Measuring Societal Welfare: Concepts and Objections.

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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.

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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 i1,..,N with Utility finctions Ui(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(U1(x), U2(x),..,UN(x)) where F is some
  increasing function of the Uis
• Fomented a huge debate over its nature and very
  existence in the mid 1900s.
• Many views as to what form it should take and how
  it should be represented. Three examples,
  Benthamite, Daltonian and Rawlsian dominate the
  stage.

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The Benthamite Tradition

• Utilitarians, 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 1700s (Hume (1711-1776) sketched the idea in
  his many philosophical writings).
• Formally articulated by Jeremy Bentham
  (1748-1832) in An Introduction to the Principles
  and Morals of Legislation (1789) and developed by
  Mill, Edgworth, Sidgwick and Pigou through the
  1800s and early 1900s.
• Requires that we can identify Good or Utility
  and can aggregate it.

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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 identical agents with U gt 0 and
  U lt 0, Utilitarianism has a Daltonian flavour
  since an equal distribution of aggregate income
  will maximize aggregate utility.

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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 gt 0 and
  U lt 0, Utilitarianism has a Rawlsian flavour
  since an equal distribution of aggregate income
  will maximize aggregate utility.

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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 Harsanyis 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?)

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The Axioms of Individual Choice over a collection
of goods q given income x and prices p
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(No Transcript)
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The Budget Constraint, Indirect Utility Function
and Cost Function

• Given an agents preference structure accords with
  the above axioms we can develop consumer demand
  functions as functions relating observable
  entities to observable entities via the budget
  constraint and the indirect utility and cost
  functions.
• The solution to Maximising Utility subject to the
  budget constraint yields Marshallian Demand
  Functions q g(x,p). When these are substituted
  into V(q) they yield an Indirect Utility Function
  U(x,p) which can be inverted to yield a cost
  function x C(U,p) which corresponds to the cost
  of achieving U when confronting prices p.
  Minimzing this cost function for a given U yields
  Hicksian demand functions qh(U,p). Duality
  theory highlights the equivalence of minimizing
  C() for a given U with maximizing U for a given
  x.
• Consistency with the axioms and the budget
  constraint will impose some structure on the
  demand functions g() through these relationships
  which can thus be examined empirically to see if
  agents conform to the rationality hypothesis.

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Implications of The Budget Constraint
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Implications of the Axioms and the budget
constraints

• Adding up x pg(x,p) ph(U,p)
• Homogeneity h(U,?p) h(U,p) g(?x,?p) g(x,p)
• Symmetry ?hi/?pk ?hk/?pi (for testability this
  has to be represented in terms of the marshallian
  demands which can be done).
• Negativity SiSkwiwk?hi/?pk 0 (again for
  testability this has to be represented in terms
  of the marshallian demands which can be done).

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Separability and Inter-temporal Welfare

• Analysis is greatly facilitated if U(q) can be
  written as U(U1(q1),U2(q2),..,Un(qn)) where
  q1,q2,..,qn 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 skjµFC?qk/?x.?qj/?x (strong separability
  implies µFCµ i.e. independent of subgroups).
• Introduces the idea of two stage budgeting, i.e.
  allocating budgets to the Uis at the first stage
  (based upon aggregated price indices) and then
  maximize Uis individually the subject to their
  respective budgets.
• Inter-temporal choice, is 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
  Ui(x) is confined to only those goods consumed by
  i.

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So Can We Contemplate an 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(U1(q),
  U2(q),,Un(q))?

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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 individuals happinesss).
• Anonymity No identifiable individual should be
  able to determine the social choice in all
  circumstances.

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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 xa
  whereas all other agents have a strong preference
  for state xb, xa and xb must be declared socially
  indifferent.
• Rules out democracies and dictatorships as a
  social planning mechanisms.

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What to do now?

• Arrows 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.
• Harsanyis veil of Ignorance postulate.
• Hiding behind Sen (it was his idea!) well reject
  the no interpersonal comparisons constraint.
  Follow Harsanyi and maximise E(U(x)).