was writing this for an email, figured I’d make it public online as might be of interest to others
We call the legal tax rates (e.g. income tax is 39% on incomes over $180,000) the statutory tax rates. We use effective tax rates to refer to ‘tax paid divided by income’ (so paying $2,000 tax on an income of $10,000 would be a 20% effective tax rate). For most countries there are substantial differences between statutory and effective tax rates, at least for income taxes, and so putting statutory tax rates into models will give taxes that do not look like those which people actually pay in reality. There are a variety of reasons why effective taxes might differ from statutory taxes but the main one in practice is the existence of tax deductions/credits. Therefore most quantitative economic models are based on effective taxes.
We typically want to use a simple functional form for these effective taxes in our models. There are a variety of motives for wanting a simple functional form: avoid overfitting the data, ease of use, and ease of interpretability.
In all cases, we will call y_b the taxable income base (which is typically not the same thing as income, y, depending on details of a tax system). Approaches all involve choosing a functional form for the ‘average tax rate’ \tau(y_b); so taxes paid would be \tau(y_b) y_b. Note that marginal tax rate (derivative of the average tax rate with respect to y_b) is easy to derive.
So what simple functional form should we use? Papers written in the 1990s and 2000s essentially all use the ‘Gouveia-Strauss’ functional form, namely,
\begin{equation*}
\tau(y_b) = c_1 [1 − (c_3 y_b^{c_2} + 1)^{−1/c_2}]
\end{equation*}
see Gouveia & Strauss (1994) and Gouveia & Strauss (1999). Observe that with this functional form c_1 defines the top (asymptotic) marginal tax rate, while c_2 and c_3 control the curvature and initial steepness. In the notation of Gouveia and Strauss (1999) c_1 = b, c_2 = p, c_3 = s.
In the mid-2010s two other functional forms became popular HSV and GKO. With the development of these the literature has largely ceased to use Gouveia-Strauss for the USA, as HSV and GKO are considered to have a better fit to the US effective tax rate data. [Obviously it does not automatically follow that these are necessarily better for other countries with different tax systems.]
HSV is,
\begin{equation*}
\tau(y_b) = (1-c_1 y_b^{-c_2} )
\end{equation*}
see Heathcote, Storesletten, and Violante (2017). The parameter c_1 controls the level of the tax rate, whereas the parameter c_2 controls the curvature, or degree progressivity in the tax schedule. If c_2=0, average and marginal tax rates are constant as income changes (flat-rate tax), whereas c_2>0 implies a progressive tax. In the notation of Heathcote, Storesletten, and Violante (2017) c_1 = \lambda, c_2 = \tau-1 [HSV use \tau(y_b) = (1-\lambda y_b^{1-\tau} )].
GKO is,
\begin{equation*}
\tau(y_b) = c_1 + c_2 y^{c_3}
\end{equation*}
see Guvenen, Kuruscu & Ozkan (2010), and is sometimes called the ‘Power’ functional form. Note that HSV is just GKO with c_1=1 (and changing signs of the other parameters; which is just changing interpretation).
Guner, Kaygusuv, and Ventura (2014) estimates all three of these (plus one other) for US tax data.
Roughly speaking, the HSV form is focused on having nice analytical properties (making it possible to do more theory with the model), while the GKO form is focused on combining parsimony (few parameters) with a good fit to US effective tax rate data.
There are no shortage of further issues: what is taxable income, and how does it differ from income?, other taxes?, what is the unit of taxation (do couples pay taxes together, or as two separate people)?