G(x), x ∈ ℜ + +
Goals of Inequality Analysis
• How to compare inequality in x vs. y or F vs. G.
• Can we (should we) say anything about the degree of
inequality in a cardinally meaningful way?
• What types of comparisons could be of interest?
4
Illustrative Example:
• A = (8, 13, 18, 25, 36)
• B = (6, 15, 15, 23, 41)
• C = (7, 14, 19, 24, 36)
How would you rank these from “most equally distributed
income” to “least equally distributed income”?
Note – some income gaps are larger, some smaller between
any pair of the above distributions.
However, most emphasis is on relative incomes and types
of transfers (progressive vs. regressive).
Pigou-Dalton principle of transfers: If an income
transfer from a poorer to a richer person in the income
distribution is made, inequality rises.
This principle has a status in income distribution analysis
almost as important as the Pareto principle in welfare
economics.
5
B = A + 3 regressive transfers
i.e., $2 from person 1 to 2
$3 from person 3 to 4
$5 from person 4 to 5
Therefore, by the Pigou-Dalton principle of transfers,
incomes in B are less equally distributed than incomes in
distribution A.
It is not so straightforward to compare the degree of
inequality in A vs. C.
C = A + one regressive transfer + one progressive transfer
i.e., $1 from person 1 to 2
$1 from person 4 to 3
6
Quintile Shares (%)
Quintile
A
B
C
Cum %
Cumulative
Shares (%)
A
B
C
Poorest
1
2
3
4
5
Cum.
8
13
18
25
36
100
6
15
15
23
41
100
7
14
19
24
36
100
.2
.4
.6
.8
1.0
8
21
39
64
100
6
21
36
59
100
7
21
40
64
100
*NOTE: The slope of the Lorenz curve at cumulative share point k
(where we are cumulating the kth percent poorest person’s income)
is in fact that person’s income xk divided by mean income µ (i.e.,
xk/µ). This can be a useful fact simply to help in sketching Lorenz
curves but also to understand related inequality measures, etc.
So, for distribution B, the Lorenz curve slope remains constant (at
15/20 = 0.75) from k=0.2 to k = 0.6.
7
Graph of Lorenz Curves
8
Lorenz Curves and Lorenz Criterion
In fact, whenever one distribution (B) can be generated
from another distribution (A) through a series of regressive
transfers, the Lorenz curve for A will lie above (weakly at
least) than for B at all points.
In such cases it is common to say that distribution A is
“unambiguously more equal” than is B and we say that
distribution “A Lorenz dominates distribution B” (A >L B).
This reflects the general acceptance of the Pigou-Dalton
principle as central to inequality analysis.
The Lorenz curves for A and C, on the other hand,
intersect. However, we will see later that when Lorenz
curves intersect, not all is necessarily lost.
Cases of intersecting Lorenz curves are empirically
important.
Eg., Kakwani (1984) found Lorenz crossings in more than
30% of 2,556 possible pairwise comparisons between 72
countries.
Also, many OECD countries (and others) have recently
introduced tax flattening reforms and these typically lead to
intersecting Lorenz curves for after-tax income
distributions (see Davies and Hoy, 1995, 2002).
9
EXAMPLE
• Impact of risk categorization (public vs. private
insurance)
Distributional Impact of Community Rating Legislation in
Insurance (see M. Hoy, 1984).
R-S-W Model:
2 states: Loss: x1 = x0 – L
No Loss: x2 = x0
2 risk types; h, l: ph > pl
Expected Loss: µh = phL , µl = plL
Proportions of each type: q h , q l
Full coverage price (actuarially fair):
A = q h µ h + ql µ l
Let r be rate of coverage (0 < r < 1).
10
If no categorization (B):
If Loss: x1 = x0 –rA – (1-r)L
No Loss: x2 = x0 –rA
x1 < x2
Let S1B, S2B be set (and number) of individuals receiving
income x1, x2, respectively
• See Lorenz curve.
Suppose categorization is allowed (A):
2 (imperfect) risk categories – H, L
Prices: AH > A > AL
→ obtain 4 income groups:
S1A : Class H, pay AH, suffer loss L
S2A : Class L, pay AL, suffer loss L
S3A : Class H, pay AH, suffer no loss
S4A : Class L, pay AL, suffer no loss
11
Formal Definition of the Lorenz Curve
For a discrete distribution x = (x1, x2, …, xn):
j
x
 j
L  = ∑ i , where 1 ≤ j ≤ n
 n  i =1 X
n
and X = ∑ xi is total income.
i =1
For a continuous distribution F(x), assume x ∈ [ x , x ] , and
frequency density f(x) is nonzero throughout the income
range. Then there is just one income level y with rank p,
which satisfies p=F(y), the income of the first 100p percent
of income recipients is
min
xmax
y
N
∫ xf ( x)dx
xmin
max
and total income is
N
∫ xf ( x)dx = Nµ
xmin
Hence, the Lorenz curve L(p) is defined by
y
p = F ( y ) ⇒ L( p ) =
∫
xmin
xf ( x)dx
µ
,
0 < p <1
We will assume throughout that xmin > 0.
12
Lemma
For 0 < p < 1, L(p) = y/µ and L (p) = 1/[µf(y)] > 0.
Proof: Differentiate L(p) using the chain rule
dL d [ L( p) / dy ] yf ( y ) / µ y
=
=
=
µ
dp
dp / dy
f ( y)
Now differentiate again for the second result, noting dp/dy
= dF(y)/dy=f(y).
So the Lorenz curve is upward sloping and convex.
Moreover, L(p) = y/µ = 1 at the percentile point
corresponding to mean income (i.e., when the pth person’s
income is µ).
At this point p the slope of the Lorenz curve is equal to the
slope of the line of perfect equality and so this marks the
point at which the Lorenz curve is furthest away from the
line of perfect inequality.
13
Graph of Lorenz Curve with Illustration of Lemma.
14
Inequality Indices
We present a formal definition of an inequality index only
for the case of continuous distributions.
Let x ∈ [ x min , x max ] , be an interval of real-valued
Let f(x) be a strictly positive, continuous, relative frequency
distribution from a space Ω over incomes.
A “valid” inequality index is a function I:Ω → R that rises
when there is regressive transfer.
15
Two commonly used inequality indices based on the
geometry of the Lorenz curve:
Schutz Index (S):
(1)
S = F(µ) – L[F(µ)]
(i.e., maximum distance between L(p) and line of complete
equality)
Gini Index (G):
1
(2a)
G = 1 − 2 ∫ L( p )dp
0
(i.e., area A divided by A+B indicated on the above graph,
or twice area A)
16
The Gini index is probably the most used inequality index
by practitioners.
Also, Gini index is a normalization of the difference
between any two randomly drawn persons’ incomes from
the distribution.
(2b)
G=∑
i
∑
j
| xi − x j |
2N 2 µ
From a normative perspective neither of these indices is
compelling.
17
Other commonly used inequality indices include:
Variance:
n
2
(3) V = (1 / n)∑ ( xi − µ )
i =1
Coefficient of Variation:
(4)
CV = V / µ
Logarithmic Variance:
n
(5)
n
L = (1 / n)∑ [log( xi ) − log( µ )] = (1 / n)∑ [log( xi /µ )] 2
2
i =1
i =1
Kolm family of inequality indices:
n
(6)
K (α ) = (1 / α ) ln[(1 / n)∑ exp(α ( µ − xi ))], α > 0
i =1
18
Atkinson family of inequality indices:
(7a)
n


Aε = 1 − (1 / n)∑ ( xi / µ )1−ε 
i =1


1 /(1−ε )


= 1 − exp (1 / n)∑ ln( xi / µ ),


, ε ≠ 1, ε > o
ε =1
Note that the Atkinson family (Aε) is based on the
utilitiarian welfare function using utilities:
x1−ε
u ( x) =
, ε ≠ 1, ε > 0
1− ε
u ( x) = ln( x), ε = 1
That is, the utility function is iso-elastic; i.e.,
%∆(u ' ( x)) x ⋅ u ' ' ( x)
=
= −ε
%∆ ( x)
u ' ( x)
19
Remarks:
• Inequality and welfare are inversely related
• I ≠ -W since normalizations for I are useful
• u(x) in Atkinson’s formulation is the CRRA utility
function used in risk theory
The Atkinson index and associated welfare function allow
one to develop a useful cardinal, albeit subjective, measure
of inequality - “cost of inequality”.
First, define equally distributed income, xe, by:
n
n
i =1
i =1
W = (1 / n)∑ u ( xi ) = (1 / n)∑ u ( x e ) = u ( x e )
Due to concavity of the utility function, xe < µ, mean
income.
The extent to which xe is less than µ is interpreted as the
“cost of inequality”. It can be shown with a little algebra
that Atkinson’s measure of inequality is also defined by
(7b):
1− ε
n


Aε = 1 − ( xe / µ ), xe = (1 / n)∑ xi 
i =1


1 /(1− ε )
20
This is illustrated by in the following graph, where µ - xe is
equivalent to the risk premium in risk theory and µAε is the
cost of inequality, or just Aε is the fraction of
income/welfare lost due to inequality:
21
<<Insert Figure 1>>
22
Cowell, F. A. (2000) "Measurement of Inequality" in
Handbook of Income Distribution, (ed. Atkinson, A.B. and
Bourguignon, F.), North Holland, Amsterdam
Hoy, M. (1984) “The Impact of Imperfectly Categorizing
Risks on Income Inequality and Social Welfare,” Canadian
Journal of Economics, vol. 17, no. 3, pp. 557-568.
Jenkins, S. (1991), “The Measurement of Income
Inequality,” in Economic Inequality and Poverty:
International Perspectives (ed., Lars Osberg), M.E. Sharpe
Ltd. Armonk, New York.
Lambert, P. J. (2001), The Distribution and Redistribution
of Income, pp. 302. Manchester University Press.
23