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Fairness and Well-Being F. Maniquet 1 Canazei Winter School, January 2015 1 CORE (UCL) F. Maniquet Fairness and Well-Being CWS 1 / 26

Introduction Based on: Fleurbaey, M. and F. Maniquet 2014, Fairness and Well-Being Measurement, in progress. Fleurbaey, M. and F. Maniquet 2011, A Theory of Fairness and Social Welfare, CUP. Decancq, K. M. Fleurbaey and F. Maniquet 2014, Multidimensional poverty measurement with individual preferences, mimeo. Fleurbaey, M. and Blanchet 2014, Beyond GDP, F. Maniquet Fairness and Well-Being CWS 2 / 26

Motivation Comparing gains and losses in well-being is necessary for policy evaluation (social indexes, fair allocation, optimal taxation, etc). F. Maniquet Fairness and Well-Being CWS 3 / 26

Motivation Comparing gains and losses in well-being is necessary for policy evaluation (social indexes, fair allocation, optimal taxation, etc). Ethical versus empirical well-being measures (Layard Oswald, Kahneman, Vickrey Harsanyi Mirrlees). Fairness: equality of resources. Human beings are autonomous moral agents (Rawls); respect (rational and well-informed) individual preferences. Typically (in the literature on fair allocation): well-being measurement and aggregation together (surveys: Thomson, 2011, Fleurbaey and Maniquet, 2011). Result (fair social orderings): extreme inequality aversion (maximin). F. Maniquet Fairness and Well-Being CWS 3 / 26

Introduction Well-Being Measurement: consumption set: X set of admissible preferences: R Well-Being Measure: W : X R R such that W(x,R) W(x,R) xrx. F. Maniquet Fairness and Well-Being CWS 4 / 26

Introduction Well-Being Measurement: consumption set: X set of admissible preferences: R Well-Being Measure: W : X R R such that W(x,R) W(x,R) xrx. Remark I: It boils down to building comparability and cardinality in the numerical representation of preferences. Remark II: x could be replaced with I: the set of indifference curves is a lattice. F. Maniquet Fairness and Well-Being CWS 4 / 26

Main ideas 1 Desirable, divisible and cardinal goods: ray utility and money-metric utility: focal ethical well-being measures. 2 Building comparability more intuitive than building cardinalization (with consequences on aggregation). 3 Multiple ways to combine RU and MMU with well-being measures in the presence of bounded, discrete, non-desirable, and/or non-cardinal commodities. F. Maniquet Fairness and Well-Being CWS 5 / 26

Step 1: desirable, divisible and cardinal goods X R K + R R: monotonic, convex, continuous. W: continuous in x. F. Maniquet Fairness and Well-Being CWS 6 / 26

Nested Contour good 2 R W(x,R) > W(x,R ) x x 0 R good 1 F. Maniquet Fairness and Well-Being CWS 7 / 26

Lower Contour Inclusion good 2 R x W(x,R ) < max{w(x,r),w(x,r )} x 0 R R x good 1 F. Maniquet Fairness and Well-Being CWS 8 / 26

Worst preferences Axiom Worst Preferences There exists R w R such that for all x X, R R, W(x,R w ) W(x,R). F. Maniquet Fairness and Well-Being CWS 9 / 26

Worst preferences Axiom Worst Preferences There exists R w R such that for all x X, R R, W(x,R w ) W(x,R). Theorem Let X be a convex and compact set. A well-being measure W over X satisfies Lower Contour Inclusion if and only if it satisfies Nested Contour and Worst Preferences. Moreover, for worst preferences R w R, the well-being measure is defined by: for all x X and R R: W(x,R) = max x L(x,R) W(x,R w ). F. Maniquet Fairness and Well-Being CWS 9 / 26

W l (ray utility) good 2 0 w l x wl x R R l Rl = R w good 1 F. Maniquet Fairness and Well-Being CWS 10 / 26

Figure : Intermediary Preferences I: W(x,R ) [W(x,R),W(x,R )]. F. Maniquet Fairness and Well-Being CWS 11 / 26 good 2 R R x R 0 good 1

Convex Hull Inclusion Intuition: if the consumption is intermediary, then the well-being is intermediary. x = λx + (1 λ)x W(x,R ) [W(x,R),W(x,R )] and the same is true for any y indifferent to x, y indifferent to x and y indifferent to x. F. Maniquet Fairness and Well-Being CWS 12 / 26

Convex Hull Inclusion good 2 x R W(x,R ) min{w(x,r),w(x,r )} R x R x 0 good 1 F. Maniquet Fairness and Well-Being CWS 13 / 26

Best preferences Axiom Best Preferences There exists R b R such that for all x X, R R, W(x,R w ) W(x,R). F. Maniquet Fairness and Well-Being CWS 14 / 26

Best preferences Axiom Best Preferences There exists R b R such that for all x X, R R, W(x,R w ) W(x,R). Theorem Let X be a convex and compact set. A well-being measure W over X satisfies Convex Hull Inclusion if and only if it satisfies Nested Contour and Best Preferences. Moreover, for best preferences R b R, the well-being measure is defined by: for all x X and R R: W(x,R) = min x U(x,R) W(x,R b ). F. Maniquet Fairness and Well-Being CWS 14 / 26

W p (money-metric utility) good 2 R 0 x p R p = R b w p 1 w p 1 x R good 1 F. Maniquet Fairness and Well-Being CWS 15 / 26

Intermediary Preferences II Axiom Intermediary Preferences II For all x,x,x X, R,R,R R, if then W(x,R ) [W(x,R),W(x,R )]. U(x,R ) = U(x,R) +U(x,R ), 2 F. Maniquet Fairness and Well-Being CWS 16 / 26

Intermediary Preferences II Axiom Homotheticity For all x,x X, R,R R H, λ R, if W(x,R) = W(x,R ) then W(λx,R) = W(λx,R ). F. Maniquet Fairness and Well-Being CWS 17 / 26

Intermediary Preferences II Axiom Homotheticity For all x,x X, R,R R H, λ R, if W(x,R) = W(x,R ) then W(λx,R) = W(λx,R ). Theorem Let X be a convex and compact set. A well-being measure W over X satisfies Lower Contour Inclusion, Intermediary Preferences I and Homotheticity if and only if it is ordinary equivalent to the Ray Utility Measure. A well-being measure W over X satisfies Convex Hull Inclusion, Intermediary Preferences II and Homotheticity if and only if it is ordinary equivalent to the Money-Metric Utility Measure. F. Maniquet Fairness and Well-Being CWS 17 / 26

Desirable and cardinal commodities: summary Lower Convex Inclusion Worst Preferences R w + Intermediary Pref. I R w has IC s + Homotheticity ray utility Nested Contour Convex Hull Inclusion Best Preferences R b + Intermediary Pref. II R b has IC s + Homotheticity money metric utility F. Maniquet Fairness and Well-Being CWS 18 / 26

(Pigou-Dalton) Transfer good 2 R x x x x W(x,R) > W(x,R ) R 0 good 1 F. Maniquet Fairness and Well-Being CWS 19 / 26

Combining WB measures with welfarist aggregators max bu i Comparability N min b i U i a i + b i U i φ i (U i ) max a +U i U a i + bu i D a + bu i Cardinalization L φ(u i ) min F. Maniquet Fairness and Well-Being CWS 20 / 26

Step 2: satiation, ordinal goods, discrete goods X = R + A Lower Contour Inclusion does not imply Worst Preferences (and Leontieff preferences not well defined). Convex Hull Inclusion not well defined. F. Maniquet Fairness and Well-Being CWS 21 / 26

W ã x R w (x,a ) w (x,a) R 0 ã A F. Maniquet Fairness and Well-Being CWS 22 / 26

Equal Well-Being at Preferred Attribute x R R x 0 a A F. Maniquet Fairness and Well-Being CWS 23 / 26

W a max x (x,a) R w (x,a ) R w 0 A F. Maniquet Fairness and Well-Being CWS 24 / 26

Four different well-being measures 1 Combining W l and W ã, we can define W lã as follows: for all (x,a) X A, all R R, W lã (x,a) = w (x,a)i (wl,ã). 2 Combining W l and W a max, we can define W la max as follows: for all (x,a) X A, all R R, W la max (x,a) = w (x,a)i (wl,a max (wl,r)). 3 Combining W p and W ã, we can define W pã as follows: for all (x,a) X A, all R R, W pã (x,a) = w (x,a)i max ( R,{(x,ã) X A px w} ). 4 Combining W p and W a max, we can define W pa max as follows: for all (x,a) X A, all R R, W pa max (x,a) = w (x,a)i max ( R,{(x,a ) X A px w,a A} ). F. Maniquet Fairness and Well-Being CWS 25 / 26

Conclusion It is possible to build ethical well-being measures based on fairness views. The nature of the goods matters. Classical goods: Two families of well-being measures; dual characterization: Worst vs Best Preferences. Fairness well-being is closely related to the ability to trade-off between goods. Axiomatic foundation to money-metric + ray utility. Sheds light on the dichotomy between money-metric and ray utility. Other goods: other measures + combination with money-metric + ray utility. Open new possibilities to the FSO literature: fairness requirements lead to constructing comparabilities rather than cardinalization, but possibilities exist: escape maximin. F. Maniquet Fairness and Well-Being CWS 26 / 26