Lorenz map, inequality ordering and curves based on multidimensional rearrangements
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2022
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| _version_ | 1866913315975856128 |
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| author | Fan, Yanqin Henry, Marc Pass, Brendan Rivero, Jorge A. |
| author_facet | Fan, Yanqin Henry, Marc Pass, Brendan Rivero, Jorge A. |
| contents | We propose a multivariate extension of the Lorenz curve based on multivariate rearrangements of optimal transport theory. We define a vector Lorenz map as the integral of the vector quantile map associated with a multivariate resource allocation. Each component of the Lorenz map is the cumulative share of each resource, as in the traditional univariate case. The pointwise ordering of such Lorenz maps defines a new multivariate majorization order, which is equivalent to preference by any social planner with inequality averse multivariate rank dependent social evaluation functional. We define a family of multi-attribute Gini index and complete ordering based on the Lorenz map. We propose the level sets of an Inverse Lorenz Function as a practical tool to visualize and compare inequality in two dimensions, and apply it to income-wealth inequality in the United States between 1989 and 2022. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2203_09000 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Lorenz map, inequality ordering and curves based on multidimensional rearrangements Fan, Yanqin Henry, Marc Pass, Brendan Rivero, Jorge A. Econometrics Methodology We propose a multivariate extension of the Lorenz curve based on multivariate rearrangements of optimal transport theory. We define a vector Lorenz map as the integral of the vector quantile map associated with a multivariate resource allocation. Each component of the Lorenz map is the cumulative share of each resource, as in the traditional univariate case. The pointwise ordering of such Lorenz maps defines a new multivariate majorization order, which is equivalent to preference by any social planner with inequality averse multivariate rank dependent social evaluation functional. We define a family of multi-attribute Gini index and complete ordering based on the Lorenz map. We propose the level sets of an Inverse Lorenz Function as a practical tool to visualize and compare inequality in two dimensions, and apply it to income-wealth inequality in the United States between 1989 and 2022. |
| title | Lorenz map, inequality ordering and curves based on multidimensional rearrangements |
| topic | Econometrics Methodology |
| url | https://arxiv.org/abs/2203.09000 |