Homothetic and translatable preferences: Utility representation and functional equations involved

I provide a full characterization of those preferences (total preorders) defined on RnRn that admit a representation by means of a continuous, homogeneous of degree one and translative utility (order-preserving) function. In addition, I pay attention to the functional form of the kind of representation obtained, including some qualifications such as being continuous and/or monotonic. In particular, a full description of the functions f:R3→Rf:R3→R that are continuous, homogeneous of degree one, translative and monotonic is presented. Further applications to utility theory and social choice theory are also provided.