Atkinson index

Computes the Atkinson inequality index, which incorporates an explicit normative judgement about inequality aversion through the parameter epsilon. Higher epsilon gives more weight to transfers at the bottom of the distribution.

Usage

iq_atkinson(
  x,
  weights = NULL,
  epsilon = 0.5,
  na.rm = FALSE,
  ci = FALSE,
  R = 1000L,
  level = 0.95
)

Arguments

x

Numeric vector of incomes (strictly positive).

weights

Optional numeric vector of survey weights.

epsilon

Numeric. Inequality aversion parameter (> 0). Default 0.5. Common values: 0.5 (moderate), 1.0 (high), 2.0 (very high aversion).

na.rm

Logical. Remove NA values? Default FALSE.

ci

Logical. Compute bootstrap confidence intervals? Default FALSE.

R

Integer. Number of bootstrap replicates. Default 1000.

level

Numeric. Confidence level. Default 0.95.

Value

An S3 object of class "iq_atkinson" with elements:

value

Numeric. The Atkinson index (0 to 1).

epsilon

Numeric. The inequality aversion parameter used.

ede

Numeric. The equally distributed equivalent income.

mean_income

Numeric. The mean income.

n

Integer. Number of observations.

se, ci_lower, ci_upper, level

Bootstrap CI fields, NULL unless ci = TRUE.

Details

The Atkinson index involves either a power transformation x^(1 - epsilon) or log(x) (when epsilon = 1) and so requires strictly positive values. Use the Gini, S-Gini, or Kolm index for distributions that include zeros or negatives.

References

Atkinson, A. B. (1970). "On the Measurement of Inequality." Journal of Economic Theory, 2(3), 244–263.

Biewen, M. and Jenkins, S. P. (2006). "Variance Estimation for Generalized Entropy and Atkinson Inequality Indices: The Complex Survey Data Case." Oxford Bulletin of Economics and Statistics, 68(3), 371–383.

Examples

d <- iq_sample_data("income")

# Moderate inequality aversion
iq_atkinson(d$income, epsilon = 0.5)
#> 
#> ── Atkinson Index ──────────────────────────────────────────────────────────────
#> • Value: 0.1506
#> • Epsilon: 0.5
#> • EDE income: 41783.21
#> • Mean income: 49190.12
#> • Observations: 1000

# With bootstrap CIs
iq_atkinson(d$income, epsilon = 0.5, ci = TRUE, R = 200)
#> 
#> ── Atkinson Index ──────────────────────────────────────────────────────────────
#> • Value: 0.1506
#> • Epsilon: 0.5
#> • EDE income: 41783.21
#> • Mean income: 49190.12
#> • Observations: 1000
#> • Bootstrap 95% CI: [0.1328, 0.1712]

# High inequality aversion
iq_atkinson(d$income, epsilon = 1)
#> 
#> ── Atkinson Index ──────────────────────────────────────────────────────────────
#> • Value: 0.2768
#> • Epsilon: 1
#> • EDE income: 35572.94
#> • Mean income: 49190.12
#> • Observations: 1000

# Very high inequality aversion
iq_atkinson(d$income, epsilon = 2)
#> 
#> ── Atkinson Index ──────────────────────────────────────────────────────────────
#> • Value: 0.4765
#> • Epsilon: 2
#> • EDE income: 25750.78
#> • Mean income: 49190.12
#> • Observations: 1000