Welfare: Fairness

1. Prerequisites Almost essential Welfare: Basics Welfare: Fairness MICROECONOMICS Principles and Analysis Frank Cowell December 2006

2. Fairness: some conceptual problems • Can fairness be reconciled with an individualistic approach to welfare? • How can fairness be incorporated into a model? • on what can we base it? • what relation to other welfare concepts? • Why introduce a concept of fairness?

3. Fairness: Concepts • Fairness as an external moral imperative • Considered further in the social welfare-function approach • Fairness as the mirror image of Pareto superiority • Use individuals’ own utility functions • Fairness based on selfishness? • Formulate fairness concept as “absence of envy” • Reason for introducing fairness as a principle • sometimes efficiency criteria alone produce disgusting results... example

4. b a a b x1 x1 x2 x2 [x′′] [x°°] [x′] [x°] [x] Fairness in the trading model Ob • The Edgeworth box • Extreme, efficient allocations • Two more efficient allocations • Another, intermediate example • Swap a's and b's allocations • Are [x°], [x°°] "obviously" unfair? • Perhaps also [x'], [x''] ? • a prefers to have b's allocation in [x] • So [x] is not fair Oa

5. Towards a definition of fairness • Recall the definition of Pareto superiority as: • allocation [x] is superior to [x′] if… • for all h: Uh(xh)³ Uh(x′h) • for some h: Uh(xh)> Uh(x′h) • Use this individualistic approach to formalise fairness as “no-envy” • compare, not with an alternative, hypothetical bundle… • ..but with the bundles enjoyed by other people • An allocation is fair if, for every pair of individuals h and k: • Uh(xh) ³Uh(xk ) • given my tastes I weakly prefer my bundle to yours

6. A result on fairness • THEOREM: if all persons have equal incomes then a competitive equilibrium is a fair allocation. • An apparently appealing result • Seems to combines two opposing principles: • individualism – embodied in competitive behaviour • egalitarianism – embodied in equal-incomes requirement • Proof is straightforward

7. Fairness result: proof • For every household h let • Ah := {xh: Si pixih  yh} • attainable set for h • If [x*] is a CE then • x*h Ah and • Uh(x*h) ³Uh(xh ) for all xh Ah • But if all incomes are equal then, for any h and k: • Ah = Ak • so x*k Ah • Therefore Uh(x*h) ³Uh(x*k ) for any households h and k • So no one would prefer another person’s bundle • CE is fair (envy free)

8. b b a a x1 x1 x2 x2 • [x*] The fair allocation Ob • The Edgeworth box • An efficient allocation • Supporting price ratio = MRS • Incomes in terms of good 1 • The allocation [x*] is a CE if incomes are as shown Oa

9. The fairness result – discussion • Is the result as appealing as it seems? • What if Alf and Bill have different needs? • Age, • disability, • family...? • Should not this be reflected in money incomes? • Would not the equal-income solution be regarded as “unfair” • Does the problem come from • competition? • individualism?

10. Summary • Consider fairness along with other general welfare principles • Efficiency • neat and simple • but perhaps limited • Potential efficiency • Persuasive but perhaps dangerous economics/politics • Fairness • nice idea but doesn't get us far • For these reasons it may be useful to examine an explicit welfare-function approach