A single-axiom cardinal characterisation of quadratic utility

In the cardinal part of their analysis of spherical preferences, Chambers and Echenique (2020) characterise quadratic-plus-linear utility using two axioms concerning changes from a status quo: Status Quo Independence (SQI) and Eventual Linearity (EL). We show that the two axioms are not independent. Specifically, SQI implies EL and, indeed, forces the additivity condition in EL to hold at every status quo, not merely at some status quo. Consequently, under continuity and normalisation, SQI alone characterises the representation, implying that their cardinal theorem remains valid when EL is dropped. The respective roles of continuity and normalisation are also clarified.