1. What is a QALY worth? Admissible utility functions for health, longevity & wealth James K. Hammitt Harvard University (Center for Risk Analysis) Toulouse School of Economics (LERNA-INRA)

2. Standard metrics for valuing health Willingness to pay (WTP) Widely used in environmental & transportation applications Quality-adjusted life years (QALYs) Widely used in public health and medical applications DALYs (disability-adjusted) Utility for health and wealth What utility functions are consistent with both concepts? Implications for WTP to reduce health risk 2

3. Louis’ contribution U(w, h) U 1 > 0, U 2 > 0 U 11 < 0, U 22 < 0 U 12 ≥ 0 Bleichrodt, Crainich, Eeckhoudt, ‘Comorbidities and the willingness to pay for health improvements’, J Public Econ 2003 3

4. Quality-adjusted life years Sum of duration-weighted “health-related quality of life” q(h) = HRQL T = duration v(∙) usually linear or present value Neglect non-health consequences What is ‘health’ (h)? Includes ‘self-care’ & ‘usual activities’? 4

5. QALYs Strong assumptions about preferences 1. Constant proportional tradeoff of duration for health HRQL independent of duration, consumption, & other factors 2. Risk preference for duration independent of health Can be generalized 5

6. Willingness to pay Compensating variation Change in money (that can be used for any purpose) that exactly offsets change in health risk Weak assumptions More money is better –Non-satiation (local) 6

7. Admissible utility functions Assume (for any level of wealth) Preferences for health and longevity are consistent with QALYs –Q(h, T) = v[q(h), T] –q(h) independent of w Then (future) lifetime utility u(h, T, w) = [Q(h, T)] a(w) + b(w) –a(w) > 0 7

8. 8 Marginal utility of wealth u(h, T, w) = [Q(h, T)] a(w) + b(w)

9. 9 Marginal WTP per QALY (  ) u(h, T, w) = [Q(h, T)] a(w) + b(w) Marginal value of QALY Effect of health & longevity on wealth (neglect)  is independent of Q (future health & longevity) iff a' = 0 → marginal utility of wealth independent of survival, health, & longevity a' > 0 →  decreases with Q Diminishing marginal WTP with severity & duration of potential illness WTP increases with age and chronic/future illness

10. 10 Value per statistical life u a = utility if survive period u d = utility if die (bequest) u a > u d u a ' > u d ' ≥ 0 Drèze, ‘L’utilité sociale d’une vie humaine’, Revue Française de Recherche Opérationnelle 1962

11. 11 QALYs and VSL Utility if survive Utility of bequest b'(w) = 0 → VSL is independent of QALYs b'(w) > 0 → VSL increases with QALYs, but less than proportionately

12. 12 Implications WTP per QALY not constant Decreases with future QALYs Value of reducing morbidity risk not proportional to expected QALY gain VSL not proportional to future QALYs Decreases with future QALYs –VSL independent of QALYs if indifferent to bequest (b ' = 0) Value of reducing mortality risk not proportional to expected QALY (or life expectancy) gain