|
Part I Lorenz Curves, Orders and Redistribution |
|
|
|
|
3 | (6) |
|
2 The Generalized Inverse of Distribution Functions |
|
|
9 | (20) |
|
2.1 A "Gentle" Derivation of the Generalized Inverse |
|
|
9 | (3) |
|
2.2 Properties of the Generalized Inverse Distribution Function |
|
|
12 | (5) |
|
2.3 Generalized Inverse and Order Relations |
|
|
17 | (1) |
|
|
18 | (4) |
|
2.5 Generalized Inverse and Computations |
|
|
22 | (7) |
|
2.5.1 Generalized Inverses from Generalized Inverses |
|
|
22 | (1) |
|
2.5.2 Generalized Inverse of the Generalized Inverse |
|
|
23 | (3) |
|
2.5.3 Generalized Inverse and Expectation Values |
|
|
26 | (2) |
|
|
28 | (1) |
|
3 Lorenz Densities and Lorenz Curves |
|
|
29 | (26) |
|
3.1 Introduction of Lorenz Densities and Lorenz Curves |
|
|
29 | (5) |
|
3.2 Some Properties of Lorenz Densities and Lorenz Curves |
|
|
34 | (3) |
|
|
37 | (4) |
|
3.3.1 Approximations Based on Distribution Functions |
|
|
37 | (4) |
|
3.3.2 Related Approximations |
|
|
41 | (1) |
|
3.4 Characterizations of Lorenz Densities and Lorenz Curves |
|
|
41 | (3) |
|
3.5 Lorenz Curves for all Distributions |
|
|
44 | (4) |
|
3.6 Lorenz Curves from Lorenz Curves |
|
|
48 | (1) |
|
3.6.1 Lorenz Curves from Two or More Lorenz Curves |
|
|
48 | (1) |
|
3.6.2 Lorenz Curves from One Lorenz Curve |
|
|
48 | (1) |
|
3.7 Lorenz Curves for Finite and Infinite Variance Distributions |
|
|
49 | (1) |
|
3.8 The Gini Index and Other Inequality Indices |
|
|
50 | (3) |
|
|
50 | (2) |
|
3.8.2 Other Inequality Indices |
|
|
52 | (1) |
|
3.9 Lorenz Curves in Higher Dimensions |
|
|
53 | (2) |
|
|
53 | (2) |
|
4 Lorenz Curves and Partial Orders |
|
|
55 | (28) |
|
4.1 Partial Orders for Lorenz Curves |
|
|
55 | (2) |
|
4.2 Lorenz Order and Majorization |
|
|
57 | (5) |
|
4.3 Lorenz Order and Integral Orders |
|
|
62 | (7) |
|
4.3.1 Lorenz Curves and Orders |
|
|
62 | (6) |
|
4.3.2 Lorenz Densities and the Increasing Convex Order |
|
|
68 | (1) |
|
4.4 Utility Functions and Lorenz Curves |
|
|
69 | (14) |
|
4.4.1 A Representation Formula |
|
|
69 | (5) |
|
4.4.2 Modifications of the Representation |
|
|
74 | (1) |
|
4.4.3 Utility of Consumption |
|
|
75 | (6) |
|
|
81 | (2) |
|
5 Transfer and Distribution Approximation |
|
|
83 | (12) |
|
5.1 Convergence in Distribution |
|
|
83 | (2) |
|
5.2 Extension of Pigou--Dalton Transfers |
|
|
85 | (4) |
|
5.3 Towards Strengthening Convergence in Distribution |
|
|
89 | (2) |
|
5.4 A Probabilistic Version of Pigou--Dalton Transfers |
|
|
91 | (1) |
|
5.5 Taxation and Transfer |
|
|
92 | (2) |
|
5.6 Further Order Relation |
|
|
94 | (1) |
|
|
94 | (1) |
|
6 Societal Utility and the Atkinson Theorem |
|
|
95 | (8) |
|
6.1 Pigou-Dalton Transfers: Revisited |
|
|
96 | (2) |
|
6.2 Pigou-Dalton Transfers and Distribution Approximations |
|
|
98 | (1) |
|
6.3 Economic Interpretation |
|
|
99 | (4) |
|
|
100 | (3) |
|
Part II Lorenz Curves and Models |
|
|
|
7 Pareto Distribution, Self-similarity and Empirics |
|
|
103 | (26) |
|
7.1 Self-similarity of Lorenz Curves |
|
|
106 | (10) |
|
7.1.1 Pure Self-similarity |
|
|
107 | (2) |
|
7.1.2 Gini Self-similarity |
|
|
109 | (2) |
|
7.1.3 Median Self-similarity |
|
|
111 | (5) |
|
|
116 | (5) |
|
|
116 | (2) |
|
7.2.2 Alternative Transformation |
|
|
118 | (3) |
|
7.3 Plato's Concept of Social Justice |
|
|
121 | (1) |
|
7.4 Empirics of Income Distributions |
|
|
122 | (7) |
|
7.4.1 Best Fit Values for Nations |
|
|
122 | (3) |
|
7.4.2 Mean Value and Median in Poverty Assessment |
|
|
125 | (1) |
|
7.4.3 Conclusion: The General Picture |
|
|
126 | (2) |
|
|
128 | (1) |
|
8 Proportionality-Induced Distribution Laws |
|
|
129 | (12) |
|
8.1 Geometrical Interpretation |
|
|
129 | (2) |
|
8.2 The Main Differential Equations of the Equity Calculus |
|
|
131 | (1) |
|
8.3 Closed Form Solutions |
|
|
132 | (2) |
|
|
134 | (1) |
|
8.5 Further Differential Equations of the Equity Calculus |
|
|
135 | (5) |
|
8.5.1 Fractured Exponents |
|
|
135 | (1) |
|
8.5.2 Proportionality Functions |
|
|
136 | (1) |
|
|
137 | (1) |
|
8.5.4 Averages over Other Income Ranges |
|
|
138 | (2) |
|
|
140 | (1) |
|
8.6 System of Proportionality Laws |
|
|
140 | (1) |
|
|
140 | (1) |
|
9 Preferences and Coalitions |
|
|
141 | (32) |
|
|
141 | (1) |
|
|
142 | (3) |
|
|
142 | (1) |
|
|
143 | (1) |
|
|
144 | (1) |
|
|
145 | (17) |
|
|
145 | (2) |
|
9.3.2 Identifying Minimum Loss Coalition Partners |
|
|
147 | (3) |
|
|
150 | (5) |
|
9.3.4 Situation After Compensation but Before Complete Redistribution |
|
|
155 | (2) |
|
9.3.5 Situation After Complete Redistribution |
|
|
157 | (3) |
|
9.3.6 Varying Majority Levels |
|
|
160 | (2) |
|
9.4 Other Income Distributions |
|
|
162 | (8) |
|
9.4.1 One-Parametric Lorenz Curves |
|
|
162 | (4) |
|
9.4.2 Two-Parametric Lorenz Curves |
|
|
166 | (2) |
|
9.4.3 Three-Parametric and Other Lorenz Curves |
|
|
168 | (2) |
|
|
170 | (3) |
|
Appendix: Bifurcation Parameters for One-Parametric Lorenz Curves---General Case |
|
|
170 | (1) |
|
|
171 | (2) |
Index |
|
173 | |