This content was previously covered in an hour long youtube video. I found it prudent to make a more concise, written version and post it here.
To make any rigorous claim, we need to start with some basic axioms, or principles so simple we can take them as given. This is especially true in a field such as social welfare theory, where the content cannot be reduced to the basic principles of number sense and operations. Any claim or thesis on this topic must start with certain axiomatic assumptions (whether explicitly listed or not) and the strength and validity of the claim can be measured in part by how obvious those axioms are. Here, our axioms are taken from the established core principles of social welfare theory, from ideas used by left-leaning sources, and from observable tendencies of risk aversion. Int he field of social welfare theory, the general assumptions discuss which of two possible scenarios is better or worse for a society, incorporating ideas such as each individual's material wellbeing, the extent to which the rules of that society are fairly applied, the extent to which promises made by the society are kept, and others. By combining those, we can describe the individual welfare of a member of that society as a function of xyr material wellbeing, whether the rules are fairly applied to xem, etc. We can then define the social welfare of the entire society as some function of the individual welfares of all individuals in the society, and are attempting to maximize that social welfare.
The axioms we assume are:
1) Any change that would decrease the individual welfare of all members will decrease the social welfare, and is therefore a bad change that should be avoided.
(This is taken as axiomatic whenever the risk of a climate catastrophe is discussed, so there should be no argument with it here.)
2) Any change that merely rearranges which members of which demographics (racial, ethnic, gender, etc,) have which individual welfares, this must be considered a neutral change.
(This idea is the basis for describing racism negatively. While right-wing extremists may take issue with it, we presume it to be axiomatic here.)
3) The social welfare should be a continuous function of individual welfares.
(Again, this is never seriously disputed,)
4) The social welfare function must be indifferent toward unaffected members.
(That is, if some people will be unaffected by the change, we can assess whether a change is good or bad purely by evaluating the people who are affected. This is frequently used when discussing changes, that we do not list off every person, even those uninvolved.)
5) When assessing which of two societies is better, we must ask ourselves “which society would one rather live in, assuming no knowledge of what our role in that society would be?”
This is called the “veil of ignorance” principle, and was developed by leftist economist John Rawls. Without it, we'd be hard-pressed to describe why social welfare is a good thing. It also allows us the powerful tool to relate questions of social welfare to questions of risk assessment.
6) Given a fixed probability of a fixed potential individual gain, there exists some potential loss small enough that the individual would consider it a worthwhile risk.
(This is an observable reality based on behaviors toward risk).
Theorem 1: If a change to a society would improve the individual welfares of one or more people, and leave the other unchanged (lowering nobody's individual welfare) then this must be considered a positive change.
Proof: Suppose that a change would improve one or more members' individual welfares leaving the others unaffected. Invoking axiom 4, we can ignore the unaffected members and treat this like a change that would benefit everyone in the (smaller) society that ignores the unaffected members. By axiom 1, this must be considered an improvement.
Corollary to theorem 1: Axiom 1) can be replaced with the stricter axiom “If a change would improve the welfare of one person and leave the others unchanged (even if that person were already the most well-off in the society) then this must be considered a beneficial change.
Clearly, the original axiom 1 and this stricter axiom 1 are not equivalent. That is not what this corollary claims. Instead, the claim is that, when taken together with the other axioms, whether we use the original axiom 1 or the stricter one, we'll be left with identical resulting sets of axioms. Two sets of axioms A and B are equivalent if and only if the following are both true:
• Anything satisfying all axioms in set A will also satisfy all axioms in set B, and
• Anything satisfying all axioms in set B will also satisfy all axioms in set A.
To prove that anything satisfying our original set of axioms will also satisfy the one with the stricter axiom 1, we invoke theorem 1.
To prove the result in the opposite direction, we note that a global improvement for everyone can be viewed as a sequence of improvements for everyone, one person at a time. Since each individual improvement would be an overall improvement, we find that the original axioms 1 becomes satisfied.
Theorem 2: A valid social welfare function cannot be one that purely minimizes disparity between individuals.
Proof: Consider a distribution of individual welfares with any disparity between at least one pair of individuals. Consider the change made by lowering each individual welfare to match the lowest individual welfare. By theorem 1, this must be a negative change. Yes it reduces any measure of disparity to 0. Therefore, this change reduces disparity and also reduces social welfare.
Theorem 3: The function that seeks to maximize the lowest individual welfare is not a valid social welfare function.
This is the most interesting theorem of the above, as some introductory works on social welfare theory present the above function as a valid social welfare function. Those works only include axioms 1 through 4, and show that such a function satisfies those four axioms. However, it can be shown that this function is inconsistent with axioms 5 and 6.
Suppose that a society is considering a change that would improve the individual welfares of the top 50% by some amount X and decrease the individual welfares of the bottom 50% by some other amount Y. According to the above function, for any fixed X, no matter how small Y was, the change would be considered a negative change. However, when viewed from the risk assessment perspective provided by axiom 5, the choice becomes “gain X 50% of the time while losing Y 50% of the time, or decline the risk.” By axiom 6, there must exist some positive Y for which the risk is considered worthwhile. This refutes what is often called the “egalitarian social welfare function.”
