What is a QALY worth? Admissible utility functions for health, longevity & wealth

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

3. Louis’ contribution • U(w, h) • U1 > 0, U2 > 0 • U11 < 0, U22 < 0 • U12 ≥ 0 • Bleichrodt, Crainich, Eeckhoudt, ‘Comorbidities and the willingness to pay for health improvements’, J Public Econ 2003

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’?

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

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)

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

8. u(h, T, w) = [Q(h, T)] a(w) + b(w) Marginal utility of wealth • b' ≥ 0 (standard, marginal utility of bequest) • a' > 0 ↔ marginal utility of wealth greater if alive than dead (standard) • → marginal utility of wealth increasing with • Health (standard?) • Longevity (highly plausible) • →

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

10. Value per statistical life ua = utility if survive period ud= utility if die (bequest) ua > udua' > ud' ≥ 0 Drèze, ‘L’utilité sociale d’une vie humaine’, Revue Française de Recherche Opérationnelle1962

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. 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