Lateral Thinking: Promising Path or Deceptive Deviation?

1. Cornell 2009 Lateral Thinking: Promising Path or Deceptive Deviation? A B Atkinson, Nuffield College, Oxford

2. Introduction: On lateral thinking • Indices covering different dimensions: income and health inequality • Keeping dimensions separate and radar diagrams • Interdependence and copulas • Conclusions: Directions for future research

3. Lateral thinking “My interest in the question of measuring inequality was originally stimulated by reading an early version of the paper by Rothschild and Stiglitz [“Increasing risk: A definition and its economic consequences”], to which I owe a great deal” (Journal of Economic Theory, 1970, p 244). CLEAR PARALLEL: Risk and Inequality • Mean preserving spread and principle of transfers (Pigou and Dalton). • Expected utility and additive social welfare function. • Risk aversion and inequality aversion. • Stochastic dominance.

4. Diagram on country rankings for different values of ε was based on the diagrams used to show the results of college rowing races (bumps).

5. Different views! “The e parameter, which is bound by the limits of 0 and 1, determines the level of inequality aversion” (US Census Bureau, 2000, p 11). LIS Key Figures uses values of epsilon of 0.5 and 1.0. “A system of weighting that appeals to me is the inverse square law, according to which the welfare weight is the inverse of the square of the individual’s income” (Mirrlees, 1978, p 134). A Sen, The Idea of Justice, allow “the possibility that even in the original position different people could [take] very different principles as appropriate for justice, because of the plurality of their reasoned political norms and values” (2009, p 90).

6. ECONOMICS HAS BENEFITED FROM BORROWING • FROM OTHER DISCIPLINES (examples) • Physics (Samuelson’s Foundations). • Psychology (behavioural economics). • Ecology (“Ecology for bankers”, Nature, 2009). • WITHIN ECONOMICS (examples) • Offer curves from international trade. • Duality applying cost function to consumer theory. • Harberger model in public finance (“This paper … was inspired by a long tradition of writings in the field of international trade” (JPE, 1962, page 215).

7. BUT • Need for care in drawing parallels • Illustrate by issues raised by move to other dimensions: • New dimensions and aggregation of sub-indices • Portraying sub-indices: different dimensions of inequality • Multi-dimensionality at individual level.

8. Introduction: On lateral thinking • Indices covering different dimensions: income and health inequality • Keeping dimensions separate and radar diagrams • Interdependence and copulas • Conclusions: Directions for Future Research

9. CAN WE APPLY THE SAME MEASURES TO HEALTH? • “It is surprising in view of … the sheer amount of literature on inequalities in health that so little attention has been paid to the question of how health inequality is best measured” (Wagstaff, Paci and van Doorslaer, 1991). • First need to distinguish • inequality of health status, h • covariance of h with income or socio-economic status,y • Much of the health literature is concerned with second question. Reserve term “health inequity” to refer to positive covariance. • Strategic Review of Health Inequalities in England post 2010

10. Health inequality: distribution of h

11. Health inequity: covariance of h and y

12. Inequalities in health status • Some authors apply same measures as income to health status or age at death: Lorenz curve and Gini coefficient. • BUT important questions • is health status purely ordinal (Allison and Foster, 2004): “not good”, “fairly good”, … • Where cardinal (as with achieved life, ℓ), should we use the Gini or constant aversion measure? • Absolute Gini seems more appropriate than usual relative Gini (compare achieved life spans of 50,70 and 90 with 55, 77 and 99). • Kolm index more appropriate for same reason, but if use epsilon, then “we should be more averse to inequalities in health than to inequalities of income” (Anand, 2004, page 16). • If we are combining W[Iy,Ih] should W be homothetic? • Does marginal value of extra year decline with ℓ? WDR 1993 shows value of extra year rising up to around age 10. • For these reasons, may be limited to first order comparisons.

13. CONCLUSIONS • For health • Need to distinguish health inequality and health inequity • Health may be ordinal variable, and limit to first order dominance. • Cardinal variables absolute rather than relative • In general • Need to treat each dimension on its own terms: (Trannoy and Muller).

14. Introduction: On lateral thinking • Indices covering different dimensions: health inequality • Keeping dimensions separate and radar diagrams • Interdependence and copulas • Conclusions: Directions for Future Research

15. How portray different dimensions? “It would always be desirable to have a snapshot view of the status of human development in various States while analysing their respective strengths and weaknesses on some relevant human development indicators. … To meet this objective the National Human Development Report introduces Development Radars” (Indian Planning Commission, 2002, page 12). Radar diagram, also known as web chart, spider chart or star plot.

16. Or as “Wind rose” (Crothers, Field Studies, 1981) Wind direction at St Ann’s Head, Pembrokeshire

17. How do we interpret these charts? “The Development Radars give a snapshot view. … They capture the relative contribution of different dimensions in overall human development. The greater the shaded area of any indicator the better is the attainment on that indicator. … the more is the shaded area corresponding to the 1990s vis a vis the area corresponding to the 1980s, the faster is the pace of human development.” (Indian Planning Commission, 2002, pages 15-16).

18. Problems: • Effect of improvement in indicator i depends on sum of indicators h and j. • Area depends on ordering of indicators: “the order of assigning variables to features in a symbol (to the rays in a star, for example) is usually rather arbitrary, yet the shape of the symbol, and to some extent the effectiveness of the whole display, can depend critically on the assignment” (Chambers, Cleveland, Kleiner and Tukey, 1983, page 164) . • Area is visually misleading (Tufte). • Advantages over histogram: • Could allow weighting (?)

19. First set of ten EU Social Indicators Income X Health X Employment Education

20. CONCLUSIONS • Need to be careful in use of radar diagrams. • May introduce unplanned interactions. • Positive message: when moving to higher level of evaluation with multiple dimensions, need to consider the full range of domains, including those not represented by indicators.

21. Introduction: On lateral thinking • Indices covering different dimensions: income and health inequality • Keeping dimensions separate and radar diagrams • Interdependence and copulas • Conclusions: Directions for Future Research

22. Interdependence at level of individual: that health status positively correlated with income. If individual’s circumstances evaluated by v(y,h), then aversion to correlation implies that vy h ≤ 0. Positive transformations of h leave the sign unchanged. This property is assumed here. Let the joint cumulative distribution be H(y,h), with marginal cumulative distributions F(y) and G(h). Where the marginal distributions are identical, the first order dominance condition is that H(y,h) be less (or identical) for all y and h.

23. Thinking laterally, how does this relate to copulas? Sklar’s theorem: given a joint distribution function H and respective marginal distribution functions, there exists a copula C that binds the margins to give the joint distribution: H(y,h) = C{F(y),G(h)}. Where the marginal distributions are identical, the first order dominance condition is that H(y,h) be less (or identical) for all y and h. Separates differences in marginal distributions from differences in interdependence.

24. G(h) C{F,G} h Only ranks matter F(y) y

25. Health inequity in European Union

26. Difference from independence

28. Digression: Understanding total income G(h) Non-earned income C{F,G} F(y) • Rise in top income shares: • Marginal distribution of earnings • Marginal distribution of non-earned income • Change in correlation Earned income

29. Conclusions: Directions for Future Research • Lateral thinking can be highly productive; good reason for reading widely. • Inequality measurement benefited. • BUT need care when extending to new dimensions: treat each dimension on its own terms. • Need care when using analytical tools developed for other purposes (e.g. radar diagrams); have to ask what they bring to the party. • There are promising paths to pursue (e.g. copulas).