Ratio-Scale Invariance and Logarithmic Aggregation

We characterize logarithmic aggregation within the class of continuous additively separable social orderings on strictly positive vectors. The key requirement is that the chosen aggregate representation respond to a common proportional renormalization by an additive shift independent of the underlying welfare profile. Under Pareto monotonicity, continuity, and additive separability, this scale-translation requirement forces each coordinate contribution to be affine in the logarithm of welfare. The ordering therefore admits a representation of the form ∑ᵢ₌₁ⁿ aᵢ log xᵢ with strictly positive weights. Our contribution is not to derive separability from primitive ordinal axioms, but to identify logarithmic aggregation as the unique separable form for which common proportional rescalings induce profile-independent additive shifts in the aggregate representation.