Editing Social welfare function (section)

==Cardinal welfare==
A '''cardinal social welfare function''' is a function that takes as input numeric representations of individual utilities (also known as [[cardinal utility]]), and returns as output a numeric representation of the collective welfare. The underlying assumption is that individuals utilities can be put on a common scale and compared. Examples of such measures include [[life expectancy]] or per capita income.

For the purposes of this section, income is adopted as the measurement of utility.

The form of the social welfare function is intended to express a statement of objectives of a society.

The [[utilitarian]] or [[Benthamite]] social welfare function measures social welfare as the total or sum of individual utilities:
:<math>W = \sum_{i=1}^n Y_i</math>
where <math>W</math> is social welfare and <math>Y_i</math> is the income of individual <math>i</math> among <math>n</math> individuals in society. In this case, maximizing the social welfare means maximizing the total income of the people in the society, without regard to how incomes are distributed in society. It does not distinguish between an income transfer from rich to poor and vice versa. If an income transfer from the poor to the rich results in a bigger increase in the utility of the rich than the decrease in the utility of the poor, the society is expected to accept such a transfer, because the total utility of the society has increased as a whole. Alternatively, society's welfare can also be measured under this function by taking the average of individual incomes:
:<math>W = \frac{1}{n}\sum_{i=1}^n Y_i = \overline{Y}</math>

In contrast, the max-min or Rawlsian social welfare function (based on the philosophical work of [[John Rawls]]) measures the social welfare of society on the basis of the welfare of the least well-off individual member of society:
:<math>W = \min(Y_1, Y_2, \cdots, Y_n)</math>
Here maximizing societal welfare would mean maximizing the income of the poorest person in society without regard for the income of other individuals.

These two social welfare functions express very different views about how a society would need to be organised in order to maximize welfare, with the first emphasizing total incomes and the second emphasizing the needs of the worst-off. The max-min welfare function can be seen as reflecting an extreme form of [[uncertainty aversion]] on the part of society as a whole, since it is concerned only with the worst conditions that a member of society could face.

[[Amartya Sen]] proposed a welfare function in 1973:
:<math>W_\mathrm{Gini} = \overline{Y} (1-G)</math>
 
The average per capita income of a measured group (e.g. nation) is multiplied with <math>(1-G)</math> where <math>G</math> is the [[Gini index]], a relative inequality measure. James E. Foster (1996) proposed to use one of [[Anthony Barnes Atkinson|Atkinson]]'s Indexes, which is an entropy measure. Due to the relation between Atkinsons entropy measure and the [[Theil index]], Foster's welfare function also can be computed directly using the Theil-L Index.

:<math>W_\mathrm{Theil-L} = \overline{Y} \mathrm{e}^{-T_L}</math>

The value yielded by this function has a concrete meaning. There are several possible incomes which could be earned by a ''person'', who randomly is selected from a population with an unequal distribution of incomes. This welfare function marks the income, which a randomly selected person is most likely to have. Similar to the [[median]], this income will be smaller than the average per capita income.

:<math>W_\mathrm{Theil-T} = \overline{Y} \mathrm{e}^{-T_T}</math>

Here the Theil-T index is applied. The inverse value yielded by this function has a concrete meaning as well. There are several possible incomes to which a ''Euro'' may belong, which is randomly picked from the sum of all unequally distributed incomes. This welfare function marks the income, which a randomly selected Euro most likely belongs to. The inverse value of that function will be larger than the average per capita income.

=== Axioms of cardinal welfarism ===
Suppose we are given a [[preference relation]] ''R'' on utility profiles. ''R'' is a weak [[total order]] on utility profiles—it can tell us, given any two utility profiles, if they are indifferent or one of them is better than the other. A reasonable preference ordering should satisfy several axioms:<ref name=moulin2004>{{Cite Moulin 2004}}</ref>{{rp|66–69}}

1. [[Monotonicity criterion|'''Monotonicity''']]: if the utility of one individual increases, while all other utilities remain equal, ''R'' should strictly prefer the second profile. For example, it should prefer the profile (1, 4, 4, 5) to (1, 2, 4, 5). Such a change is called a [[Pareto optimality|Pareto improvement]].

2. [[Anonymity (social choice)|'''Symmetry''']]: [[Permutation|reordering or relabeling]] the values in the utility profile should not change the output of ''R''. This axiom formalizes the idea that every person should be treated equally in society. For example, ''R'' should be indifferent between (1, 4, 4, 5) and (5, 4, 4, 1), since these profiles are reorders of each other.  

3. '''Continuity''': for every profile ''v'', the set of profiles weakly better than ''v'' and the set of profiles weakly worse than ''v'' are [[closed set]]s.{{Technical inline|date=March 2024}}

4. '''Independence of unconcerned agents:''' ''R'' should be independent of individuals whose utilities have not changed. For example, if ''R'' prefers (2, 2, 4) to (1, 3, 4), it also prefers (2, 2, 9) to (1, 3, 9); the utility of agent 3 should not affect the comparison between two utility profiles of agents 1 and 2. This property can also be called '''locality''' or '''separability'''. It allows us to treat allocation problems in a local way, and separate them from the allocation in the rest of society.

Every preference relation with properties 1–4 can be represented as by a function ''W'' which is a sum of the form:
:<math>W(u_1,\dots,u_n) = \sum_{i=1}^n w(u_i)</math>
where ''w'' is a continuous increasing function.

=== Harsanyi's theorem ===
Introducing one additional axiom—the nonexistence of [[Coherence (philosophical gambling strategy)|Dutch Books]], or equivalently that social choice behaves according to the [[Von Neumann–Morgenstern utility theorem|axioms of rational choice]]—implies that the social choice function must be the [[utilitarian rule]], i.e. the weighting function <math>w(u)</math> must be equal to the utility functions of each individual. This result is known as [[Harsanyi's utilitarian theorem]].

=== 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.