Optimal allocations to heterogeneous agents with an application to stimulus checks

A planner allocates discrete transfers of size $D_g$ to $N$ heterogeneous groups labeled $g$ and has CES preferences over the resulting outcomes, $H_g(D_g)$. We derive a closed-form solution for optimally allocating a fixed budget subject to group-specific inequality constraints under the assumption that increments in the $H_g$ functions are non-increasing. We illustrate our method by studying allocations of "support checks" from the U.S. government to households during both the Great Recession and the COVID-19 pandemic. We compare the actual allocations to optimal ones under alternative constraints, assuming the government focused on stimulating aggregate consumption during the 2008--2009 crisis and focused on welfare during the 2020--2021 crisis. The inputs for this analysis are obtained from versions of a life-cycle model with heterogeneous households, which predicts household-type-specific consumption and welfare responses to tax rebates and cash transfers.