Equity Principles Expository Note

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Diagrammatic Representation of Equity Principles∗

Ram Sewak Dubey†

June 6, 2011

The purpose of this note is to explain the equity notions commonly studied in the social

choice literature through a diagram. Broadly speaking the equity principles are categorized

in two groups, procedural and consequentialist. Suppes-Sen equity which is also known

as Anonymity or (finite) Permutation Invariance is an example of procedural equity. Ham-

mond Equity, Pigou-Dalton Equity, Strong Equity and Hammond Equity for the Future are

examples of the consequentialist equity.

The note is organized as follows. In Section 1, we define the equity principles. In Section

2, we explain these definitions with the help of a diagram.

1 DefinitionsLet R be the set of real numbers and N the set of positive integers. Suppose Y ⊂R is the set

of all possible utilities that any generation can achieve. Then X = Y N is the set of all possible

utility streams. If xn ∈ X , then xn = ( x1, x2, · · · ), where, for all n ∈ N, xn ∈ Y represents

the amount of utility that the generation of period n earns. For all y, z ∈ X , we write y ≥ z if

yn ≥ zn, for all n ∈ N; we write y > z if y ≥ z and y = z; and we write y >> z if yn > zn for

all n ∈ N.

Definition. Hammond Equity (HE): If x, y ∈ X , and there exist i, j ∈N, such that y j > x j >

xi > yi while yk = xk for all k ∈ N\ {i, j}, then x y.

Definition. Strong Equity (SE): If x, y ∈ X , and there exist i, j ∈ N, such that y j > x j >

xi > yi while yk = xk for all k ∈ N\ {i, j}, then x ≻ y.

Definition. Pigou-Dalton Equity (PDE): For all x, y ∈ X , if there exists i, j ∈ N such that

(i) yi < xi x j < y j and xi − yi = y j − x j; and (ii) xk = yk for all k ∈N\ {i, j}, then x ≻ y.

Definition. Suppes Sen Equity or Anonymity Axiom (AN): For all x, y ∈ X , if there exist

i, j ∈ N such that xi = y j and x j = yi, and for every k ∈ N\ {i, j}, xk = yk , then x ∼ y.

∗I thank Tapan Mitra for helpful conversations.†Department of Economics, Uris Hall, Cornell University, Ithaca, NY 14853, USA;

Email : [email protected]

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Definition. Hammond Equity for the Future (HEF): If x, y ∈ X , are such that x ≡ {a,d ,d , · · · }and y ≡ {b,c,c, · · · } and d < c < b < a, then x ≻ y1.

2 Explanatory Figure

We combine all the equity notions in one diagram. Observe that in the definitions of HE,

SEA, PD and AN, only a pair of elements of sequences x and y are different. Therefore

it is possible to depict the ranking of such sequences by showing the relationship between

the corresponding pair of elements of two sequences. This is useful as we can show the

operation of these equity principles in two dimensional diagram. Also observe that in HEF,

the sequences which are comparable contain only two distinct elements each. Therefore, itis possible to show HEF ranking also in a two dimensional diagram.

2.1 Hammond Equity

Consider allocation A ≡ (a,d ) which is below the 45◦ line. Set of allocations weakly pre-

ferred to A are all the points contained in the interior of the △ ADE . Take any point, say

P ≡ (b,c), in the interior of the △ ADE . Then a > b > c > d implies (b,c) (a,d ).

Set of allocations weakly worse than A are all the points contained in the interior of

the blue rectangle. Take any point, say P ≡ (b,c), in the interior of the rectangle. Then

b > a > d > c implies (b,c) (a,d ).Similarly consider allocation B which is above the 45◦ line. Set of allocations weakly

preferred to B are all the points contained in the interior of the △ BDE . Set of allocations

weakly worse than B are all the points contained in the interior of the red rectangle.

2.2 Strong Equity

Set of allocations strictly preferred to A are all the points contained in the interior of the

△ ADE . Take any point, say P ≡ (b,c), in the interior of the △ ADE . Then a > b > c > d

implies (b,c) ≻ (a,d ). Set of allocations strictly worse than A are all the points contained

in the interior of the blue rectangle. Take any point, sayP

≡ (b,c

), in the interior of therectangle. Then b > a > d > c implies (b,c) ≺ (a,d ).

Set of allocations strictly preferred to B are all the points contained in the interior of the

△ BDE . Set of allocations strictly worse than B are all the points contained in the interior of

the red rectangle.

1This equity notion was introduced in a paper by Asheim and Tungodden (2004).Reference for the other

equity notions are Hammond (1976) (for Hammond Equity), d’Aspremont and Gevers (1977) (for Strong

Equity), Asheim et al. (2007) (we have followed this paper for the definitions in this note).

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2.3 Pigou-Dalton EquitySet of allocations strictly preferred to A are all the points on the diagonal AC excluding A

itself. Take any point, say P ≡ (b,c), on the line segment AC . Observe that the sum of the

two coordinates is same on the entire line segment GF ,(since slope of this line is −45◦). So,

b + c = a + d or a − b = c − d . Then a > b > c > d implies (b,c) ≻ (a,d ). Set of allocations

strictly worse than A are all the points on the diagonal AF excluding A itself. Take any point,

say P ≡ (b,c), on the line segment AF . Then b > a > d > c implies (b,c) ≺ (a,d ).

Set of allocations strictly preferred to B are all the points on the diagonal BC which does

not include B itself. Set of allocations strictly worse than B are all the points on the diagonal

BG which does not include B itself.

2.4 Suppes-Sen Equity or Anonymity Axiom

Consider allocation A which is below the 45◦ line. Point B (the mirror image of point A

with respect to the 45◦ line) is the sole point which is comparable to A and both points are

indifferent to each other.

2.5 Hammond Equity for Future

Let s = 1 and t = 2, then the only points which are comparable are those lying below the 45◦

line. Point A refers to an allocation {a,d ,d , · · · }, with xs being consumption in period 1 and xt being the consumption in each of the remaining periods. Consider allocation A which is

below the 45◦ line. Set of allocations strictly preferred to A are all the points contained in

the interior of the triangle ADE . Take any point, say P ≡ (b,c), in the interior of the △ ADE .

Then a > b > c > d implies (b,c) ≻ (a,d ).

Set of allocations strictly worse than A are all the points contained in the interior of

the blue rectangle. Take any point, say P ≡ (b,c), in the interior of the rectangle. Then

b > a > d > c implies (b,c) ≺ (a,d ). Notice the HEF lacks symmetry, in the sense that

points above 45◦ line are not comparable by HEF.

References

G. B. Asheim and B. Tungodden. Resolving distributional conflicts between generations.

Economic Theory, 24(1):221–230, 2004.

G.B. Asheim, T. Mitra, and B. Tungodden. A new equity condition for infinite utility streams

and the possibility of being paretian. In J. Roemer and K. Suzumura, editors, Intergener-

ational Equity and Sustainability, volume 143, pages 55–68. (Palgrave) Macmillan, 2007.

C. d’Aspremont and L. Gevers. Equity and informational basis of collective choice. Review

of Economic Studies, 44(2):199–209, 1977.

P. J. Hammond. Equity, Arrow’s Conditions, and Rawls Difference Principle. Econometrica,

44(4):793–804, 1976.

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xt

xsO

A

B

C

D

E

F

G

45◦

45◦

45◦

AN: A ∼ B

SE(HE): A ≺() ∆ ADE; A ≻() Blue Rectangle

SE(HE): B ≺() ∆ BDE; B ≻() Red Rectangle

PDE: A ≺ AC; B ≺ BC; A ≻ AF; B ≻ BG

HEF: A ≺ ∆ ADE

Figure 1 : Commonly used equity notions in Social Choice Literature4

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Equity Principles Expository Note