Measuring Societal Welfare: Concepts and Objections. This course is about measuring and comparing aspects of wellbeing. Before we do so we need to look at some theoretical foundations for, and objections to, the activity.

The Social Welfare Function A long tradition of the notion of “The Common Good” comparatively recently articulated in the Samuelson (1947) – Bergson (1938) Social Welfare Function. N agents i=1,..,N with Utility finctions U i (x) defined over all social states x (a list of all of the things that define a particular state for all individuals) and a benign impartial administrator who chooses x to maximize SWF = F(U 1 (x), U 2 (x),..,U N (x)) where F is some increasing function of the U i ’s Fomented a huge debate over its nature and very existence in the mid 1900’s. Many views as to what form it should take and how it should be represented. Three examples (with modifications), Benthamite, Daltonian and Rawlsian dominate the stage.

The Benthamite Tradition Utilitarian’s, the first welfarists, did not contemplate measurement as a problem Utilitarianism – “The Greatest Good for the Greatest Number” (Just add it up across the population!) Ideas born in the Scottish Enlightenment in the mid 1700’s (Hume ( ) sketched the idea in his many philosophical writings). Formally articulated by Jeremy Bentham ( ) in An Introduction to the Principles and Morals of Legislation (1789) and developed by Mill, Edgworth, Sidgwick and Pigou through the 1800’s and early 1900’s. Requires that we can identify “Good” or “Utility” and can aggregate it.

The Daltonian Principle: “Inequality is a bad thing”. Dalton (1925) Inequality of Incomes, Pigou(1912) Wealth and Welfare For Constant Aggregate Income Levels, a more equal distribution of utility is to be preferred. Captured in the Pigou-Dalton Principle of Transfers (any transfer from a poor man to a rich man increases inequality). For a society of identically preferenced agents with U’ > 0 and U’’ < 0, Utilitarianism has a Daltonian flavour since an equal distribution of aggregate income will maximize aggregate utility.

One recent modification: Equality of Opportunity “The conception of social justice held by many, perhaps most, citizens of the Western democracies is that of equality of opportunity. Exactly what that kind of equality it requires is a contested issue, but many would refer to the metaphor of ‘leveling the playing field’, or setting the initial conditions in the competition for social goods as to give all, regardless of their backgrounds an equal chance of achievement.” Roemer (2006). Accepting of differential outcomes (inequality) if it is a result of effort but not if it is a result of circumstance. Late 1970’s, Arneson, Dworkin and Roemer the proponents.

Another Recent Modification: Polarization Really a notion about groups, characterized as increased commonality (identification) within groups together with increased differences (alienation) between groups. More a trend than a State. Societal inequalities can either increase or decrease with increased polarization. Characterization due to Esteban and Ray (1994).

The Rawlsian Principle Social Welfare is simply the welfare of the poorest person.(Rawls Theory of Justice 1971) The Maximin Rule. Maximize the income of the poorest person. For a society of identical agents with U’ > 0 and U’’ < 0, Utilitarianism has a Rawlsian flavour since an equal distribution of aggregate income will maximize aggregate utility. Sometimes modified to the poorest group – focus on the poor.

The Pretense toward Scientific method and Measurement Problem Happiness, felicity, satisfaction, utility, well being, ophelimity – we have many words for something (U(x)) we can not measure cardinally! Robbins, Hicks, Samuelson argued that since only ordinal measurement made sense statements like “Agent i is happier than agent j under x” did not make sense. NO INTER-PERSONAL COMPARISONS OF UTILITY! All is not lost, to understand the actions of the individual we only require that she be able to order or rank states in order of preference. Utility Theory does the rest. To compare societies we can borrow ideas from utility theory for analyzing social states providing we can get round the interpersonal comparisons thing. One such attempt was Harsanyi’s “veil of ignorance”, i.e. which state would you choose if you did not know which agent you were all you knew was that you would be randomly allocated one of the positions in that society. (Ethically Defensible?)

Axioms of Welfare (choice) Notation (~ indifference) (≤ at least as good as) (< preferred to) Reflexivity: q ~ q Completeness: for any q 1 and q 2 either q 1 ≤q 2 or q 2 ≤q 1 Transitivity: if q 1 ≤q 2 andr q 2 ≤q 3 then q 1 ≤q 3 Continuity: for A(q 1 )={q|q≤q 1 } and B(q 1 )={q|q 1 ≤q} A and B are closed, i.e. they contain their own boundaries. (Ensures welfare function may be represented by something like a utility function V()). Nonsatiation: V() non-decreasing in all arguments and strictly increasing in at least one. Convexity: For q 1 < q 2 q 3 = λq 1 +(1-λ)q 2 is such that q 1 ≤q 3

Separability and Inter-temporal Welfare Analysis is greatly facilitated if U(q) can be written as U*(U 1 (q 1 ),U 2 (q 2 ),..,U n (q n )) where q 1,q 2,..,q n mutually exclusive and exhaustive sub-vectors of q. Types of Separability: Weak and Strong (or Additive) separability. Has considerable implications for the structure of preferences. For any goods k and j from two different subgroups F and C respectively weak separability implies s kj =μ FC ∂q k /∂x.∂q j /∂x (strong separability implies μ FC =μ i.e. independent of subgroups). Introduces the idea of two stage budgeting, i.e. allocating budgets to the U i ’s at the first stage (based upon aggregated price indices) and then maximize U i ’s individually the subject to their respective budgets. Inter-temporal choice, i’s associated with time periods where the budget constraint is lifetime wealth. Note the similarity of these structures with the SWF. Utilitarianism is like strong separability in individual utilities if the the x in each U i (x) is confined to only those goods consumed by i.

So Can We Contemplate a SWF (Social Welfare Function)? Robbins argued that rigorous adherence to the strictly ordinal notion of utility precluded any interpersonal comparisons of utility. We cannot add up utilities like the Utilitarians did. The only societal welfare improvements or deteriorations that economists could proclaim were Paretian ones: no one is worse of and at least one person is better off or no one is better off and at least one person is worse off. The problem is we cannot give a numeric value to U(q). Suppose we could, what about SWF = F(U 1 (q), U 2 (q),…,U n (q))?

Arrows Impossibility Theorem: The “Reasonable” Conditions Collective Rationality: The collective choice is represented by an ordering over all states that is complete and transitive. Universal Domain: The domain of the welfare function should contain all logically possible orderings of individuals. Pareto Inclusiveness: If all individuals prefer a to b then the welfare function should prefer a to b. Independence of Irrelevant Alternatives: The ranking of two alternatives depends solely on information on how individuals rank those alternatives (i.e. not on a direct comparison of the individual’s happiness’s). Anonymity: No identifiable individual should be able to determine the social choice in all circumstances.

The Implications The Only SWF satisfying all these conditions must make all Pareto Incomparable States socially indifferent, i.e. Pareto comparisons are the only basis for social choice. Even if one agent has a mild preference for x a whereas all other agents have a strong preference for state x b, x a and x b must be declared socially indifferent. Rules out democracies and dictatorships as a social planning mechanisms.

What to do now? Arrow’s theorem makes Pareto Judgments the only basis for social choice. If we want to be Democrats or Dictators, Utilitarians, Daltonians or Rawlsians for that matter, we have to reject at least one of the principles. Harsanyi’s veil of Ignorance postulate. Hiding behind Sen (it was his idea!) we’ll reject the no interpersonal comparisons constraint. Follow Harsanyi and maximise E(U(x)).