Single-Peaked Preferences and economics of welfare

Single-Peaked Preferences:

• By relaxing Universal Domain, might we to overturn Arrow’s result for a narrower class of individual preferences?

• If allocations can be ordered such that each individual’s ranking (or utility) is a singlepeaked function of the allocations afterwards social preferences exist. For example, suppose the alternatives are summarized by the size of government from small to big. If each of us has our own “most preferred” size of government and utility falls away from this peak, then a complete and transitive social ranking of the size of government exists. Indeed, majority rule would work as a social welfare function.

• There is no cause to believe it’s sensible in terms of ethics.

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(Single-Peaked Preference)

Benevolent Dictator:

• Arrow’s assumption of No Dictator is important. Without it complete as well as transitive social preferences would exist as the preferences of any one individual – the dictator – and would thus be complete and transitive.

• This pretences a problem for social engineering all the weight of social policy would be directed to this one individual.

• Nevertheless if society’s dictator were benevolent caring for others like a parent cares for his or her children, then a social ranking of alternatives would exist.

• Presume a benevolent dictator cares about how the alternative x impacts each individual i and he summarizes each individual i’s preferences by some utility function ui .

• Still with ordinal utility our dictator can represent his – the social – preferences with a social welfare function.

• In stipulations of social choice our dictator chooses from all the feasible alternatives x the one that maximizes the social welfare function.

• As we asserted above, varying the weights i α - e.g. caring more for Bob and less for Alice – traces out the Pareto set.

• So, with explicit weights, the benevolent (non-paternalistic) dictator chooses one alternative
from the Pareto set.

• Without explicit weights, solving the benevolent dictator’s problem amounts to finding the
Pareto set (his is how it is used in much of modern economics).

• The empirically relevant case is the one with no explicit weights, because the benevolent dictator does not exist.

• The benevolent dictator approach is equal – in terms of virtue as well as incompleteness – to the Pareto principle.

Cardinal Utility:

• A third attempt to revivify social preferences allows for interpersonal comparisons of utility.

• Replace IIA with cardinal utility as well as the utilities ui in the social welfare function would carry cardinal significance.

• Specified the premise of cardinal utility complete and transitive social preferences exist, can be summarized by a social welfare function, and could guide social choices over social policies, economic systems, etc.

• These results build from a basis of cardinal utility, which is no foundation at all.

• Nothing obvious in behaviour can provide any evidence for or against cardinal utility.9

• So it is rational to conclude that cardinal utility – like a magic elf – does not exist Utility is ordinal.

• For matters of social engineering one can’t escape from interpersonal comparison of value.

• But intensity of preference is not limited to cardinal utility; we compare valuations across agents without reference to utility. We use a extra measure such as the compensating or equivalent variation.

• Dollar-valuations are comparable therefore perhaps summing valuations across agents would give a social welfare function.

Robustness of Arrow’s Impossibility Theorem:

• Going further than the Pareto principle to a complete ranking of social alternatives is difficult if not impossible.

• Although the analysis doesn’t constitute a proof that no relaxation of Arrow’s assumptions could overturn the Impossibility Theorem, the theorem is quite robust.

• We are left a restricted role for social welfare analysis – use the social welfare function to generate the Pareto set.

Robustness of the First Welfare Theorem:

• Exploring strength of the First Welfare Theorem remains.

• Can it manage externalities and public goods?

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