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)
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
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
3
Quality-adjusted life years
Sum of duration-weighted “health-related
quality of life”

• q(h) = HRQL
QALYs  v q (h)
• T = duration
• v(∙) usually linear or present value
T
Neglect non-health consequences
• What is ‘health’ (h)?
• Includes ‘self-care’ & ‘usual activities’?
4
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
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
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
Marginal utility of wealth
u(h, T, w) = [Q(h, T)] a(w) + b(w)
U
 Qa  w   b  w 
w
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)
→ 𝑢12 =
𝜕2 𝑢
𝜕𝑤𝜕ℎ
>0
8
Marginal WTP per QALY (m)
u(h, T, w) = [Q(h , T)] a(w) + b(w)
a  w
dw
w
m


dQ Qa  w   b  w  Q
Marginal value
of QALY
Effect of health & longevity
on wealth (neglect)
m is independent of Q (future health & longevity) iff a' = 0
→ marginal utility of wealth independent of survival, health, & longevity
a' > 0 → m decreases with Q
Diminishing marginal WTP with severity & duration of potential illness
WTP increases with age and chronic/future illness
9
Value per statistical life
ua  w   ud  w 
u
VSL 

1  p  ua  w   pua  w  Eu
ua = utility if survive period
ud = utility if die (bequest)
ua > ud
u a' > u d' ≥ 0
Drèze, ‘L’utilité sociale d’une vie humaine’, Revue
Française de Recherche Opérationnelle 1962
10
QALYs and VSL
ua w  Qa w  bw
ud w  bw
Utility if survive
Utility of bequest
Qa w
VSL 
1  p Qaw  bw
b'(w) = 0 → VSL is independent of QALYs
b'(w) > 0 → VSL increases with QALYs, but less
than proportionately
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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
12