The government seeks to maximise a utilitarian social welfare function subject to these constraints.
It faces a tradeoff between efficiency and equity: Emmanuel Saez in his article titled "Using Elasticities to Derive Optimal Income Tax Rates" derives a formula for optimal level of income tax using both the compensated and uncompensated elasticities.
[2] Saez writes that the tradeoff between equity and efficiency is a central consideration of optimal taxation, and implementing a progressive tax allows the government to reallocate their resources where they are needed most.
This type of tax creates a large distortion, disfavoring women from the labour force during years when the couple has the greatest child care needs.
Eytan Sheshinski has studied a simplified income-tax model, in which the tax is a linear function of the income:[5]
The goal is to find the values of a and b such that the social welfare (the sum of individual utilities) is maximized.
The before-tax income y is determined by the amount of labor l and an innate ability factor n, where the relation is assumed to be linear too:
Under certain natural assumptions, it is proved that the optimal linear tax has a>0, i.e., it provides a positive lump-sum to individuals with zero income.
Additionally, the optimal tax rate is bounded above by a fraction that decreases with the minimum elasticity of the labour supply.