Logarithmically homogeneous preferences

An extended-real-valued function on R₊ⁿ is called logarithmically homogeneous if it is given by the logarithmic transformation of a homogeneous function on R₊ⁿ. Specifying a consumer's preference on the consumption set by a difference comparison relation, this paper provides some axioms on the relation under which the full class of utility functions representing the relation are logarithmically homogeneous. It is also shown that all the utility functions are strongly concave and all the indirect utility functions are logarithmically homogeneous. Moreover, the additively separable logarithmic utility functions are derived by strengthening one of the axioms.