Transcript Chapter 1
Social choice theory and composite indicators: In defense of linearity Overview Composite indicators vs MD social choice Axioms & results for MD social choice Implications for composite indicators 2 CI versus MD social choice Illustration: we want to measure performance of 3 European countries (be,nl,lu) 1 benchmark country (us) via 2 performance dimensions (only) GDP/h: GDP per hour worked SSR: Schooling Success Rate 3 CI versus MD social choice SSR : 2005 : 2006 nl lu us be Composite indicators allow us to compare performance of countries, but not of groups of GDP/h countries ↔ MD social choice allows both 4 Axioms for MD social choice For simplicity we stick to the previous example X x be x nl x lu x be GDPh be x SSR nl x GDPh x nlSSR x luGDPh x luSSR assuming a fixed number of countries & equal population size Purpose of MD social choice: find attractive rule to judge whether one situation X is better or worse than another, say Y But what is attractive? introduce axioms: create simple imaginary situations X and Y in which it is (relatively) easy to judge whether one situation is better than the other. All simple axioms together leads to a rule (or a family of rules) which also allow(s) us to judge more complex real-world situations MD social choice axioms might also impose structure on CI’s 5 Three technical axioms Completeness: either X is at least as good as Y, or Y is at least as good as X (or both) Transitivity: if X is at least as good as Y and Y is at least as good as Z, then also X must be at least as good as Z Continuity: (technical) small changes in a situation X cannot lead to large changes in its comparison with other situations Result 1 (Debreu, 1954) If a rule satisfies Completeness, Transitivity as well as Continuity then there exists a continuous function f s.t. X is at least as good as Y if and only if f(X) ≥ f(Y). 6 Separability Separability: countries with the same performance in two situations X and Y do not matter when evaluating X and Y SSR : 2005 : 2006 Result 2 (Debreu, 1954; Blackorby, Donaldson & Auersperg, 1981; Tsui, 1995) nl If a rule satisfies Separability in addition to lu Completeness, Transitivity and Continuity then there be must exist continuous functions gbe, gnl and glu s.t. X is at least as good as Y if and only if gbe(xbe)+gnl(xnl)+glu(xlu) ≥ gbe(ybe)+gnl(ynl)+glu(ylu) GDP/h 7 Monotonicity & Anonymity Monotonicity: if all countries perform at least as good in X compared to Y (& some better), then X is better than Y Anonymity: the name of a country does not matter Result 3 SSRIf a rule satisfies Monotonicity SSR and Anonymity in : 2005 : 2005 : 2006Completeness, Transitivity and : 2006 addition to Separability, nl lu Continuity then there must exist a lu be strictly nl increasing & continuous as lu good as Y if be function g s.t. X is at least be and only if g(xbe)+g(xnl)+g(xlu) ≥ g(ybe)+g(ynl)+g(ylu) g is the implicitGDP/h CI-function of our rule which GDP/h measures the performance of countries! 8 Pigou-Dalton Result 2006) SSR 4 (Bosmans, Lauwers and Ooghe,: 2005 If a rule satisfies Pigou-Dalton in : 2006 addition to Separability, Completeness, Transitivity, Continuous Differentiability, Monotonicity and Anonymity then nl lu there exist weights wGDPh,wSSR > 0 and a function h with h’ > 0 and h” <be0 s.t. the CI-function g equals g : xGDPh , xSSR hwGDPh xGDPh wSSR xSSR GDP/h 9 Implications for CI’s Perfect Substitutability between dimensions SSR : 2005 : 2006 nl lu be GDP/h 10 Conclusion Composite indicators vs MD social choice If we want to be able to compare groups of countries EU versus benchmark group EU over time Old EU versus new EU members and if we care about convergence of countries, then the implicit CI should be linear, i.e., a weighted sum of the performance in the different dimensions. 11