Journal of Health Economics 29 (2010) 541–548
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Journal of Health Economics
journal homepage: www.elsevier.com/locate/econbase
Increasing marginal utility of small increases in life-expectancy?
Results from a population survey
Maria Knoph Kvamme a,∗ , Dorte Gyrd-Hansen b,c , Jan Abel Olsen a,d , Ivar Sønbø Kristiansen a,b
a
Institute of Health Management and Health Economics, University of Oslo, Norway
Insitute of Public Health - Health Economics Research Unit, University of Southern Denmark, Winsløwsvej 9, DK-5000, Denmark
c
Danish Institute for Health Services Research, Dampfærgevej 27-29, DK-21000 Denmark
d
Department of Community Medicine, University of Tromsø, 9037 Tromsø, Norway
b
a r t i c l e
i n f o
Article history:
Received 25 November 2008
Received in revised form
18 December 2009
Accepted 30 March 2010
Available online 9 April 2010
JEL classification:
A13
I19
a b s t r a c t
The standard practice in cost-effectiveness analyses of health care is to assign a linear value to increasing
lifetime gains. The aim of the current study was to examine the possible existence of non-linear utility for
short life extensions. A representative sample of the Norwegian population, aged 40–59 years (n = 2402),
was asked to imagine that they had a limited remaining lifetime (1 year or 10 years) and were offered
a treatment that would increase lifetime by a specified amount of time from 1 week to 1 year. In all
scenarios, the price per week of life extension was held constant. The proportion of respondents that
accepted the treatment increased with increasing extensions, indicating a convex utility function. The
result suggests increasing marginal utility for life extensions up to 1 year.
© 2010 Elsevier B.V. All rights reserved.
Keywords:
Cost-effectiveness-methodology
Linear models
Willingness to pay
1. Introduction
In cost-effectiveness analyses (CEA) the health gains in the
denominator of the cost-effectiveness ratio (CER) are valued linearly. What matters is the cost per unit of health outcomes,
independent of the size of this outcome; e.g. D 500,000 for 10 QALYs,
and D 5000 for 0.1 QALY are both recalculated to a CER of D 50,000
per QALY. However, while everybody would agree about linearity in the numerator, i.e. that D 500,000 represents 100 times as
much money as D 5000, the question is whether we agree with
the assumption of linearity in the denominator, i.e. do we value
10 QALYs gained 100 times as much as 0.1 QALY gained? If people value QALYs gained linearly, CEA is consistent with people’s
preferences. If they do not, it is not.
The assumption of linearity becomes an issue only if we take
the “welfarists” view that CEA should reflect people’s preferences.
∗ Corresponding author at: Institute of Health Management and Health Economics, PO Box 1089 Blindern, N-0317 Oslo, Norway. Tel.: +47 22 45 15 00;
fax: +47 22 84 50 91.
E-mail addresses: [email protected] (M.K. Kvamme),
[email protected] (D. Gyrd-Hansen), [email protected] (J.A. Olsen),
[email protected] (I.S. Kristiansen).
0167-6296/$ – see front matter © 2010 Elsevier B.V. All rights reserved.
doi:10.1016/j.jhealeco.2010.03.010
If QALYs are considered to be a sophisticated measure of health,
rather than a reflection of people’s preferences for different sizes
of health gains, then clearly 10 QALYs are measured to be 100
times as much health than is 0.1 QALY. However, if the existence of
non-linear preferences over QALYs is to be accounted for, two alternative views emerge on how such preferences should be elicited: (i)
QALYs should reflect individual utility of increasing health gains or
(ii) QALYs should reflect people’s social value of the distribution of
QALYs gained. Clearly, people may express different views on this
depending on whether choices are being framed in an individual
or a social context.
This paper seeks to elicit the possible existence of non-linear
individual preferences for increasing health gains. Rather than
expressing different health gains in an ex ante context of different
probabilities, we use riskless life extensions to avoid the cognitive
problems of interpreting differences in small probabilities. Differences in units of time are much simpler to imagine than differences
in probabilities, and there is evidence to suggest that people are
more consistent regarding preferences for differences in time units
than for differences in risks (Dahl et al., 2007; Kristiansen et al.,
2002; Kristiansen and Gyrd-Hansen, 2006).
The most widely used way of measuring preferences for different quantities of goods is through willingness to pay (WTP). In this
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M.K. Kvamme et al. / Journal of Health Economics 29 (2010) 541–548
context, WTP is used to elicit how much money one is willing to
sacrifice for increased lifetime. Given that people can make rational
trade-offs between increasing lifetime gains and increasing sacrifices of income, the money metric may be applied as a useful metric
for valuing lifetime gains. However, when the price to pay for a
good (in this case life extension) increases, the resulting disposable income diminishes. The valuation of the remaining income
available for spending on alternative goods increases when the
price of the life extension increases with longer extensions. Hence,
respondents with relatively low income may reach a highest possible amount of what they are able to pay for a life extension and thus
face a budget constraint problem. In our case we risk that the relationship between WTP measured in money and actual valuation
changes as the amounts approaches the disposable income.
One severe problem with WTP is that the absolute values
expressed for hypothetical goods are sensitive to framing. In this
study, it is the relative values that matter, in that we seek to elicit
whether people have linear or non-linear marginal utilities of small
increases in health gains. The aim is not to elicit a true value of a
unit of a time gain, but rather to discover how the relative values of
different sizes of time gains change.
Theoretical studies of QALY models usually assume that life
extensions are in the order of months and years rather than days
and weeks. In practice, however, most health interventions prolong life by less than 1 year, and frequently in the order of days and
weeks (Wright and Weinstein, 1998).
A recent literature review indicates threshold effects both in
terms of quality and quantity of life, and the authors recommend that further research should be directed toward this question
(Dolan et al., 2005). The hypothesis of a life extension threshold was
put forward by Olsen (2000), and has later achieved some empirical support both in terms of individual preferences (Gyrd-Hansen
and Kristiansen, 2007; Rodriguez-Miguez and Pinto-Prades, 2002)
and societal preferences (Olsen, 2000; Rodriguez-Miguez and
Pinto-Prades, 2002; Gyrd-Hansen and Kristiansen, 2007). Previous
studies have examined valuations of health benefits in years (Dolan
and Jones-Lee, 1997; Johannesson et al., 1994; Rodriguez-Miguez
and Pinto-Prades, 2002). Given that many health care resources
are devoted to interventions with much shorter lifetime gains, we
chose to investigate the valuation of life extensions from 1 week to
1 year.
Different QALY models, in which the assumption of linearity is
relaxed, do exist. These models allow for curved utility functions
for duration, but are not commonly used in economic evaluations
(Abellán-Perpiñán et al., 2006; Doctor et al., 2004). Two specifications of the utility function for life-years have been suggested.
The first is the log/power group, W(t) = tr and the second is the
linear/exponential group W(t) = e−ct (Bleichrodt et al., 1999). The
log/power model has been used by several authors (Miyamoto and
Eraker, 1988; Pliskin et al., 1980; Stiggelbout et al., 1994; AbellánPerpiñán et al., 2006; Bleichrodt and Pinto, 2000; Doctor et al.,
2004). The latest study’s empirical findings indicate that the power
QALY model yields the best predictive validity, and the authors conclude that the best-fitted power coefficient is 0.65. This parameter
indicates that the utility function for life duration is concave with
diminishing marginal utility of life-years. The power coefficient
may reflect risk aversion, diminishing marginal utility, or risk neutrality (Abellán-Perpiñán et al., 2006). A linear/exponential model
has also been proposed (Moore and Kip Viscusi, 1990).
The aim of the present study is to examine the possible existence of non-linear utility of short life extensions in an individual
perspective. Based on previous research we hypothesised a function including increasing marginal utility of life extensions in the
start (convex shape) up to a certain amount of time gains, followed
by a concave shape when the marginal utility becomes diminishing.
This study is limited to the start of this function, i.e. do people have
increasing marginal utility of small increases in life-expectancy?
2. Methodology
The study was designed to have three successive willingness to
pay valuations of time gains that increased fourfold from the initial
gain to the second, as well as from the second to the third gain, and
with opening bids that increased accordingly. Respondents were
thus encouraged to think about the relationship between different
sizes of time gains and their valuations of these gains. While not all
respondents may notice this proportionality between opening bids
and size of gains, this framing was deliberately intended to set the
respondents’ minds in linearity mode. In that way, any departure
from linearity would reflect a conscious non-linear preference, or
an inability to increase linearly due to budget restrictions.
There were four versions of the questionnaire (Q1 –Q4 ), differing
in terms of the time perspective without treatment (T0 = 1 year in
Q1 and Q2 ; T0 = 10 years in Q3 and Q4 ), and the length of time gains
(tG = 1 week, 1 month, 4 months in Q1 ; tG = 2 weeks, 2 months, 8
months in Q2 and Q3 , and; tG = 3 weeks, 3 months, 1 year in Q4 )
(Table 1).
The willingness to pay valuation had a two-step procedure: (i)
yes/no to an opening bid, followed by; (ii) a payment card with nine
options ranging from 0, via the opening bid as the mid-point, to a
top-end value six times the value of the opening bid, plus an openended option to fill in an alternative value. The amounts were at
least doubled from one alternative amount to the next, with exception for the last amount which was increased by 50% from the one
before. It was not possible to make only minor deviations unless
respondents chose their own amount (6.8% of the respondents did
so).
Prior to the study on which the paper reports, a pilot study was
conducted to test the appropriateness of the questions for valuing life extensions and to identify relevant prices per week of life
extension. The pilot respondents were asked to imagine that they
were suffering from a fatal disease with 1 or 10 years remaining
lifetime, and were offered a hypothetical treatment that would
increase their lifetime by various durations. The pilot sample consisted of 38 persons (22 male, 16 female) aged 23–85 years.1 The
main questions concerned whether the respondent were willing to
pay NOK 100 (1D = 8 NOK) per week to increase their lifetime, and
what amount they at maximum would be willing to pay for the
same increase in life-expectancy. A low initial amount was used
to reduce the influence of income effect in the subsequent valuation of the larger gains. The amount per week was the same for the
perspectives of T0 = 1 and 10. The values from the 50th percentile
for T0 = 1 (NOK 2500) and the 46th percentile for T0 = 10 (NOK 500)
were used as the amounts in the main study. The 46th percentile
for T0 = 10 was chosen instead of the 50th percentile to obtain a
round figure.
A logistic regression was performed to test the null hypothesis of a linear valuation of life extensions, i.e. the same rate of
acceptance for all offers of life extension within each questionnaire (Table 1). We also used logistic regression analysis to explore
predictors of responses to this opening bid question (Table 3).
To account for interdependence in responses (each respondent
valued three treatments) we used a robust cluster technique to esti-
1
The age range and spread of our pilot sample was not optimal for our study
because of the difference in range and spread of ages of the respondents in the pilot
study compared to the range and spread of ages of the respondents in the main
study. In our pilot there were only four individuals between 40 and 59 years which
mean that our results cannot be said to be robust to the definition of the pilot sample.
M.K. Kvamme et al. / Journal of Health Economics 29 (2010) 541–548
543
Table 1
Survey design in the four versions of the questionnaires (Q1 –Q4 ): no-treatment profile (T0 ); time gains (tG ); opening bids, and; acceptance rates to an offer of life extensions
for a fixed amount per week (NOK 2500 per week when gained after T0 = 1 year, and NOK 500 when gained after T0 = 10 years).
Time gain tG
T0 = 1 year
T0 = 10 years
Q1 (N= 550)
1 week
2 week
3 week
1 month
2 months
3 months
4 months
8 months
1 year
Q2 (n = 538)
Opening bid
Acceptance rate
2500
48%
10,000
Q3 (n = 525)
Q4 (n = 530)
Opening bid
Acceptance rate
Opening bid
Acceptance rate
5000
54%
1000
55%
Opening bid
Acceptance rate
1500
60%
6000
66%
25,000
76%
55%
20,000
40,000
59%
4000
62%
59%
80,000
p-Trend
<0.001
61%
16,000
65%
0.001
<0.001
<0.001
Table 2
Percentage of respondents with zero WTP and percentage respondents in different categories of reasons why they were unwilling to pay.
Offer of life extension
n
Respondents with zero WTP
Total
Perspective T0 = 1 year
Time gain tG
1 week
2 weeks
4 weeks
2 months
4 months
8 months
Offer of life extension
Q1
482
481
481
481
479
480
n
33.4%
27.2%
24.7%
22.8%
18.3%
14.6%
Time gain tG
2 weeks
3 weeks
2 months
3 months
8 months
1 year
Q3
435
446
433
446
432
444
No value
Cannot afford
Other or unknown
21.2%
16.4%
14.6%
11.9%
9.7%
4.6%
9.3%
9.6%
7.3%
8.8%
5.5%
6.9%
0.6%
0.6%
1.0%
1.3%
1.7%
2.7%
2.3%
0.6%
1.7%
0.6%
0.8%
0.2%
Respondents with zero WTP
Total
Perspective T0 = 10 years
Benefit not large enough
Q2
Benefit not large enough
No value
Cannot afford
Other or unknown
29.3%
22.9%
22.3%
16.6%
14.7%
6.5%
9.7%
9.2%
8.4%
7.9%
4.4%
4.7%
0.0%
0.0%
0.0%
0.5%
0.0%
1.8%
0.7%
2.2%
0.9%
2.3%
1.2%
1.4%
Q4
40.0%
34.3%
31.9%
27.3%
20.6%
14.4%
mate confidence intervals of the regression model parameters. We
analysed percentages of zero WTP responses (Table 2), and respondents’ individual preference patterns for variations in life-gains
(Table 4).
The value of the opening bid was the equivalent of NOK 2500
per week when gained after T0 = 1 year, and NOK 500 per week
when gained after T0 = 10 years. By comparing Q2 versus Q3 , we can
identify any degree of income effects, in that the absolute value of
the opening bid of the third and highest gain is NOK 80,000 in Q2
as compared with NOK 16,000 in Q3 .
These amounts would represent a significant slice of the
household budget particularly for low income respondents. A consequence of a strong budget constraint is a high marginal utility
of income. Hence, a decreasing valuation in monetary terms of
additional gains in life-expectancy may be due to higher marginal
utility of income, and not decreasing utility of additional health
gains. Since the aim of the present study is to elicit the functional
form of the utility function for health gains, merely using WTP as a
convenient instrument, any large discrepancies in marginal utility
of income across intervention scenarios may invalidate our task.
For the open-ended WTP question which follows the initial price
bid, this problem is most pronounced. Consequently, we expected
that the response to the initial price bid is a more valid measure of
relative valuations than is the maximum WTP. Another valid measure of relative valuations is changes in zero bids across increasing
magnitudes of health gains.
2.1. Data
A random sample from the Internet-based panel of TNS Gallup
Norway was used in the main study, stratified by gender and age.2
The survey was performed in February 2007. The total number
of respondents was 2402, all from the age group 40 to 59 years
old (Appendix A). It was believed that younger age groups would
be less interested in longevity questions, while older respondents
may have difficulties with this type of hypothetical questions or
they may feel that the 10 years perspective of expected remaining
lifetime would be too long to have personal relevance.
2
TNS Gallup had information on age, gender, level of education, household
income, personal income, marital status and whether the respondent was living
with children under 15 years old.
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M.K. Kvamme et al. / Journal of Health Economics 29 (2010) 541–548
The respondents indicated on a 5-point scale how certain they
were about their answers. They were also asked to indicate their
health status on a 5-point scale, ranging from “very good” to “very
poor”. Data were analysed in STATA/SE 10.0.
2.2. Invalid answers
For the analyses of the opening dichotomous bid WTP questions
(yes/no), the following groups of answers were excluded: Respondents who stated they could not relate to the question (n = 121)
were omitted. Respondents who rejected the life extension offers
and had zero WTP for them were asked to state a reason for their
answer. They were omitted if they gave a reason showing they
had not understood the question, encompassing protest against
the study design, or giving answers such as “I think the government
should pay for health care” (n = 71). Finally, answers were classified
inconsistent and omitted if a respondent had been willing to pay
a positive amount of money for a short life extension but nothing
for a longer (n = 67). In all 259 were excluded, leaving us with 2143
respondents included in the analyses of the opening bid-questions.
In the follow-up questions of maximum WTP, some additional answers were considered invalid. Respondents who: (i) were
‘scope-insensitive’ (the same WTP-amount no matter the size of
the health gain, n = 6) or (ii) had stated a smaller max WTP for a
longer life extension than for a shorter life extension (n = 273) were
excluded. There were 1864 valid responses for the analysis based
on max WTP values.
The four reduced respondent groups (one for each of the questionnaire versions) were analysed with respect to differences in
terms of age, gender, education, household income and marital status (Appendix B). There were no statistically significant differences
between the groups.
3. Results
3.1. Acceptance rates
All four versions of the questionnaire showed increasing acceptance rates to the opening bid with increasing time gains (Table 1).
Except for the shortest gain (1 week in Q1 ), all other opening bids
gave majority acceptance among respondents. At the other end,
the highest gain (1 year in Q4 ) represents the highest absolute
acceptance rate, but more importantly: the highest acceptance rate
relative to the acceptance rate of the first choice where 3 weeks is
on offer (76% vs. 60%).
When background variables were included in the analysis
(Table 3), the results from Table 1 were confirmed. Life extension
was entered as a continuous variable in an alternative model to the
one presented in Table 3, yielding highly statistically significant
(p < 0.001) coefficients of 1.014 in T1 and 1.018 in T10 .
Because the price per week was kept constant (NOK 2500 for
T0 = 1 and NOK 500 T0 = 10) in the different life extension offers (tG ),
the percentage acceptance would be the same within each questionnaire if the respondents had linear utility functions. In the first
logistic regression analysis, we tested the trend of same acceptance
rates for the three offers of life extensions within each questionnaire. The odds for acceptance were increasing significantly for
all offers of life extension with increasing duration in both time
perspectives, i.e. indicating that the average respondent displayed
increasing marginal utility of increasing gains. Note, however, that
the lowest relative increase in acceptance rates occurred in Q2 ,
something which is likely to be explained by an income effect, i.e.
that the highest bid-values had a deterrent effect. Respondents in
Q3 were presented with identical gains as those in Q2 , but the relative increase in acceptance was higher, although the difference was
not statistically significant.
3.2. Zero bids
The zero bids gave the reverse picture of the acceptance rates,
and were as such consistent with a pattern of increasing marginal
utility of increasing gains. The proportions of respondents that
were not willing to pay anything at all declined with increasing
life extension offers (Table 2). For T0 = 1, the proportion of respondents with zero WTP was 33.4% for 1 week, and 14.6% for 8 months.
For T0 = 10, the proportions were 40.0% for 2 weeks and 14.4% for 1
year extension. While the zero bids showed a consistent pattern in
all questionnaire versions in that they diminished with increasing
gains, it is worth noting that one out of seven respondents (14.4%
and 14.6%) were not willing to pay anything for such significant
gains as 8 months extra lifetime on top of 1 year, or 1 year extra on
top of 10 years.
Respondents’ justifications for their zero WTP are given in the
last four columns of Table 2. The main reasons are that the benefit
is not large enough, or that the health gain on offer has no value.
While it might make sense to provide such answers for small gains,
it appears somewhat strange when the gain is 8 months on top of a
1-year life-expectancy. Still, when questioned as to why they were
unwilling to pay, respondents might feel obliged to give a reason, no
matter how (un)convincing it may sound. Note that the proportions
saying they could not afford are small, with the largest proportion
(2.7%) relating to the highest opening bid (NOK 80,000).
3.3. Explaining variations in acceptance rates
A logistic regression model (Table 3) with the acceptance
rates as the dependent variable showed that standard sociodemographic variables, such as age, gender, education, and living
with children, had no explanatory power. However, the influence of
health status was close to being statistically significant (p = 0.071 in
T1 and 0.077 in T10 ) indicating that the odds for accepting the offer
decreased with poorer health state. It is quite intuitive that people in a good health state are more inclined to pay for additional
lifetime than people in a poor health state. The lack of impact of
the age variable might be explained by the narrow age span of the
respondents (40–59 years) in this survey.
Household income and life extension on offer had a statistically
significant effect on acceptance rates. Respondents with a higher
household income were more inclined to accept the intervention.
In comparison to the intervention which offered the smallest gain
in life-expectancy (1 and 2 weeks, respectively), the interventions
which offered larger health gains tended to result in higher odd
ratios. In an additional model an interaction variable between levels
of household income and life extension was included. The coefficient was positive and statistically significant, indicating that those
with higher income to a greater extent exhibit increasing marginal
utility over life extensions. This is a pattern that is expected if we
assume that the rate of increase in marginal utility of income is
higher for low income groups.
Logistic regressions with dummies for all the life extensions
on offer were performed in order to examine whether acceptance
rates for consecutive following life extensions were statistically
significantly different from zero or not. In these tests we did not
find a monotonically increasing marginal WTP for increasing life
extensions. However, when life extension was included as one
(continuous) variable it was significantly increasing. We cannot
infer the exact form of the life extension function with our sample. In the 1-year perspective, the only significant difference was
between 1 week and 2 weeks life extension (p = 0.035). In this time
gap we found the highest increase of respondents who accepted the
life extension offer (from 48% to 54%). In the 10-year perspective,
there were two statistically significant differences but here they
M.K. Kvamme et al. / Journal of Health Economics 29 (2010) 541–548
545
Table 3
Logistic regression of responses to an offer of life extensions for a fixed amount per week (NOK 2500 or NOK 500).
Explanatory variables
Perspective = 1 year 957 clusters in respid
Perspective = 10 years 915 clusters in respid
Odds Ratio
p
95% Conf. Interval
Odds Ratio
p
95% Conf. Interval
Age (in years)
Gender (0 = female, 1 = male)
Education: University (0 = no, 1 = yes)
Household income (7 levelsa )
Living with children <15 years (0 = no, 1= yes)
Health status (1 = very good, 5 = very poor)
0.994
0.860
1.095
1.314
1.109
0.874
0.641
0.201
0.457
<0.001
0.460
0.071
(0.971; 1.019)
(0.682; 1.084)
(0.862; 1.391)
(1.185; 1.458)
(0.843; 1.460)
(0.755; 1.012)
0.983
0.925
1.014
1.213
0.861
0.874
0.212
0.529
0.910
<0.001
0.345
0.077
(0.957; 1.010)
(0.725; 1.180)
(0.792; 1.299)
(1.088; 1.351)
(0.631; 1.175)
(0.753; 1.015)
Life extension
1 week
2 weeks
3 weeks
1 month
2 months
3 months
4 months
8 months
1 year
1 (reference)
1.323
0.035
(1.019; 1.716)
Not included in 1 year perspective
1.353
<0.001
(1.191; 1.537)
1.636
<0.001
(1.260; 2.124)
Not included in 1 year perspective
1.569
<0.001
(1.313; 1.875)
1.807
<0.001
(1.391; 2.347)
Not included in 1 year perspective
Not included in 10 years persp.
1 (reference)
1.293
0.060
Not included in 10 years persp.
1.326
<0.001
1.748
<0.001
Not included in 10 years persp.
1.574
<0.001
3.230
<0.001
Log pseudolikelihood = −1911.7875Pseudo-R2 = 0.0313
Log pseudolikelihood = −1721.6699
(0.990; 1.689)
(1.170; 1.503)
(1.331; 2.297)
(1.311; 1.890)
(2.407; 4.334)
Pseudo-R2 = 0.0330
a
Household income in NOK, level 1: <200,000, level 2: 200,000–399,999, level 3: 400,000–599,999, level 4: 600,000–799,999, level 5: 800,000–999,999, level 6:
1,000,000–1,999,999, level 7: ≥1,200,000.
Table 4
Preference patterns for increasing gains, as inferred from differences in individual
maximum WTP for the three gains being valued.
Individual preference
Increasing marginal utility
Constant marginal utility
Diminishing marginal utility
Others
Scope insensitive
Inconsistent
n
1 year
10 years
Q1
Q2
Q3
Q4
22%
31%
24%
7%
1%
14%
558
21%
30%
26%
9%
2%
12%
555
19%
34%
17%
7%
3%
19%
544
26%
30%
17%
8%
2%
18%
545
Total
22%
31%
21%
7%
2%
16%
2202
Valid total
27%
38%
26%
9%
1807
were between 2 and 3 months (p = 0.049) and between the two
longest life extension offers; 8 months and 1 year (p < 0.001). In the
last time gap the percentage points of respondents who accepted
the offer increased from 65% to 76%. This might be explained by the
interpretation of 1 year as a considerable amount of time where
anyone easily can imagine what they would be able to do.
3.4. Linear or non-linear preferences as reflected by individuals’
max WTP
The advantage of presenting respondents with an open-ended
WTP questions is that a point of indifference and thus a maximum
WTP per week is identified for each individual and for each question presented to the individual. Consequently, it is informative to
analyse the data at a disaggregated level, since the shape of each
individual’s preference function can be identified through the three
responses given by each respondent.
Table 4 shows the distribution of respondents into various
‘preference camps’ depending on how their three maximum
willingness-to-pay values differed. WTPG1 refers to respondents’
max WTP per week for the first gain presented to them, WTPG2
refers to the second gain, and WTPG3 refers to the third and largest
gain. Assuming constant marginal utility of income across scenarios, we can categorize individuals into preference camps depending
on their pattern of WTP responses:
Three combinations suggest increasing marginal utility:
WTPG1 < WTPG2 < WTPG3
WTPG1 = WTPG2 < WTPG3
WTPG1 < WTPG2 = WTPG3
Constant marginal utility involves identical WTP per week:
WTPG1 = WTPG2 = WTPG3
Three combinations suggest diminishing marginal utility:
WTPG1 > WTPG2 > WTPG3
WTPG1 = WTPG2 > WTPG3
WTPG1 > WTPG2 = WTPG3
The following combinations suggest S-shaped or inverse Sshaped preferences (Others):
WTPG1 < WTPG2 > WTPG3
WTPG1 > WTPG2 < WTPG3
Table 4 suggests that the distribution of respondents across the
possible ‘preferences camps’ is fairly similar across the four versions of the questionnaire. The only exception seems to be in Q4
where a higher proportion signaled increasing marginal utility (26%
as compared with 20.5% in the remaining), a result which suggests
that a time gain of 1 year can be interpreted as being some sort of
a threshold.
The last column shows that among the valid total, 27% of respondents demonstrate increasing WTP over life extensions, whereas
38% exhibit constant WTP. It was argued above that maximum WTP
may be an invalid instrument for eliciting the functional form of
utility over life-expectancy gains, when these gains are as high as 8
months and 1 year. The argument was that increasing budget constraints may cause the marginal utility of income to increase, which
would result in decreasing WTP per added week of life-expectancy
despite valuations of added life-extension being at least as high as
initial gains. Table 4 demonstrates that despite the likely increase in
marginal utility of income, 65% of respondents express constant or
increasing WTP for added health gains, something which suggests
that at least 65% would most likely exhibit increasing marginal utility over lifetime gains within the range explored in this survey (from
1 week to 1 year).
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M.K. Kvamme et al. / Journal of Health Economics 29 (2010) 541–548
4. Discussion
The aim of the present study was to test for the possible existence of non-linear utility over small life-expectancy gains. We
hypothesised that individuals have increasing marginal utility of
own lifetime gains when these gains are of a smaller magnitude. We
examined a range for added lifetime of between 1 week and 1 year,
and asked respondents to value these life extensions in monetary
units.
The sample in the main study, and in particular the reduced
sample in the main study after removal of invalid answers, is
not necessarily representative for the Norwegian population aged
40–59 years. However, no statistically significant differences were
found in socio-economic characteristics between the reduced and
the total sample.
The exercise involved a close-ended WTP question in which the
price per week gained was held constant across all scenarios. This
design is unique, due to its simplicity. Rather than seeking to elicit
the absolute valuations of different outcomes, our focus was to
employ a test for non-linearity by measuring the relative valuations
of different sizes of lifetime gains. An increasing rate of acceptance
of an intervention provided at a given price per week gained would
imply an increasing valuation. And assuming a constant marginal
utility of income, a decreasing propensity to accept, would imply
a lower valuation of the intervention. The results (Table 1) suggest that respondents are more inclined to accept an intervention,
which offers a larger lifetime gain, than one which offers a shorter
lifetime gain. When price per week is held constant, the total price
to be paid is high for larger gains. A price of for example NOK
80,000 (D 10,000) will most likely strain the budget, and represent
higher than average opportunity costs. Despite the likelihood of an
income effect, respondents are still more prone to accept the large
gain at the higher price. There is a statistically significant increasing propensity to accept the intervention as the lifetime gain on
offer is increased, but the acceptance rate increases at a lower rate,
suggesting that the increase in expressed marginal WTP over lifetime gains is affected by increases in marginal utility of income.
More generally, the result that individuals experience increasing
marginal utility over lifetime is a robust result, since we show that
a similar pattern of preferences exist for various ranges of time
gains (from 1 week to 1 year) and for different time perspectives
(1 year or 10 years into the future).
In order to further validate our result, respondents were asked
to indicate their maximum WTP for the intervention in a follow-up
question. For each individual we have three consecutive maximum
WTP statements for interventions, which offer still larger health
gains. This allowed us to define the individual respondent’s utility function as increasing, constant or decreasing in health gains.
The results from the close-ended WTP questions were confirmed,
since it was shown that a likely majority of respondents exhibited
increasing or constant utility over health gains (Table 4). Due to possible budget constraints, which may affect the monetary valuation
of large life-gains, constant WTP across life-gains may well signify
increasing marginal utility over lifetime gains and decreasing WTP
over life-gains does not necessarily reflect decreasing marginal utility over lifetime. Assuming that there exists some degree of income
effect, the conservative conclusion is that at least 65% (27% plus
38%) of the respondents exhibit increasing marginal utility over
health gains.
The conclusion that a significant fraction of the population
experience increasing marginal utility for larger lifetime gains is
further supported by analysing the frequency of zero bid responses
(Table 2). Irrespective of setting (time perspective and time gain
range) we observe a greater inclination to respond with a zero bid
when the time gain is small, and the frequency of zero bidders are
clearly reduced as the time gain increases. Moreover, the zero bid
is more frequently justified by the benefit not being large enough
when smaller gains are offered. This justification is more frequently
invoked when the gains are on offer in the future.
No clear minimum threshold value for when a gain in lifetime is
considered worthwhile could be inferred as the increases in acceptance rates were rather smooth. The answers on acceptance of
different offers of life extensions showed a clear increasing trend
for longer life extension offers in both the 1- and the 10-year perspectives. Because the “yes”-answers increased from 48% to 61% for
T0 = 1 when the offers increased from 1 week to 8 months, and from
55% to 76% for T0 = 10 when the offers increased from 2 weeks to 1
year, this could mean that the respondents had different threshold
values. The offer of 1 week was the only offer given in the study
with an acceptance level below 50% and could represent a threshold value. This may suggest that 1 week was considered too short
to be a real gain in life extension. However, the trend of increasing acceptance for longer life extensions as well as the declining
level of respondents not willing to pay anything when offers of life
extensions were increased, both point to individual thresholds.
The design of our study can give a “yea-saying” bias in the tendency to accept the offer of life extension, but this effect would
probably apply equally to all extensions and would therefore give
a bias toward linearity in utility. The association between acceptance and both income and health status (Table 3) further supports
the validity of the results.
These results confirm a priori assumptions that income will
affect WTP, and better health states increases the utility associated with life extensions, suggesting that responses are based on
some degree of reflection. Interestingly, other socio-economic factors such as gender, level of education, age and family situation did
not have any impact on acceptance rates.
Previous studies have shown that people generally have been
unwilling to trade off life-expectancy for small improvements
in quality of life (Miyamoto and Eraker, 1988). We find that
respondent is less willing to trade money for life-expectancy
when life-expectancy gains are small suggesting that small
life-expectancy gains are of little value. These results seem contradictory, since the former observation implies that even small
life-expectancy gains have a significant value to individuals, while
the latter implies the opposite. The underlying explanation for the
apparent contradiction may be a lack of symmetry in losses and
gains in life-expectancy.
The overall conclusion of our analysis is that irrespective of
whether we focus on the relative valuations (responses to constant price per week), absolute valuations (preference camps) or
zero bids, the pattern in responses implies that a majority of
respondents have increasing marginal utility over time gains. This
conclusion holds regardless of time perspective, and for various
time gain ranges within the bounds of 1 week and 1 year. Given
the strong design of our analysis, the conclusion is equally strong.
Small time gains are not valued highly, and much less than larger
time gains. Our results clearly suggest that if QALYs are meant to
reflect individual utility of health gains, focusing only on the sum
of QALYs or life-years gained poorly reflects the preferences of the
population. Our results suggest that the distribution of health gains
is equally important, and that the small time gain given to the
many is valued lower than the larger time gain to the few. The
outcome of this study suggests that less weight should be given to
interventions that provide only short time gains in life-expectancy.
This result opposes earlier suggestions of diminishing marginal
utility over lifetime and a concave utility function over lifeyears (Abellán-Perpiñán et al., 2006; Bleichrodt and Pinto, 2000;
Stiggelbout et al., 1994) The power QALY models previously proposed: U(q,t) = H(q)tr (r = 0.65, 0.78 and 0.74) indicate that the utility
M.K. Kvamme et al. / Journal of Health Economics 29 (2010) 541–548
function for life duration is concave rather than linear (AbellánPerpiñán et al., 2006; Bleichrodt and Pinto, 2000; Stiggelbout et
al., 1994). The results in our study indicate an increasing marginal
utility for lifetime up to 1 year, which is the opposite of a power
function for t with a coefficient of r < 1.
The difference in our results and the results of previous studies
is most likely explained by the fact that different magnitudes of life
extensions have been examined. We have examined extensions of
up to 1 year while the other researchers have used greater extensions, most of them up to 10 years. We used a riskless procedure
with the aim of focusing on preferences while the elicitation methods in the earlier studies were performed as decisions under risk
(except for Stiggelbout et al. which included decisions without risk
in addition to decisions with risk) (Abellán-Perpiñán et al., 2006;
Bleichrodt and Pinto, 2000; Stiggelbout et al., 1994). Although more
research is needed to reveal the influence of risk posture, the bias
from using a WTP-technique versus decision under risk would be
toward a decreasing marginal utility of increasing gains due to
income effects. Thus, using a procedure including risk would most
likely strengthen our results. Another difference between our study
and that of Abellán-Perpiñán et al. (2006) is that their respondents
had to imagine EQ-5D health states varying from perfect health
to worst possible health state and most respondents in our sample reported “very good” to “neither good nor poor” health. In our
sample from the Norwegian population we did not observe any significant influence on the valuation of life extensions due to health
state. Further research is needed to cover individuals with poor
and very poor health since we had few representatives from these
groups.
If the utility function for duration is not linear but has an increasing marginal utility at low values of t, this can affect economic
evaluations in health care at different levels. First, in the generalized QALY formula: U(q,t) = H(q)W(t) t might in reality neither
reflect a linear function, nor an exponential or power coefficient
but a different shape which allows for an increasing marginal valuation of duration up to 1 year. Second, if TTO is used for deriving the
QALY weights, these could be under- or over-valued depending on
the time periods used for eliciting the weights. TTO utilities could
be biased upwards if the marginal utility for duration is increasing
and downwards if it is diminishing (Miyamoto and Eraker, 1985).
Third, this could affect cost-effectiveness analyses using life-years
gained as outcome measure. Fourth, the non-linear utility function
has the implication that while single programmes offering small
health increments may be deemed not worthwhile, eliciting preferences for a bundle of these single programmes may produce a
different result.
CEA/CUA is based on a linear valuation of increasing lifetime
gains. Most of the previous literature that has challenged this practice suggests diminishing marginal utility of increasing lifetime
gains. However, such gains have been examined for greater life
extensions (up to 45 years). The results from the current study
indicate that for small life extensions (up to 1 year), people have
increasing marginal utility of lifetime gains. The seemingly conflicting results are compatible with an S-shaped function. Other
researchers have proposed models which might be compatible
with our findings. Saha (1993) introduced a model called an “expopower utility function” which allows for changing marginal utility
at different parameter values (Saha, 1993). This family of functions
allows both convex and concave curvatures and has been applied in
recent research on health utility (Abdellaoui et al., 2007; Bleichrodt
et al., 2005; Doctor and Miyamoto, 2003). Abdellaoui et al. (2007)
showed that there are no systematic differences between risky and
riskless utility for money if data are analysed under prospect theory. Certainly, more empirical research is needed for the estimation
of utility functions for health gains.
547
Acknowledgements
This research was supported by a grant from the Health Economics Research programme at the University of Oslo (HERO).
The authors would like to thank Arna Desser for language corrections.
Appendix A. Characteristics of all respondents by
questionnaire
Questionnaire
Number of respondents
Age (mean)
Min
Max
Female (%)
Q1
Q2
Q3
Q4
Total
600
49.1
40
59
49.7
600
49.1
40
59
50.0
602
48.8
40
59
48.7
600
49.3
40
59
49.5
2402
Education (%)
Nine years
Secondary
Univ ≤ 4 years
Univ > 4 years
5.67
39.8
32.2
22.3
6.00
39.0
31.5
23.5
6.64
42.9
29.6
20.9
7.33
39.3
29.8
23.5
6.41
40.3
30.8
22.6
Household Inc (%)
<200,000
2–400,000
4–600,000
6–800,000
8–1,000,000
1–1,200,000
>1,200,000
2.00
18.5
28.7
29.0
14.5
3.50
3.83
2.17
16.5
29.8
30.2
15.2
3.17
3.00
2.82
15.3
30.1
32.6
11.6
4.49
2.33
2.50
18.3
26.8
33.2
11.3
4.83
3.00
2.37
17.2
29.1
31.2
13.2
4.00
3.04
Marital status (%)
Married
Partner
Unmarried married
Previously married
65.5
12.3
10.8
11.3
64.2
13.7
9.50
12.7
64.2
14.3
7.82
13.6
63.8
12.8
10.8
12.5
64.4
13.3
9.75
12.5
Appendix B. Characteristics of respondents by
questionnaire after the removal of invalid answers (steps 1
and 2)
Questionnaire
Q1
Q2
Q3
Q4
Total
ANOVA
Number of respondents
Age (mean)
Min
Max
Female (%)
Education (%)
Nine years
Secondary
Univ ≤ 4 years
Univ > 4 years
Household Inc (%)
<200,000
2–400,000
4–600,000
6–800,000
8–1,000,000
1–1,200,000
>1,200,000
Marital status (%)
Married
Partner
Unmarried married
Previously married
486
48.9
40
59
49.6
487
48.9
40
59
49.9
440
48.8
40
59
49.3
451
49.0
40
59
49.0
1864
Prob > F
0.45
5.35
39.5
32.7
22.4
5.75
38.8
30.8
24.6
5.91
42.7
29.6
21.8
5.10
39.9
31.5
23.5
5.53
40.2
31.2
23.1
2.06
18.1
28.6
27.4
15.6
3.91
4.32
1.85
15.2
30.0
30.8
15.8
3.08
3.29
3.18
14.6
31.8
33.6
10.7
4.32
1.82
1.77
18.0
25.1
34.6
12.4
4.88
3.33
2.20
16.5
28.9
31.5
13.7
4.02
3.22
65.8
11.9
10.5
11.7
64.1
13.8
9.03
13.1
64.7
14.4
7.06
13.9
65.4
12.0
10.2
12.4
65.0
13.0
9.23
12.8
0.82
0.95
0.08
0.94
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