| We extend the Quadratic Almost Ideal Demand System to encompass commodity and leisure demand, estimate it with CEX data, use it to formulate the Mirrlees joint taxation problem, and solve the problem using a First Order Gradient algorithm. We take a Bayesian approach to labor supply uncertainty, based on multiple data imputations, to assess the sensitivity of our conclusions to missing values, sample selection, and measurement error. Our estimated QUAIDS model is locally integrable without the imposition of inequality constraints, though non-separability is strongly rejected by the data, pointing to the value of computing commodity and income taxes simultaneously. Optimal commodity tax rates are 8%, 2%, and 7% for fuel, housing, and all other goods respectively. The marginal income tax schedule qualitatively reflects features found in other numerical work: it is U-shaped, slightly negative at low levels of ability, rising very quickly and peaking at 33% still at relatively low levels of ability, and falling monotonically for the rest and by far the larger portion of the ability spectrum until it is slightly negative again at the highest level of ability. We use monetary measures of welfare to compare the outcomes of the four major optimal tax models (joint taxation, non-linear income taxation, commodity taxation, and linear income taxation) and find that models allowing for commodity as well as income taxes perform little better than models that only allow for income taxation, though we find that non-linear income taxation produces significantly larger welfare benefits for all individuals relative to linear income taxation. We assess the sensitivity of optimal non-linear income taxes to the planner's objectives and find that increasing the planner's aversion to inequality raises marginal tax rates by no more than 8% while reducing it lowers them by no more than 7%, assuming aversion to inequality is non-negative and finite. When the planner is Rawlsian, the optimal taxes on the least able are positive, and the highest marginal tax rates peak at over 80% for low ability workers. When the planner's objectives satisfy Cardinal Full Comparability, optimal taxes are similar to those implied by Cardinal Ratio Scale objectives with high aversion to inequality. |
