<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$Z^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>Z</mi> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> be a set of n-person income profiles over two time periods. The notion that a profile <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$z\in Z^n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>z</mi> <mo>∈</mo> <msup> <mi>Z</mi> <mi>n</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> exhibits higher mobility than <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$z'$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>z</mi> <mo>′</mo> </msup> </math></EquationSource> </InlineEquation> is expressed as <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$z\succsim z'$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>z</mi> <mo>≿</mo> <msup> <mi>z</mi> <mo>′</mo> </msup> </mrow> </math></EquationSource> </InlineEquation>. Cowell and Flachaire give a set of principles, stated as formal axioms, we wish <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\succsim $</EquationSource> <EquationSource Format="MATHML"><math> <mo>≿</mo> </math></EquationSource> </InlineEquation> to fulfill. Numeric measures, <i>m</i>, are sought to represent <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\succsim $</EquationSource> <EquationSource Format="MATHML"><math> <mo>≿</mo> </math></EquationSource> </InlineEquation> so that <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$z\succsim z'$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>z</mi> <mo>≿</mo> <msup> <mi>z</mi> <mo>′</mo> </msup> </mrow> </math></EquationSource> </InlineEquation> corresponds with <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$m(z)\ge m(z')$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>≥</mo> <mi>m</mi> <mrow> <mo stretchy="false">(</mo> <msup> <mi>z</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. We report that the Jensen differences of strictly convex functions are useful in constructing examples of measures that meet their first four axioms. Their fifth axiom is found incompatible with the first four.</p>

Measuring movement of incomes and income mobility

期刊论文
发表日期：

2025年9月22日

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Che Tat Ng

摘要

Let $Z^n$ $Z^{n}$ be a set of n-person income profiles over two time periods. The notion that a profile $z\in Z^n$ $z \in Z^{n}$ exhibits higher mobility than $z'$ $z^{'}$ is expressed as $z\succsim z'$ $z ≿ z^{'}$ . Cowell and Flachaire give a set of principles, stated as formal axioms, we wish $\succsim $ $≿$ to fulfill. Numeric measures, m, are sought to represent $\succsim $ $≿$ so that $z\succsim z'$ $z ≿ z^{'}$ corresponds with $m(z)\ge m(z')$ $m (z) \geq m (z^{'})$ . We report that the Jensen differences of strictly convex functions are useful in constructing examples of measures that meet their first four axioms. Their fifth axiom is found incompatible with the first four.