Editing Social welfare function (section)

=== Non-utilitarian ===
By Harsanyi's theorem, any non-utilitarian social choice function will be incoherent; in other words, it will agree to some bets that are unanimously opposed by every member of society. However, it is still possible to establish properties of such functions. 

Instead of imposing rational behavior on the social utility function, we can impose a weaker criterion called '''independence of common scale''': the relation between two utility profiles does not change if both of them are multiplied by the same constant. For example, the utility function should not depend on whether we measure incomes in cents or dollars.

If the preference relation has properties 1–5, then the function ''w'' must be the [[Isoelastic utility|isoelastic function]]:

<math>\frac{c^{1-\eta} - 1}{1-\eta}</math>

This family has some familiar members:
* The limit when <math>\eta \to -\infty</math> is the ''leximin'' ordering.
* For <math>\eta = 0</math> we get the [[Nash bargaining solution]]—maximizing the product of utilities.
* For <math>\eta = 1</math> we get the [[utilitarian]] welfare function—maximizing the sum of utilities.
* The limit when <math>\eta \to \infty</math> is the ''leximax'' ordering.

If we require the '''[[Pigou–Dalton principle]]''' (that inequality is not a positive good) then <math>\eta</math> in the above family must be at most 1.