Inequality Measurement - A unifying framework for dynamics, multidimensionality, and uncertainty

We propose a new unifying framework for the measurement of economic inequality that incorporates dynamics, multidimensionality, and uncertainty. A dynamic social evaluation function aggregates the individual value functions, which are the solutions to the individual dynamic programming problems. The proposed inequality measure is a dynamic and multivariate version of the well-known Atkinson index. For each state variable, we minimize the total resources necessary to keep current social welfare constant. Shadow values corresponding to these optimization problems capture the trade-off across dimensions in a common metric and define the weights of each state dimension for the multivariate inequality index. This linearly decomposable index measures the weighted minimum share of the total amount of the state variables necessary to attain the current level of social welfare. The construction of this new inequality index is illustrated in a structural household life-cycle model with child care which is estimated on PSID data for married couples in the US in 2000. State variables include household wealth, and stochastic male and female wages. We show how the states do not exclusively determine the value of the inequality measure but also the future actions that arise from them, leading to important differences between the dynamic and static inequality measures. Finally, we demonstrate how a reduced-form approach can be used instead of a structural model. We deploy offline reinforcement learning techniques, specifically offline Monte Carlo prediction, to estimate the value functions, and show that in this empirical setting, reduced-form and structural approaches yield almost identical dynamic inequality measures.