An explicit solution to an optimal tax problem

Under specific but perhaps not over-restrictive assumptions on social welfare and consumer preferences, an explicit closed-form solution for an optimal linear income tax is derived. Specifically, given linear income supply functions and a rank-order social welfare function, the optimal tax rate and benefit level are characterized by four parameters: I, a measure of pre-tax inequality in the ability (wage) distribution; r, the fraction of potential total income required for (non-redistributed) government revenue; σ, the fraction of potential total income required for consumer subsistence expenditures; and a disincentive parameter, δ, the marginal propensity to spend on leisure or the amount by which earned income is reduced in response to a unit increase in unearned benefit. Defining , the ratio I/(1 - σ - r), the optimal tax rate τ is given by: The formula is used to fully characterize τ in terms of the parameters. Results include the following: τ = 0 if I = 0; τ = 1 if δ = 0; τ is increasing in I, σ and r; τ may be increasing or decreasing in δ depending on the value of ; when disincentive effects are large, τ becomes close to so that, in such economies, if σ and r are small, the optimal tax rate is equal to the measure of pre-tax inequality. Formulae for the deadweight loss associated with the tax are derived and some observations are offered on the empirical issues associated with the model.