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Social Welfare Functions in DCEA

Atkinson Social Welfare Function

The Atkinson SWF evaluates population health by penalising inequality according to the parameter eta (η). Higher η = stronger aversion to inequality.

The Equally Distributed Equivalent (EDE) health is the key output: the level of health that, if equally distributed, would give the same social welfare as the actual distribution.

Choosing eta: evidence from the UK

Robson et al. (2017) elicited public preferences for health inequality aversion in England using a questionnaire. Their central estimate was η ≈ 1, with a range of 0.1 to 4.8 across the sample.

NICE (2025) does not mandate a specific η but expects sensitivity analysis across a range including 0, 1, and higher values.

EDE calculation

health  <- c(52.1, 56.3, 59.8, 63.2, 66.8)
weights <- rep(0.2, 5)

# eta = 0: no inequality aversion (arithmetic mean)
calc_ede(health, weights, eta = 0)
#> [1] 59.64

# eta = 1: moderate aversion (geometric mean)
calc_ede(health, weights, eta = 1)
#> [1] 59.41665

# eta = 5: strong aversion
calc_ede(health, weights, eta = 5)
#> [1] 58.51938

EDE profile

profile <- calc_ede_profile(health, weights, eta_range = seq(0, 10, 0.1))
library(ggplot2)
ggplot(profile, aes(eta, ede)) +
  geom_line(colour = "steelblue", linewidth = 1) +
  labs(x = expression(eta), y = "EDE (years)",
       title = "EDE Profile") +
  theme_minimal()

Equity weights

ew <- calc_equity_weights(health, weights, eta = 1)
ew  # Q1 (most deprived) gets highest weight
#> [1] 1.1361198 1.0513649 0.9898301 0.9365798 0.8861054

Social welfare decomposition

post_health <- health + c(0.5, 0.6, 0.5, 0.4, 0.3)
calc_social_welfare(health, post_health, weights, eta = 1)
#> $ede_baseline
#> [1] 59.41665
#> 
#> $ede_post
#> [1] 59.88513
#> 
#> $delta_ede
#> [1] 0.4684784
#> 
#> $efficiency_component
#> [1] 0.46
#> 
#> $equity_component
#> [1] 0.008478412

References

Robson M et al. (2017). Health Economics 26(10): 1328-1334. https://doi.org/10.1002/hec.3386

These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.
Health stats visible at Monitor.