# Independence Axiom for Linear Utility - Who proved this first?

Who first proposed the following axiomatization of linear utility using Independence? I remembered that it was Debreu but I am not so sure. What was the first paper proving this?

Consider a preference relation $$\succeq$$ on $$X=\mathbb{R^2_{+}}$$. If $$\succeq$$ satisifies: \begin{align} &1.\mbox{ }(a_1,a_2)\succeq (b_1,b_2)\implies(a_1+t,a_2+s)\succeq (b_1+t,b_2+s),\forall t,s\\ &2.\mbox{ }a_1\geq b_1 \mbox{ and } a_2\geq b_2 \implies (a_1,a_2)\succeq (b_1,b_2)\mbox{ (and the analogous for }\succ\mbox{)}\\ &3.\mbox{ Continuity } \end{align} Then: exists a linear representation for $$\succeq$$.

I've tried to read the Debreu (1960) on "Topological method in cardinal utility" (which is very difficult to read, in the sense of both typeset and math). It seems be saying that if we have the following independence axioms (in 3D or higher): for any index $$i$$, any $$c$$,

1. $$(a_i,a_{-i})\succsim (b_i,a_{-i})$$ implies $$(a_i,c_{-i})\succsim (b_i,c_{-i})$$

Then, with assumption that the preference is not constant on each direction, there is a separable utility $$U=\sum_ja_ju_j$$ that represents $$\succsim$$.