# 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