Lorenz curve

Lorenz curve
Reidpath · Public domain · source
Name	Lorenz curve
Field	Vilfredo Pareto, Max O. Lorenz
Introduced	1905
Related	Gini coefficient, Pareto distribution, Atkinson index

Contents

Lorenz curve The Lorenz curve is a graphical representation used to depict distributional inequality within a population. It was introduced by Max O. Lorenz and has been widely applied by scholars and institutions such as Vilfredo Pareto, Simon Kuznets, World Bank, International Monetary Fund, and Organisation for Economic Co-operation and Development to analyze income, wealth, and other resource allocations. The curve underpins many measures including the Gini coefficient, Theil index, and Atkinson index and is cited in work by researchers affiliated with Harvard University, London School of Economics, University of Chicago, University of Oxford, and Stanford University.

Definition and interpretation

The Lorenz curve plots the cumulative share of a resource against the cumulative share of a population, enabling visual comparison across groups such as those studied by United Nations, World Health Organization, European Commission, Federal Reserve System, and Bank for International Settlements. It is interpreted alongside benchmarks derived from thinkers like John Maynard Keynes, Adam Smith, Karl Marx, Joseph Stiglitz, and Paul Krugman, and is used in policy discussions by bodies including United States Department of the Treasury, UK Treasury, German Federal Ministry of Finance, International Labour Organization, and Organisation for Economic Co-operation and Development. The area between the Lorenz curve and the line of equality is commonly converted into inequality measures referenced in reports by Brookings Institution, Peterson Institute for International Economics, National Bureau of Economic Research, International Monetary Fund, and World Bank.

Mathematically, the Lorenz curve L(p) can be defined for a nonnegative distribution with cumulative distribution functions studied in mathematical economics by Paul Samuelson, Kenneth Arrow, Amartya Sen, Anthony Atkinson, and Angus Deaton. For population quantile p in [0,1], L(p) = (1/μ) ∫_0^p F^{-1}(t) dt where μ is the mean and F^{-1} is the quantile function; similar formulations appear in works by Milton Friedman, Robert Solow, Gary Becker, James Tobin, and Franco Modigliani. Discrete sample versions rely on order statistics and summations used in empirical studies at National Bureau of Economic Research, Institute for Fiscal Studies, IZA Institute of Labor Economics, Centre for Economic Policy Research, and CEPR. Transformations and normalizations connect L(p) to indices developed by Corrado Gini, Gunnar Myrdal, Simon Kuznets, Anthony Atkinson, and Erik Thorbecke.

Key properties include nondecreasing convexity and boundary conditions L(0)=0 and L(1)=1, properties discussed in theoretical treatments by Kenneth Arrow, Amartya Sen, Anthony Atkinson, John Nash, and Frank Ramsey. The Lorenz curve yields the Gini coefficient, defined as twice the area between the line of equality and the curve, a concept introduced by Corrado Gini and used by OECD, World Bank, United Nations Development Programme, International Labour Organization, and European Central Bank. Related measures such as the Theil index and Atkinson index were developed by Henrik Theil, Anthony Atkinson, Amartya Sen, John Rawls, and Kenneth Arrow; inequality decompositions citing Simon Kuznets, Angus Deaton, Thomas Piketty, Emmanuel Saez, and Gabriel Zucman build on Lorenz properties. Stochastic dominance and majorization concepts associated with John von Neumann, Oskar Morgenstern, Alfred Marshall, Vilfredo Pareto, and Maurice Allais relate to Lorenz ordering.

Empirical estimation uses sample data and techniques promoted by researchers at National Bureau of Economic Research, Statistics Canada, Office for National Statistics, Eurostat, and U.S. Census Bureau; methods include kernel smoothing, bootstrap inference, and jackknife variance estimation outlined by Bradley Efron, Carl Friedrich Gauss (historical methods), John Tukey, Leo Breiman, and Donald Rubin. Microdata applications appear in studies by Thomas Piketty, Emmanuel Saez, Anthony Atkinson, Angus Deaton, and institutions like World Bank, International Monetary Fund, United Nations, OECD, and European Commission. Survey design, top-coding adjustments, and tax-record linkage techniques referenced in work by Alan Auerbach, Joel Slemrod, Richard Murphy, Raj Chetty, and James Poterba affect Lorenz estimates. Policy evaluation contexts include analyses produced by Brookings Institution, American Enterprise Institute, Center on Budget and Policy Priorities, Cato Institute, and Urban Institute.

Classic empirical graphs compare Lorenz curves for countries and periods featured in reports by World Bank, OECD, United Nations Development Programme, International Monetary Fund, and World Inequality Lab led by Thomas Piketty and Emmanuel Saez. Sectoral examples appear in studies of wealth distribution by Gabriel Zucman, labor income by David Autor, human capital by James Heckman, and firm-level productivity by Zvi Griliches. Graphical techniques draw on visualization best practices from Edward Tufte, William Cleveland, Hadley Wickham, Leland Wilkinson, and John Tukey and are implemented in software by teams at R Project, Python Software Foundation, StataCorp, SAS Institute, and Mathematica.

Extensions include multidimensional Lorenz-type tools studied by Amartya Sen, Michael Shorrocks, Frank Cowell, Bernard van Praag, and John Foster; concentration curves and generalized Lorenz curves are used by Anthony Atkinson, Alan Auerbach, Henrik Theil, Joseph Stiglitz, and James Meade. Applications in welfare economics, social choice, and public finance cite Kenneth Arrow, Amartya Sen, John Rawls, James Mirrlees, and Tony Atkinson. Spatial and network generalizations relate to research by Paul Krugman, Daron Acemoglu, Esther Duflo, Abhijit Banerjee, and Gariani Acemoglu (note: see individual works for correct authorship), while stochastic and dynamic versions appear in work by Robert Lucas Jr., Edmund Phelps, Christopher Sims, Robert Shiller, and Olivier Blanchard.

Criticisms highlight that Lorenz curves compress distributional detail into a single curve, limiting identifiability as noted by Anthony Atkinson, Amartya Sen, Thomas Piketty, Emmanuel Saez, and Gabriel Zucman. Concerns about data quality, tax evasion, and offshore wealth are raised in analyses by Gabriel Zucman, Raymond Baker, Pascal Saint-Amans, Richard Murphy, and James Henry. Methodological debates over welfare interpretation and normative weighting invoke John Rawls, Kenneth Arrow, Amartya Sen, Tony Atkinson, and Daniel Kahneman. Empirical sensitivity to sampling, top-coding, and undercoverage is discussed in work from United Nations, World Bank, OECD, U.S. Census Bureau, and Eurostat.

Category:Income distribution