# What utility functions are equivalent to additive functions?

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?

• I don't think there is a general rule which applies for the use of PMT but for example, you can not use this one if you have Constant Elasticity of Substitution (CES) utility function. By the way, the specification of utility functions is generally made for analytical reasons. People use them wheter it makes a tractable model or not. Aug 16, 2015 at 10:42
• This is why we have axioms on preference when definition a utility function. Without axioms we don't know if a utility function is equivalent to another one. Aug 17, 2023 at 12:34

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.