Measuring Trends in U.S. Income Inequality

This book is the culmination of roughly seven years of joint research be­ tween us. We have both been interested in income inequality measurement for a considerably longer period of time. One author (Ryu) has a back­ ground in physics. While he was working on his Ph. D. in Physics at M. I. T. he became acquainted with Robert Solow. Professor Solow introduced Ryu to economics. After finishing his Ph. D. in physics, Ryu went on to the Uni­ versity of Chicago where Arnold Zellner guided him to a dissertation on using orthonormal basis and maximum entropy as estimation methods in econometric applications. The precise definition and examples of orthonormal basis (ONB) and maximum entropy (ME) methods will be given in the book. As it turns out, a natural application of these methods is the study of income distribution. Professor Zellner suggested that Ryu look at some of my joint work with Robert Basmann on functional forms of Lorenz curves as one starting place to do his own research. Ryu requested some of our data and asked for several of our papers with the express pur­ pose of introducing functional forms of Lorenz curves that Ryu felt would do a better job of approximating the empirical Lorenz curve. Thus, our first introduction was essentially one of Ryu trying to invent a better mousetrap. The interested reader can review the results given in Basmann et al. (1990) and Chapter Four of this book to see if Ryu succeeded.

1 Introduction.- 1.1 Introduction.- 1.2 A Brief Review of the Literature.- 1.3 An Overview of Recent Trends in Income Inequality in the U.S..- 1.4 The Plan of the Book.- 2 The Maximum Entropy Estimation Method.- 2.1 Review of Jaynes’ (1979) Concentration Theorem.- 2.2 Determination of a Maximum Entropy Density Function Given Known Moments.- 2.3 Estimation of the Maximum Entropy (ME) Density Function When Moments are Unknown.- 2.3.1 Estimation of the Exponential Density Function (N = 1).- 2.3.2 Estimation of the Normal Density Function (N = 2).- 2.4 Estimation of the Exponential Density Function for N>2.- 2.5 Asymptotic Properties of the Maximum Entropy Density function.- 2.6 Maximum Entropy Estimation of Univariate Regression Functions.- 2.7 Model Selection for Maximum Entropy Regression.- 3 Capabilities and Earnings Inequality.- 3.1 Introduction.- 3.2 The Theory.- 3.2.1 The Earnings Potential Accumulation Model.- 3.2.2 A Semiparametric Transformation from Earnings to Capability.- 3.2.3 The Loglinear Transformation function.- 3.3 Empirical Results.- 3.3.1 A Transformation from Earnings to Normally Distributed Capability.- 3.3.2 A Loglinear Transformation without the Normality Assumption for Capability.- 3.4 Summary and Concluding Remarks.- Appendix 3.A: Derivation of an Earnings Distribution With Maximum Entropy Method.- 4 Some New Functional Forms for Approximating Lorenz Curves.- 4.1 Introduction.- 4.2 A Flexible Lorenz Curve with Exponential Polynomials.- 4.3 Approximation of the Empirical Lorenz Curve.- 4.4 A Comparison of Two Alternative Derivations of the Lorenz Curves.- 4.5 Choosing an Exponential Series Expansion Rather Than a Plain Series Expansion.- 4.6 About Expanding the Inverse Distribution Rather Than a Lorenz Curve in a Series.- 4.7 Orthonormal Basis Expansion for Discrete Ordered Income Observations.- 4.8 A Flexible Lorenz Curve with Bernstein Polynomials.- 4.9 Applications with Actual Data.- 4.10 Summary and Concluding Remarks.- 5 Comparing Income Distributions Using Index Space Representations.- 5.1 Introduction.- 5.2 The Theory.- 5.3 Theil’s Entropy Measure.- 5.4 Maximum Entropy Estimation of Share Functions.- 5.5 Motivation for Decomposing the Share Function Through the Legendre Functions.- 5.6 A Comparison of Index Space Analysis to Spectral Analysis.- 5.7 Empirical Results.- 5.8 Summary and Concluding Remarks.- Appendix 5.A: A Review of the Concepts of Completeness, Orthonormality, and Basis.- 6 Coordinate Space vs. Index Space Representations as Estimation Methods: An Application to How Macro Activity Affects the U.S. Income Distribution.- 6.1 Introduction.- 6.2 The Theory.- 6.3 An Index Space Representation of the Share function.- 6.4 A Comparison of the Index Space Representation with the Coordinate Space Representation.- 6.5 The Impact of Macroeconomic Variables on the Share function.- 6.6 Inequality Measures Associated with the Legendre Polynomial Expanded Share function.- 6.7 The Empirical Results.- 6.7.1 The Performance of the Lorenz Curve Expanded in an Exponential Polynomial Series.- 6.7.2 The Impact of Macroeconomic Variables on the Share function.- 6.8 Summary and Concluding Remarks.- 7 A New Method for Estimating limited Dependent Variables: An Analysis of Hunger.- 7.1 Introduction.- 7.2 Model Specification.- 7.3 Posterior Odds Ratios to Compare Alternative Regression Hypotheses.- 7.4 The Empirical Results.- 7.5 Summary and Concluding Remarks.- Author Index.