The Atkinson index measures income (or consumption) inequality by asking how much average income society would be willing to give up to achieve an equal distribution while keeping the same level of social welfare.
Equally distributed equivalent (EDE) income
The Atkinson framework starts with the equally distributed equivalent (EDE) income: the income level (y^{\text{EDE}}) such that, if everyone had (y^{\text{EDE}}), society would be as well off (under a chosen social welfare function) as under the actual unequal distribution.
If (\mu) is mean income, the Atkinson index is:
[ A = 1 - \frac{y^{\text{EDE}}}{\mu}. ]
- (A=0) means perfect equality (EDE equals the mean).
- A larger (A) means greater inequality (EDE falls further below the mean).
The inequality-aversion parameter
In the most common version, Atkinson uses a parameter (\varepsilon) that controls how sensitive the index is to the lower tail of the distribution.
For (\varepsilon \ne 1):
[ A_{\varepsilon} = 1 - \frac{\left(\frac{1}{n}\sum_{i=1}^{n} y_i^{1-\varepsilon}\right)^{\frac{1}{1-\varepsilon}}}{\mu}. ]
For (\varepsilon = 1):
[ A_{1} = 1 - \frac{\exp\left(\frac{1}{n}\sum_{i=1}^{n} \ln y_i\right)}{\mu}. ]
Higher (\varepsilon) places more weight on changes among low-income households, making the normative assumptions explicit.
Why economists use it
Unlike purely descriptive dispersion measures, the Atkinson index is welfare-based. This makes it useful for:
- comparing inequality under different social preferences (different (\varepsilon)),
- evaluating distributional effects of taxes, transfers, and price shocks,
- reporting “cost of inequality” as a fraction of mean income.