Increasing marginal utility of small increases in life-expectancy?

Journal of Health Economics 29 (2010) 541–548

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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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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-Perpinán 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; Abellán˜ Perpinán 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 neu˜ trality (Abellán-Perpinán 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.

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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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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

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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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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 life˜ years (Abellán-Perpinán 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 (Abellán˜ Perpinán 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-Perpinán 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-Perpinán 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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