What utility functions are equivalent to additive functions?

Question

Call a utility function $u(x,y)$ additive if there exist functions $v_x,v_y$ such that: $$u(x,y)=v_x(x)+v_y(y)$$

Consider the function $u(x,y)=xy$. It is not additive, but, it can transformed using a positive-monotonic-transformation (PMT) to the function: $u'(x,y)=\log u(x,y) = \log{x}+\log{y}$, and the function $u'$ is additive.

My question is: what conditions on a function $u(x,y)$ guarantee that it can be transformed using a PMT to an additive function?

I.e, if I see a function $u(x,y)$, how can I know whether it represents a preference relation which can also be represented by an additive utility function?

Answer 1

Ted Bergstrom has Lecture Notes on Separable Preferences that seem to have what you are looking for. For example:

When are preferences additively separable?

The most useful necessary and sufficient condition for preferences to be additively separable is that every subset of the set of all commodities is separable. The proofs that I know of for this proposition are a bit more elaborate than seems appropriate here. A somewhat more general version of this theorem can be found in a paper by Gerard Debreu (Topological methods in cardinal utility). Debreu’s paper seems to be the first satisfactorily general solution to this problem. Other proofs can be found in (Foundations of Measurement) and (Utility theory for decision making).

Answer 2

If and only if (If $(x1, x2)R(y1, y2)$ and $(y1, z2)R(z1, x2)$, then $(x1, z2)R(z1, y2)$) then with a two good functions, then the variables are separable. Hence additively separable. This is the Debreu theorem.