We derive a general optimal income tax formula when individuals respond along both the intensive and extensive margins and when income effects can prevail. Individuals are heterogeneous across two dimensions: their skill and their disutility of participation. Preferences over consumption and...

We derive a sufficient statistics optimal income tax formula in a general model that incorporates unemployment and endogenous wages, to study the shape of the tax and transfer system at the bottom of the income distribution. The sufficient statistics are the macro employment response to taxation...