In medical treatment and elsewhere, it has become standard to base treatment intensity (dosage) on evidence in randomized trials. Yet it has been rare to study how outcomes vary with dosage. In trials to obtain drug approval, the norm has been to specify some dose of a new drug and compare it...

Economists have long studied policy choice by a social planner who aims to maximize welfare in democracies or other political systems where, in some sense, welfare is intended to express the well-being of a society rather than the personal preferences of a dictator. The motivation for studying...