Theoretical Background
A complete derivation of the Saez (2001) optimal income tax formula, the Mirrlees (1971) nonlinear schedule, and the empirical literature on parameter estimates.
The dominant analytical framework for deriving optimal top marginal tax rates in modern public finance is the Saez (2001) formula. The key insight is that the revenue-maximising top rate depends on only two parameters: the elasticity of taxable income ε and the Pareto parameter a of the upper tail of the income distribution.
The formula is derived by considering a small reform that raises the top marginal rate τ above income threshold z* by dτ. This reform has two effects:
Setting dM + dB = 0 and using the Pareto property that z̄* = a · z* / (a − 1) for the mean income above z*, yields the revenue-maximising top rate:
Saez and Stantcheva (2016) generalise the formula to allow the government to place a social welfare weight g on top earners. When g = 0, the government is indifferent to the welfare of top earners (Rawlsian in the limit) and the formula reduces to the revenue-maximising case. When g = 1, the government weights all individuals equally (utilitarian), and the optimal rate is lower.
The welfare weight g is a normative parameter. It cannot be estimated from data — it reflects a value judgement about how much the government cares about redistribution. The tool labels all results with g > 0 as "Welfare-Maximising" and includes a disclosure warning in the citation panel.
The Mirrlees (1971) model derives the optimal marginal tax rate at every income level, not just the top bracket. Diamond (1998) provides a tractable characterisation:
The term [1 − F(z)] / [z · f(z)] is the inverse hazard rate of the income distribution. It is high at the bottom and top of the distribution (where density is low relative to the survival function) and low in the middle. This produces the U-shaped pattern of optimal marginal rates documented by Diamond (1998): high rates at the bottom (for redistribution), lower rates in the middle, and high rates at the top (Laffer considerations).
The Full Schedule tab implements this formula numerically using fitted lognormal + Pareto income distribution parameters from Atkinson, Piketty and Saez (2011). Results are illustrative; empirically reliable optimal schedules require country-specific microdata.
The following table summarises the empirical literature on the key parameters.
| Parameter | Low | Central | High | Key source |
|---|---|---|---|---|
| ε (ETI) | 0.1 | 0.25 | 0.5 | Chetty (2012) |
| a (US) | 1.5 | 1.75 | 2.5 | Saez (2001); Atkinson et al. (2011) |
| a (France) | 1.8 | 2.1 | 2.5 | Atkinson et al. (2011) |
| a (UK) | 1.7 | 1.95 | 2.3 | Atkinson et al. (2011) |
| g | 0.0 | 0.0 | 0.5 | Normative choice |
The US central estimates (a = 1.75, ε = 0.25, g = 0) yield τ* ≈ 69.6% ≈ 70%, which matches the Saez–Stantcheva (2016) estimate cited in the Alexandria Ocasio-Cortez 70% top rate debate of 2019.