The generalised anti-inverse elasticity rule: A concise result

For an optimal commodity taxation problem under a consumption target with many consumer types, Minagawa and Upmann (2021) prove the existence of an optimal solution under three assumptions on (i) utility, (ii) welfare, and (iii) total demand. In this note we prove that the assumption on utility is sufficient for the stipulated property on total demand, so that the third assumption can be dropped from the existence result.