Optimal Allocations to Heterogeneous Agents with An Application to Stimulus Checks

A planner allocates discrete transfers of size Dg to N heterogeneous groups labeled g and has CES preferences over the resulting outcomes, Hg(Dg). 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 Hg 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.