expect to derive increasing marginal utility from reducing health risks that come to ...... (7), division by the absolute risk change produces a VSL equal to â(α/β0).
Mortality and Morbidity Risk Reduction: An Empirical Life-cycle Model of Demand with Two Types of Age Effects J.R. DeShazo and Trudy Ann Cameron1,2 Abstract We develop and test an empirical model of individuals’ intertemporal demands for health risk-mitigation programs over the remaining years of their lives. We estimate this model using data from an innovative national survey of demand for preventative health care. We find qualified support for the Erhlich (2000) life-cycle model, which predicts that individuals expect to derive increasing marginal utility from reducing health risks that come to bear later in their lives. However, we also find that as individuals age, there appears to be a systematic downward shift in this schedule of marginal utility for risk reduction at future ages. Our model improves upon earlier work by differentiating between the respondent’s current age and the future ages at which they would experience adverse health states. Using age-specific demand schedules, we simulate age-specific values for avoided future statistical health states for risk mitigation policies with various latency periods. Our empirical results contribute to the debate about whether a “senior death discount” should be employed in public policy-making with respect to health risks. JEL Classifications: I12, J17, J28, J78 Keywords: mortality risk, morbidity, VSL, choice experiment, senior death discount
1
Senior authorship is not assigned. Nominal lead authorship will rotate though the series of papers associated with this project. 2 Department of Policy Studies, UCLA, and Department of Economics, University of Oregon, respectively. We thank Vic Adamowicz, Richard Carson, Maureen Cropper, Baruch Fischhoff, Jim Hammitt, Alan Krupnick, Kerry Smith and Kip Viscusi for helpful reviews of our survey instrument and/or comments on our early results. Rick Li implemented our survey very capably. Ryan Bosworth and Graham Crawford have provided able assistance. This research has been supported by the US Environmental Protection Agency (R829485). It was was also encouraged by Paul De Civita of the Economic Analysis and Evaluation Division of the Healthy Environments and Consumer Safety Branch (HECS) of Health Canada and has been supported in part by Health Canada (Contract H5431-010041/001/SS). This work has not yet been formally reviewed by either agency. Any remaining errors are our own.
1
Mortality and Morbidity Risk Reduction: An Empirical Life-cycle Model of Demand with Two Types of Age Effects Abstract We develop and test an empirical model of individuals’ intertemporal demands for health risk-mitigation programs over the remaining years of their lives. We estimate this model using data from an innovative national survey of demand for preventative health care. We find qualified support for the Erhlich (2000) life-cycle model, which predicts that individuals expect to derive increasing marginal utility from reducing health risks that come to bear later in their lives. However, we also find that as individuals age, there appears to be a systematic downward shift in this schedule of marginal utility for risk reduction at future ages. Our model improves upon earlier work by differentiating between the respondent’s current age and the future ages at which they would experience adverse health states. Using age-specific demand schedules, we simulate age-specific values for avoided future statistical health states for risk mitigation policies with various latency periods. Our empirical results contribute to the debate about whether a “senior death discount” should be employed in public policy-making with respect to health risks. JEL Classifications: I12, J17, J28, J78 Keywords: mortality risk, morbidity, VSL, choice experiment, senior death discount
2
1
Introduction
Empirically, scholars know very little about how demands for risk-mitigation programs that enhance health and extend life vary over an individual’s lifecycle. Programs that respond to these demands range from publicly provided environmental, safety and health programs to privately available preventative care and medical therapies. Several developments have increased the need for reliable estimates of demand for these risk-mitigation programs. First, in the U.S., in Canada, and in Europe, policymakers have proposed modifying the benefit estimates used in benefit-cost analysis of regulations to reflect a lower value for reducing the risk of death for senior citizens as compared to adults.3 These proposals have come under intense scrutiny, and in some cases have been retracted, because of a lack of reliable empirical evidence.4 Second, there is a theoretical debate emerging between competing lifecycle models of health-risk mitigation. Some models predict an upward trend in the value of health-risk mitigation with age (Erhlich and Chuma, 1990; Erhlich, 2000), while other models predict a flat or downward trend with age (Sheppard and Zeckhauser, 1984). Third, scholars have begun to appreciate that life-cycle patterns of health-risk mitigation affect life-cycle consumption and health trends which, in turn, affect not only the timing of transitions to Social Security, Medicare, and Medicaid, but also determine an individual’s life expectancy (Hamermesh, 1995; Hurd et al., 1995, Gan et al., 2003). Our central contribution in this paper is an empirical model of individuals’ intertemporal allocation of health-risk-mitigation expenditures over their remaining lives. Our model defines the consumer’s multi-period planning problem into which we incorporate utility pa3
For an overview of the US proposal and retraction see Skrycki (2003). For Canada, see Canada-Wide Standards Development Committee for PM and Ozone (1999). 4 Until very recently no empirical studies focused explicitly upon age-varying effects on demand. Recent analyses include Evans and Smith (2003), Aldy and Viscusi (2004), Alberini et al. (2004), Knieser et al. (2004).
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rameters that may vary with age now and with the age at which each future health state is experienced. The model controls for income, the latency of program benefits and each individual’s years of remaining life. We estimate this model using data from an innovative national survey of demand for preventative health care. By eliciting information on how individuals would allocate risk-mitigating expenditures across future years of their life, we are able to estimate their willingness to pay for marginal risk reductions with respect to specific time profiles of illness. The development and empirical testing of our model contributes to the literature in several ways. First, we provide the first evidence in the literature that individuals intertemporally allocate risk-mitigating expenditures across future years of life. We find empirical support for the hypothesis that individuals derive increasing marginal utility from reducing risks that come to bear later in life, when both their shadow value of health and their risk of mortality are likely to be greatest. However, we also find support for the hypothesis that as individuals age, there is a systematic downward shift in willingness to pay for statistical risk reductions at future ages. As individuals age, they appear to revise downward their assessment of the value of consumption at very advanced ages; this in turn reduces the shadow value of risk mitigation at those ages. Second, this evidence about the age dependence of the willingness to pay for risk mitigation enables us to assess which of two competing theoretical regimes is most pertinent. The first modeling tradition examines life-cycle demand for mortality risk mitigation. This modeling approach predicts that the value of risk mitigation should be flat or fall with age, depending upon the presence or absence of complete markets (Arthur, 1981; Sheppard and Zeckhauser, 1984; Rosen, 1984). The second modeling tradition examines the risk of both morbidity and mortality and, in contrast to the first, predicts that the value of health risk
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mitigation will rise with age, with or without complete markets (Ehrlich and Chuma, 1990; Ehrlich, 2000). Our empirical analysis finds qualified support for the prediction of the second modeling tradition — that the value individuals place on health risk mitigation rises with the age at which the adverse health states would occur. Third, our empirical model represents an attempt toward bridging the gap between dynamic life-cycle models and current empirical methods. Current empirical methods are based predominantly upon a static theoretical model in which the individual considers a single risk that is reduced in the current period (Dreze, 1962; Jones Lee, 1974).5 Although our empirical model is not formally dynamic, it expands upon current empirical methods by incorporating elements of these dynamic models. In our model, for example, demand in any period depends not only on consumption in that period, but also on the time path of future consumption and health states as well as the individual’s discount rate and life expectancy. Rather than being constrained to a single planning period, as most state-of-the-art valuation methods have been, our model permits individuals to allocate risk-mitigating expenditures intertemporally in response to intertemporal risk gradients. These improvements not only expand the purview of risk valuation methods, but also reduce sources of unobserved heterogeneity in current demand estimates. Finally, our estimates of age-specific schedules of demand for future risk mitigation enable us to improve upon earlier theoretical assessments of policies that reduce risk in different future periods (Cropper and Sussman, 1990). For a variety of policy time profiles, we show that latency effects vary across age-cohorts (since each cohort has a specific schedule of marginal utility for future risk reductions). We believe this to be the most comprehensive empirical assessment of policies with variable risk latencies that is currently available in the 5
These single-risk, single-period models have motivated hundreds of empirical demand analyses, including those currently used to evaluate the social benefits of life-saving public policies (Viscusi, 1993).
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literature. We assume in the design of our structural empirical model that individuals face a portfolio of competing sources of risks that may affect them at different times during their lives. An individual’s problem is to choose the set of risk mitigation programs that maximizes the present discounted value of the expected utility that she derives from her remaining life. This maximization is subject to an income constraint, a discount rate, and the price of risk mitigation programs and activities. When making this choice the individual considers her indirect utility function in each future year of her life. Although she is choosing across risk mitigation programs, the more fundamental element of her choice is the probability of alternative health-states in future years. The model explicitly identifies the marginal value of avoiding episodes in three adverse health states: morbidity, post-morbidity and mortality. Individuals choose among risk mitigation programs that yield uncertain future benefits, so we adopt an option price framework. We define a measure of willingness to pay to avoid probabilistic future health states that depends upon the current age of the individual, her age in each future period when an adverse health state would be experienced, her discount rate, as well as the expected value of the future stream of income and program costs. We would prefer to be able to estimate our model using market data on individuals’ choices of risk-mitigation programs that alter their health risks at different future ages. However, given the absence of adequate market data, we have chosen to administer a national survey that elicits individuals’ stated choices over alternative risk-mitigation programs. Each individual is presented with an illness profile that describes a probabilistic time pattern of health states that the individual could experience as they age. Cast in terms of specific major illnesses, each profile describes the individual’s age at onset, the severity and duration of treatments and morbidity, the age at recovery (if there is any), and the number of lost
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life years (if there are any). We then present individuals with illness-specific preventative programs that involve diagnostic screening, remedial medications and life-style changes that would reduce their probability of experiencing that illness profile. An annual fee would be necessary to participate in each risk-reducing program. In the course of these choice experiments we observe individuals strategically allocating expenditures on risk mitigating programs across the current year and future years of their remaining lives. To analyze individuals’ program choices, we estimate a simple indirect utility function using data from a representative national survey of 1,619 U.S. citizens. Our estimated model recovers an individual’s marginal utility of avoiding a year spent in each of three health states: morbidity, post-morbidity and mortality. Controlling for the individual’s current age, we find that the marginal utility of avoiding a future lost life-year rises with the age at which that lost life-year would be experienced. In contrast, controlling for the ages at which the undesirable future health states would potentially be experienced, we find that the choices made by older individuals imply that their values of risk reductions at each future age are lower. We then evaluate the net effect of these two types of age dynamics by simulating the fitted distributions of demand for policies with different risk latencies and different illness and mortality profiles. For age cohorts of 25, 35, 45, 55, and 65-year olds, we describe each cohort’s willingness to pay to mitigate risk at their current age, in ten years, at the common age of 70, and (for the 25-year-old cohort) in 5, 15, 25, 35 and 45 years. In section 2, we compare the two dominant modeling traditions and explain their pertinence to the specification of the indirect utility function that we estimate and the hypotheses that we test. We describe the survey and the data that we use to estimate our model in Section 3. In Section 4 we develop our model and in Section 5 we present our results before concluding in Section 6..
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2
Predicting Life-cycle Patterns of Risk Mitigation
Although the fact is not universally recognized, there are two distinct theoretical traditions that predict different life-cycle patterns of health risk mitigation.
2.1
Models of Disembodied Mortality Risks
Scholars in this first tradition have focused on modeling static demand (Dreze, 1962; Schelling, 1968; Jones-Lee, 1974; Arthur, 1981) and, later, life-cycle demand (Sheppard and Zeckhauser, 1984; Cropper and Sussman, 1990; and Johansson, 2002) for the mitigation of mortality risks. In the case of a discrete life-cycle model, an individual at age i derives expected utility, Vi , from the remainder of his lifetime:
Vi =
XT
t=i
qi,t (1 + r)i−k ut (Ct )
(1)
where ut (Ct ) is the utility of consumption in year t of life, multiplied by the probability (qi,t ) that the individual at age i survives to age t, discounted to the present at a subjective discount rate r. T is the maximum length of life used in the planning period. The individual chooses qi,t and Ct to maximize the present value of Vi , subject to wealth constraints that reflect opportunities for lending and borrowing. Notice that risk does not enter the utility function as an argument, which is why we call this a model of “disembodied risk.” Relative to the next model that we review, this disembodied risk represents a limitation of this class of models. Most health risks produce a concatenation of health states, nearly all of which involve some form of morbidity rather than just sudden death. Furthermore, at advanced ages, choices about future health states will increasingly involve trade-offs between time spent in different morbidity states and the 8
risk of premature mortality. The sole focus on mortality risk, which may also be interpreted as assuming that morbidity risk is separable from mortality risk, limits the ability of this class of models to represent relevant substitution possibilities between health states over an individual’s lifetime. The first order conditions for this model illustrate that an individual’s willingness to pay at age i for a risk reduction at age k depends upon two components (see Cropper and Sussman, 1990). The first and most celebrated component represents the expected utility of consumption across time, from age k + 1 onward, converted to dollars by dividing by the marginal utility of income at age i.6 The present value of future consumption varies ambiguously with age (Johansson, 2000). The second component is the probability of survival. As this falls with age, willingness to pay for risk mitigation is assumed to increase. In terms of the theory, then, the net effect of these two components is ambiguous. To develop predictions from these models, scholars have undertaken simulations. Sheppard and Zeckhauser’s (1984) analyses are representative. If the utility of consumption is assumed to be constant over future years, the present value of lifetime consumption would be proportional to the discounted remaining life expectancy, which decreases unambiguously with age. Under these assumptions, the simulations predict the willingness to pay for risk reduction will fall with age. In absence of perfect markets, simulations reveal per-year consumption rising and then falling as individuals advance in age. This pattern of consumption leads to predictions that the demand for risk mitigation will follow a similar inverted U-shaped time path. 6
These models recognize that a reduction in risk may affect the individual’s income constraint, since they may work longer, but may also affect the willingness to pay for risk mitigation, since they extend their income over a longer life expectancy.
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2.2
Models of Health-Embodied Risk
A second modeling tradition has arisen from scholars modeling life-cycle demand for health (Grossman, 1972). Motivated by a concern with risks to health, Ehrlich (2000)7 portrays health risks (q) in year t as affecting the flow of health (h(qt )) in year t. In contrast to the mortality-risk-only models discussed above, here q may represent any type of risk that diminishes the individual’s level of health. Let hmin represent the minimum flow of health needed to survive. A health risk that results in (ht ) > hmin causes morbidity, while a health risk that results in (ht ) < hmin causes mortality. This more general representation allows a wider array of risks to be modeled. The flow of health (h(qt )) becomes an argument in the individual’s utility function alongside the consumption of a composite commodity (zt ). For the purposes of our cursory review, we present the objective function in a discrete and additive separable form (with respect to time):
Vi =
XT
t=i
(1 + r)i−k ut (zt , h(qt ))
(2)
The individual chooses Ct , qt and by implication ht (·) to maximize the present value of Vi , subject to wealth constraints. The first order conditions describe how an individual’s willingness to pay at age i for a risk reduction at age k depends upon three basic components (Ehrlich, 2000, p. 346-48). The first and second components correspond to the shadow value of future life and the shadow value of future health. As in the disembodied risk model, the first component equals the loss in the expected utility of consumption across time, from age k + 1 onward, converted to dollars by dividing by the marginal utility of income at age i. As before, it varies with 7
Also see the erratum in Ehrlich (2001).
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age in a theoretically indeterminate fashion. The second component represents the value of health protection. The shadow value of health is positive because the value of saving (future consumption) is assumed to be positive (unless the hazard rate becomes sufficiently large with age to cause negative returns to saving) (Ehrlich, 2000, p. 348). Third, the time path of risk mitigation is determined by the probability of morbidity-mortality events as individuals age. A rising time profile of risks produces a positive incentive as the marginal disutility of declining health rises with age. The shadow value of health may be determined by: a) foregone labor earnings due to poor health, b) disutility from an increase in morbidity due to poor health, which grows rapidly at advanced ages, c) a diminished value of consumption if health is a strong complement to consumption. For the case without annuities (i.e. savings opportunities), Ehrlich concludes that “the value of life and health protection is seen to be rising over a good part of the lifecycle because aging raises the benefits of protection except in the late phases of the planning horizon” (2000, p.348 ). With annuities, the time path of self-protection is expected to rise more steeply with age (p. 352). Only a strong bequest value (expressed via life insurance markets) would temper this upward trend in the value of health and life protection with age.
2.3
From the Theoretical Debate to Model Specification and Hypotheses Testing
We view the health-embodied risk models (Chuma and Ehrlich, 1990; Ehrlich, 2001) as a generalization of the mortality-only risk models (Sheppard and Zeckhauser, 1984). Conceptually, the health-embodied model treats risk more generally by allowing for a wide range of health states.8 As a consequence, risk plays a “dual” role: it contributes to utility in 8
Alternative health states range from perfect health to short periods of mild morbidity to extended and severe morbidity that results in mortality to immediate mortality without morbidity. This stands in contrast
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each year via its effect on morbidity (i.e., health) and it affects the probability of future consumption via its effect on mortality. The individual considers a wider array of substitution possibilities (between years in healthy states, morbid states, and death). This model also recognizes that the shadow value of risk mitigation may be determined by both the shadow value of health and the shadow value of future consumption. When specifying the individual’s indirect utility function in each future year of their life, our empirical model endeavors to incorporate these general features. Our first hypothesis test provides empirical support for these generalizations, while our second hypothesis test qualifies this support. The health-embodied models of risk predict that the marginal value of health risk mitigation will rise with the future age at which adverse health states will be experienced. Empirically, we find that individuals allocate progressively increasing amounts of risk-mitigation to more-distant future years of their lives. Individuals expect their shadow value of health to rise in future years as their health state declines, so the expected marginal utility of risk mitigation in future years rises. Second, we find support for the hypothesis that as individuals grow older (e.g. age i to age i + 1), there is a systematic downward shift in their schedule of marginal utility for health risk mitigation over future ages. Individuals appear to revise downward the expected utility that they will derive from risk mitigation as they approach age k. There maybe several possible explanations for this downward revision. First, as they age, individuals may revise their expectations downward about their future time path of health, their life expectancy or their expected lifetime income. Second, if the complementarities between health and all other consumption increase with age, then declining health will, in turn, diminish the marginal utility of all other consumption goods. It is reasonable to assume to the model of disembodied mortality risk in which the individual demand for risk mitigation results in only two health states: perfect health and death.
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that individuals learn about the extent of this complementarity as they age. If so, rational individuals will respond to this new knowledge by reallocating income to earlier years of life where its marginal value in consumption is higher than it will be in later years. This, in turn, will decrease the value of health risk mitigation in later years.9
3
A Structural Option Price Model
Our structural model will interpret individual’s choices as revealing their option price for programs that mitigate the risks of uncertain future health states. The model allows a great deal of flexibility in characterizing how future health states impact future income and program costs in each future year.10 The discount rate is explicit, and may vary both across individuals and across time. Although program choices have inter-temporal consequences, our model is one of static decision-making, with present value calculations applied to future period costs and benefits. In our empirical application the individual’s planning horizon is assumed to be bounded by their nominal life expectancy. Constructing and estimating empirical models of individual demand for future health states is one of the most challenging tasks economists face. Traditionally, scholars have 9
There is also a third possible explanation for this downward shift. Individuals may have believed that the illness-specific risk-mitigation programs had longer latency periods than we stated. This concern would arise if a program to reduce a major disease was described as immediately reducing the individual’s risk but the individual did not believe that the program would produce an immediate risk reduction. This belief might be held because either the effectiveness of the remedial medication was discounted or they doubted whether they would promptly comply with the life-style changes necessary to reduce their risk. 10 To appreciate why we opt to develop a structural model of choice, it may be useful to note several limitations of employing a standard conjoint specification. Consider a specification in which attributes (such as years of morbidity, years of premature mortality, latency, risk reductions, program prices) enter the utility function in an additively separable fashion. Such a specification does not explicitly accommodate the timevarying dimensions of the consumer’s choice problem. Nor does it allow for discounting streams of income, program costs and future utility. Further, it does not adequately characterize the role of uncertainty in the choice problem. Both the disutility of the illness profiles and the utility of the risk mitigation programs are uncertain. Furthermore, future uncertain health states may affect the individual’s income, costs and utility in future years.
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focused on the individual’s marginal rate of substitution between just two health states, life and death, during just one current period. In this standard model, the individual considers her utility in both states, weighted by the probability that each of these states will occur (Dreze 1967; Jones-Lee, 1974). One may then calculate the individual’s willingness to trade wealth for a reduced probability of death, holding utility constant. This simple model has served as the theoretical basis for dozens of empirical studies designed to estimate the ”value of a statistical life,” (VSL) (Viscusi, 1993; Aldy and Viscusi, 2002; Mrozek and Taylor, 2002) including those used to evaluate the appropriateness of many life-saving public policies (US EPA, 1999). In the context of the consumer’s problem, we generalize the standard model in two ways. Subject to an income constraint, the consumer’s problem is one of choosing quantities of costly risk-management programs that reduce the likelihood of the undesirable health states. As a first generalization, rather than assuming the individual faces only two homogeneous health states, our model assumes the individual faces a large set of multi-dimensional and time-denominated health states. Second, rather than assuming the individual makes a single lump-sum payment for a program in the current period, we recast the individual’s demand for risk management programs in an option price framework. This accommodates programs with a future stream of certain costs and uncertain benefits. Our more general model offers a number of advantages over the simplest conventional specification. First, it eliminates several currently problematic sources of heterogeneity bias in estimates of willingness to pay (WTP) for risk reductions. These biases are present when scholars omit relevant substitute states or dimensions of states (such as time) along which substitution may occur. Second, our model enables scholars to estimate the demand for programs that affect any feasible time sequence of states that involves periods of health,
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morbidity, recovered status, or mortality. Third, because both the costs and health states are time-denominated, our model produces specific estimates of constructs such as an individual’s marginal willingness to pay to avoid a sick-year or a prematurely lost life-year. The first step in our model’s development is to more completely describe the set of health states and the time-sequence in which the individual faces them. We allow for a wide range of morbidity states, in addition to states of perfect health and sudden death. The reference state from which future states are being evaluated, the individual’s status quo health, may be a state of morbidity rather than one of perfect health. The admission of morbidity states to the individual’s choice set also allows for the possibility that there may be fates (states) worse than death for some individuals. Failing to recognize either of these cases, when they hold, is likely to distort estimates of the WTP for risk reductions. We allow individuals to express their WTP to avoid concatenations of health states, which we call “illness profiles.” Each profile is defined as a probabilistic sequence of health states in the context of the individual’s current age and remaining nominal lifespan. Each profile describes the future age of onset of an illness, sudden death if it occurs, and if it does not, the level and duration of pain and disability that follow. The remainder of the profile is described in terms of either full recovery or premature death after a number of years of pain and disability. Describing future health states in terms of illness profiles is useful for several reasons. First, individuals actually think and talk about future health states in terms of illnesses (and their health-state profiles). Health profiles offer the appropriate lexicon and unit of analysis. Second, many risk management programs target specific illnesses. When choosing across programs, we allow individuals to substitute across illness profiles. Third, when choosing the quantity of a program to consume, it is quite likely that the individual is substituting
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among periods of health states within an illness profile. Our model explicitly allows for intertemporal substitution among health states within an illness profile. We recover individual preferences over the latency periods associated with the onset of an illness, the duration of symptoms, the period of post-illness recovered status (if there is one), and the years lost to premature death. In contrast, most existing studies evaluate the individual willingness-topay for mortality risk reductions in the current year only. Fourth, our use of illness profiles captures the reality that mortality states are often positively correlated with morbidity states. WTP estimates from models that omit the associated morbidity will tend to be biased (probably upward) since the WTP from such a model conflates the values of avoided mortality and avoided morbidity. Our second major methodological contribution is the derivation of option prices for programs that reduce the risk of experiencing illness profiles. An option price framework accommodates choices involving the intertemporal distribution of a future stream of certain costs and uncertain benefits (Graham, 1981). By explicitly discounting future periods of net benefits, we are able to define the constant current-value payment that makes an individual just indifferent concerning a costly annual treatment program to reduce the risk of a future illness. (If scaled to correspond to a 1.00 change in risk, the present discounted value of these payments produces a the multi-period optimization analog of a conventional VSL.) Our estimating specification, cast in terms of the indirect utility function, is designed to support the empirical phase of this research project. In this conjoint choice study, each illness profile is characterized as one of twelve major illnesses which represent significant causes of death in society today. Each illness profile in the conjoint choice exercise is described in richer detail than will be carried through in the development of the following estimating specification.11 11
Subsequent papers will explore additional dimensions of heterogeneity. This paper focuses on the age
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Together, our adaptation of Ehrlich’s theoretical framework, our estimating specification, and our empirical results will provide two methodological contributions to the literature on valuing future health states. First, our option price formulas lead naturally to multivariate WTP functions for heterogenous risks that contribute to heterogeneneous health profiles experienced by heterogeneous populations. Any given health threat is associated with a distribution of specific illness profiles that a given population will experience (e.g., morbidity with full recovery, chronic morbidity with no premature death, morbidity followed by premature death, and sudden premature death). When a public policy reduces the incidence of an illness by controlling its causes, it is altering the joint distribution of these illness profiles in the population. Our framework captures the value of avoiding all of the prospective illness profiles for a particular illness, not just those associated with sudden death. Second, by applying the option price framework to an illness profile we offer an important methodological advance: we estimate the marginal value of a life-year in various health states. Specifically, we estimate the marginal value of an additional life year spent in illness, spent in a recovered state, and lost to premature death. This innovation contributes substantially to both the risk-valuation literature and, ultimately, to meeting the needs of policymakers. Policymakers often evaluate policies that postpone, but do not prevent, specific health outcomes. Until now they have had no estimates of the value of incrementally delaying the onset of an illness or death, or of hastening recovery.
3.1
Indirect utility from different health states
We expand upon most earlier empirical treatments by considering four distinct health states: 1) a pre-illness healthy state, 2) an illness (sick) state, 3) a post-illness recovered state and dimensions of the problem.
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4) a dead state.12 We define each of these states as a time segment. Within each segment, the individual’s health status is assumed (for now) to be relatively homogeneous. Let i index individuals and let t index time periods. To capture an illness profile, we use sets of dummy variables that collectively exhaust the period of time between the individual’s present age and the end of his nominal life expectancy. The dummy variable preit take a value of 1 in years when the individual enjoys a healthy state. When the healthy state ends, the value of preit changes to 0 and remains there for the rest of the individual’s expected lifespan. At the end of the healthy period the individual may die suddenly or become sick. Let the dummy variable illit take on a value of 1 at this point and remain equal to 1 for the years during which the individual is ill. When he is not sick, it takes a value of zero. If the individual recovers from this illness, he may not recover to the exact state of health he experienced prior to the illness. The dummy variable labeled rcvit takes on value of 1 in the years between the conclusion of the illness and the individual’s time of death. Finally, we define lylit (life-year lost) to distinguish the extent to which death is premature (that is, the time segment between death and what would otherwise would have been the end of the individual’s nominal lifespan). For each health-state period, we assume initially that the indirect utility derived per unit of time from that particular health state is constant within that period. In our simplest model, the individual’s future undiscounted indirect utility is linear and additively separable in the utility from an arbitrary function of income and the utility derived from each distinct health state: Vit = βf (Yit ) + α0 preit + α1 illit + α2 rcvit + α3 lylit + η it 12
(3)
Within our data, the illness states are further differentiated into specific named illnesses, each of which can exhibit a wide variety of different symptom-treatment profiles that may last for widely differing periods of time.
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where β is the undiscounted marginal utility of some function of current income, f (Yit ). Utility in the pre-illness state, α0 , will be normalized to zero. Let α1 be the undiscounted (dis)utility from a future year of illness, α2 be the (dis)utility from a year of the post-illness recovered state, and α3 be the (dis)utility from a year of being prematurely dead. Policy-makers understand that individual demands for risk-reducing policies should logically depend upon the age of the individual. However, recent attempts by the US Environmental Protection Agency to employ age-differentiated VSLs in some of their alternative benefit-cost analyses, for example, have run afoul of public opinion by employing a one-third lower VSL for seniors than for other adults (Skrzycki, 2003). This coarse tailoring of benefits estimates to the age group of beneficiaries was dubbed the “senior death discount.” Environmental advocacy groups, in particular, seem to have exploited the common misinterpretation of VSLs as measures of the intrinsic worth of different age groups in order to prevent these adjustments. Of course, if the policy goal is to maximize environmental quality, then there is a strong incentive to characterize the aggregate benefits as being as large possible within the range of defensible alternative assumptions. However, if the broader goal of maximizing social welfare is sought, environmental benefits estimates should take into account differing WTP across different age groups. The use of different VSLs for seniors was so controversial, however, that it even led to amendments to appropriations bills for the US EPA.13 The provocative question of whether environmental policies ought to employ a “senior death discount” hinges on how the WTP for risk reductions varies systematically with age– both the respondent’s age at the time they are making their stated choices and the respondent’s future age at which the illness profile suggests they will be experiencing each adverse 13
Allen Amendment (House) — SEC. 418. None of the funds provided in this Act may be expended to apply, in a numerical estimate of the benefits of an agency action prepared pursuant to Executive Order No. 12866 or section 812 of the Clean Air Act, monetary values for adult premature mortality that differ based on the age of the adult.
19
health state. Equation (3) assumes that the undiscounted (dis)utility of a year of illness or injury is a constant. The simplest ways to accommodate heterogeneous preferences involve allowing these marginal (dis)utilities to depend upon characteristics of the individual at the time they are asked to make program choices. The individual’s current age, agei0 , is one of a wide variety of personal characteristic that we might allow to shift the marginal (dis)utility of each adverse health state. In this paper, however, we wish to allow the indirect utility in each future period to depend upon the age of the individual while they are experiencing the health state corresponding to that period, ageit . During the development of this specification, we will abstract from any systematic effects on preferences of differences in the individual’s current age, agei0 but we will allow future ageit to shift the marginal undiscounted utility from each distinct health state: £ ¤ (4) Vit = [β 0 + β 1 Yit ] f (Yit ) + α10 + α11 ageit + α12 age2it illit £ £ ¤ ¤ + α20 + α21 ageit + α22 age2it rcvit + α30 + α31 ageit + α32 age2it lylit + η it . Throughout our analysis, the disutility of each of these states will be interpreted as being the same as the utility associated with avoiding them. The role of the time-indexed dummy variables, ill it , rcv it , and lyl it will be simply to adjust the limits of the summations used for the present value of future intervals of new illness, recovered time, and life-years lost. We will assume that the individual uses the same discount rate, r, to discount both future money costs and the future disutility from either illness or premature mortality.14 In this 14
Empirically estimated discount rates for future money as opposed to future health states are suspected to differ to some extent. Discount rates also differ across individuals and across choice contexts, time horizons and sizes and types of outcomes at stake. No comprehensive empirical work has been undertaken
20
paper, the marginal utility of income is not invited to vary with future age (although it will ultimately vary with current age). More-elaborate specifications for the marginal utility of income that involve future age tend to produce, with our data, a number of implausible predictions. Thus we keep the income terms simple, even letting f (Yit ) = Yit , and examine in more detail the illness profile terms. With this set-up, we can develop a structural model of the ex ante option price that an individual will be willing to pay for a program that reduces his/her risk of a morbidity/mortality profile over the future. Define the present discounted value of indirect utility Vijk for the ith individual when j = A if the program is chosen and j = N if the program is not chosen. The superscript k will be S if the individual suffers the illness (or injury) and H if the individual does not suffer the illness.
3.2
Present Discounted Indirect Utility Differences
The present value of indirect utility if the individual does choose the program and does suffer the illness takes the following form. All summations below will run from 0 to Ti , the remaining number of years in the individual’s nominal life expectancy:
P DV (ViAS ) = β 0
X
δ t f (Yit∗ − cA∗ it ) + β 1
X
∗ A∗ A AS δ t (Yit∗ − cA∗ it )f (Yit − cit ) + ptermsi + εi
that conclusively demonstrates the relationships between money and health discount rates. If we were to choose hyperbolic discounting for our specification, all of the discount factors in the expressions for present discounted value, below, would need to be changed from 1/(1 + r)t to 1/(1 + t)λ . Other than this, the formulas will be the same.
21
where the terms capturing the details of the illness profile are: X X δ t illitA + α11 δ t ageit illitA + α12 δ t age2it illitA X X X +α20 δ t rcvitA + α21 δ t ageit rcvitA + α22 δ t age2it rcvitA X X X δ t lylitA + α31 δ t ageit lylitA + α32 δ t age2it lylitA +α30
= α10 ptermsA i
X
and where δt = (1 + r)−t . The imputed pattern of income and program costs under the four different health states will be relevant to the individual’s indirect utility in each state. We A A A A∗ A A A A A define Yit∗ = Yi (preA it + γ 1 illit + rcvit + γ 2 lylit ) and cit = ci (preit + γ 3 illit + rcvit + γ 4 lylit ).
Yit∗ and cA∗ it are sufficiently general to allow for a number of different assumptions about how individuals view their potential income and how they view their cost obligations under each program in different health states. We will maintain the hypothesis that (γ 1 , γ 2 , γ 3 , γ 4 ) = (1, 0, 0, 0). In words, usual income is expected to be sustained through periods of illness by health and disability insurance, but not after death (there is no life insurance and there are no bequests). Program costs are expected to be paid only while the individual is alive and healthy. The present value indirect utility, if the individual does choose the program but does not suffer the illness, involves no illness, recovery, or reduced lifespan. Thus, the expression for indirect utility takes the following form:
P DV (ViAH ) = β 0 f (Yi − cA i )
X
A δ t + β 1 (Yi − cA i )f (Yi − ci )
X
δt + εAH i
In this case, both income and the annual costs of program will continue until the end of the individual’s nominal life expectancy. However, there are no corresponding benefits such as illness-years or lost life-years avoided. 22
Present value indirect utility, if the individual does not choose the program but does suffer the illness, is given by:
P DV (ViNS ) = β 0
X
δt f (Yit∗ ) + β 1
X
NS δ t (Yit∗ )f (Yit∗ ) + ptermsA i + εi
The individual’s lifespan is potentially reduced, so future income continues only until the time of death, and ptermsA i , capturing the avoided disutility of the illness, any recovery period, and any life-years lost, will again be relevant. Present value indirect utility, if the individual does not choose the program and does not suffer the illness, is:
P DV (ViNH ) = β 0 f (Yi )
X
δ t + β 1 (Yi ) f (Yi )
X
δ t + εNH i
Recall, the individual assumes that his current income level will be sustained until the end of his lifespan in the absence of premature mortality.
3.3
Expected indirect utility
In deriving the individual’s option price for Program A, given the ex ante uncertainty about future health states, we need to calculate expected utilities. In this case, the expectation is taken across the binary uncertain outcome of getting sick, S, or remaining healthy, H. The probability of illness or injury differs according to whether the respondent participates in if the the risk-reducing intervention program. Let the baseline probability of illness be ΠNS i if the individual individual opts out of the program, and let the reduced probability be ΠAS i opts in. The risk change due to program participation, ∆ΠAS i , is presumed to be negative.
23
Expected utility if the individual buys program A is: ¡ £ ¤ ¢ × P DV (ViAS ) + 1 − ΠAS E ViA S,H = ΠAS × P DV (ViAH ) i i Expected utility if the program is not purchased (i.e. “Neither Program”, N), with the expectation again taken over the uncertainty about whether the individual will suffer the illness, is: ¡ ¢ £ ¤ × P DV (ViNS ) + 1 − ΠNS × P DV (ViNH ) E ViN S,H = ΠNS i i
£ ¤ £ ¤ Details concerning the simplification of the expected utility difference E ViA S,H −E ViN S,H
are provided in an Appendix available from the authors.
We make use of a number of notational abbreviations in getting to our empirical expected ¡ ¢ S = ΠAS − ΠN . In addition, there are many utility difference formula. First, let ∆ΠAS i i i distinct present discounted value terms, each signalled by the prefix pdv. Variables that consist of the present discounted interactions between the individual’s future age at the time each health state is experienced are prefixed by agepdv for linear age interactions and age2pdv for squared age interactions. The complete definitions of these abbreviations are given in Appendix I. A mathematical appendix available from the authors shows that the expected utility difference driving the individual’s choice between Program A and the Neither Program alternative can then be written as a quadratic in cA i (there will be an analogous utility-difference for Program B versus the Neither Program alternative): £ A¤ £ ¤ £ ¤ 2 ] + B ci + C E ViA − E ViN = A[cA i where the coefficients of this quadratic form are:
24
(5)
A = β1 ⎡
⎢ B=⎣ C =
£¡ ¢ ¤ AS A 1 − ΠAS pdvcA i i + Πi pdvppi
£¡ ¢ ¤ A AS A −β 0 1 − ΠAS + Π pdvp pdvc i i i i
⎤
⎥ £¡ ¢ ¤ ⎦ AS A AS A −β 1 2Yi 1 − Πi pdvci + Πi pdvypi
£ ¡ ¢ ¡ ¢¤ β 0 Yi pdvyiA − pdvcA + β 1 Yi2 pdvyyiA − pdvcA ∆ΠAS i i i £ ¤ +∆ΠAS αtermsA i i + εi
where the expression for ptermsA i used earlier is now expressed using our abbreviations as: A A = α10 pdviA αtermsA i i + α11 agepdvii + α12 age2pdvii
+α20 pdvriA + α21 agepdvriA + α22 age2pdvriA +α30 pdvliA + α31 agepdvliA + α32 age2pdvliA
3.4
Ex ante option prices
The respondent’s implied ex ante option price for program A can be determined by setting that makes the the expected utility difference equal to zero and solving for the vale of cA i equality hold.
First however, the unknown utility parameters must be estimated, which
requires that a number of variables be constructed. Each of the variables in the αtermsA i expression must be multiplied by the (negative) risk change, and all terms involving each of the β 0 and β 1 parameters must be combined. These constructed variables, β0termA i and
25
β1termA i are calculated from the raw data as: £ ¤ £¡ ¢ ¤ AS A − cA 1 − ΠAS pdvcA i i i + Πi pdvpi ¡ ¢ + Yi ∆ΠAS pdvyiA − pdvcA i i £¡ ¢ ¤ AS A = −2[cA 1 − ΠAS pdvcA i ]Yi i i + Πi pdvypi £¡ ¢ ¤ 2 AS A AS A + [cA ] + Π pdvpp 1 − Π pdvc i i i i i ¡ ¢ + Yi2 ∆ΠAS pdvyyiA − pdvcA i i
= β0termA i
β1termA i
The complexity of these terms stems from the effects of different illness profiles on income and program costs. Once the parameters have been estimated, we can revert to the expression for the utility difference as a quadratic function in the payment, cA i , that would make the utility-difference exactly zero.15 The model thus far accommodates the effects of the future ages, agei0 , at which adverse health states may be experienced. The baseline marginal utility parameters β 0 , α10 , α20 , and α30 will also be allowed to shift with the respondent’s current age, agei0 , at the time he or she is being asked to make these tradeoffs. By employing both agei0 and age2i0 as shifters, we make indirect utility potentially fully quadratic in the respondent’s current age.
The
coefficients on the linear age-at-health-state terms, α11 , α21 , and α31 , can also be allowed to shift with agei0 , which completes the specification by allowing for interactions between current age and the age at which a particular health status is to be experienced. From the simple undiscounted indirect utility function in equation (4), it is thus necessary 15
Of course, the squared term in cA i will be activated only if β 3 6= 0. If the error term takes on its expected value of zero, the systematic portion of the difference in expected utilities can be solved to yield point estimates of the option price.
26
to go through several steps to achieve the estimating form that can be used to explain respondent’s choices among risk-reduction programs. We see from equation (5) that the difference in expected present value indirect utilities associated with choosing a risk-reduction program is a function of income, program costs, the illness profile (as captured by the pdviA i , pdvriA ,and pdvliA terms), and implicitly of the individual discount rate ri assumed for each respondent.16 All choices posed to respondents are three-way choices (Program A, Program B, or Neither Program), so the models will be estimated using McFadden’s conditional logit estimator (actually the fixed-effects variant of this model, to control for any unobserved heterogeneity across individuals that may be correlated with age).
3.5
From maximum annual payment to PDV of payment stream
The option price for the program that accomplishes this decrease in health risks is the common certain payment, regardless of which way the uncertainty about contracting the illness is resolved, that makes the individual just indifferent between paying for the program and enjoying the risk reduction, or not paying for the program and not enjoying the risk £ A ¤ N = 0. The amount of money reduction. This payment, cA∗ i , will make E Vi ] − E[Vi
is the maximum constant annual payment that the individual will be willing to make, cA∗ i regardless of whether he suffers the illness, in order to purchase the program that reduces S to ΠAS his probability of suffering the illness from ΠN i i .
However, these annual payments cbA i are necessary for the rest of the individual’s life, so
their present value must be calculated.17 We will use the expected present value of this time profile of costs, with the expectation taken over whether or not the individual suffers the 16
In this analysis, we assume ri = r, the same for each respondent, and we conduct sensitivity analyses with respect to the magnitude of this discount rate. 17 In this context, however, there is some uncertainty over just what will constitute ”the rest of the individual’s life,” since this may differ according to whether the individual suffers the illness or not.
27
illness when they are participating in the program. i h £¡ ¢ ¡ ¢¤ AS b A 1 − ΠAS E P V (ci ) = (cbA pdvcA pdvpA i i + Πi i i ) To simplify our outline of the necessary calculations, we can revert to the case where the marginal utility of income is constant, so that β 1 = 0. In this case, the denominator of the £¡ ¢ ¡ ¢¤ AS option price formula is just β 0 1 − ΠAS pdvcA pdvpA , so that capitalizing this i i + Πi i payment over the rest of the individual’s life allows the terms in square brackets to cancel.
The formula for the present value of the streams of annual maximum payments willingly made to avoid a specified health profile reduces to: i h £ ¡ ¢ ¤ −1 ) = ∆ΠAS β 0 Yi pdvyiA − pdvcA + αtermsA E P V (cbA i β0 i i + εi i
(6)
From this result, it is clear that if the marginal utility of income is constant across the population, the expected present value of the lifetime stream of maximum annual payments is merely proportional to the size of the risk reduction, given individual preferences, income and the illness profile in question, as captured by the present discounted health state variables and their age-at-health-state interaction terms. This proportionality is a common assumption in much empirical work on WTP to avoid health risks and has been used to justify reliance on a single number for the VSL across very different risk reductions.
Indeed, if the risk reduction involved and the cost of the
program pertained only to a single year (as is the case in a number of existing VSL studies) there would be no difference between pdvyi and pdvci , so that the first term in the square brackets in equation (6) would disappear.
Furthermore, if all illness profiles were to be
treated as identical and no dependency on age was being assumed, all of the terms involving
28
α parameters would collapse into a single constant parameter, α, multiplying a dummy variable, say DiA , that indicates whether the single adverse health state occurs in alternative A. This new parameter would describe the marginal utility of the generic health outcome to be avoided. This health outcome is “sudden death this year” in many existing empirical studies. In this case, we would have: i h £ A¤ −1 b A αDi = (α/β 0 )∆ΠAS E P V (ci ) = ∆ΠAS i β0 i
(7)
to avoid death (DiA = 1). For the “no program” alternative, where (DiA = 0), the expected present value would be zero. When the marginal utility of income is heterogeneous across individuals, these simplifications are of course not possible.
In that case, the process of
calculating the expected present value of program costs does not produce a term that cancels everything but β 0 . The expected present value can still be calculated, but the formulas will ¡ ¢ and the other arguments of the B term in remain functions of both 1 − ΠAS and ΠAS i i equation (5).
3.6
Value of a statistical illness (VSI)
The expected present discounted value in equation (6) pertains to the maximum annual willingness to pay for a small risk reduction, ∆ΠAS i . The proportionality assumption leads to a tradition in the mortality valuation literature of ignoring the size of the risk difference involved, ∆ΠAS i , and scaling each expected present value option price to the amount that would correspond to a 100% risk difference. In the very special case of the variant in equation (7), division by the absolute risk change produces a VSL equal to −(α/β 0 ). To convert our more-general expected present value option price to something that might be termed the
29
“value of a statistical illness” (VSI), we could also divide our richer formula by the absolute size of the risk reduction. Using the same abbreviations B and C for the detailed expressions defined for equation (5), but again for the simpler case where β 1 = 0, the formula reduces to:
¤ £ ∙ ¸ E P V (cA∗ εi i ) −1 A A = β 0 β 0 Yi pdvli − αtermsi + (8) V SI = |∆ΠAS |∆ΠAS i | i | ¡ ¢ A A where we take advantage of the fact that pdvyiA + pdvliA = pdvcA = i so that pdvyi − pdvci −pdvliA and αtermsA i is defined following equation (5).
Across the distribution of the logistic error term, εi , the expectation is zero, so the expected value of a statistical illness depends only on the systematic portion of equation (8). The V SI in this case will depend upon the different marginal utilities of avoided periods of illness, recovered status, and premature death and on the way these marginal utilities vary with age at the time each health status is to be experienced. It will also depend upon the A A time profiles for each of these states as embedded in the terms pdviA i , pdvri , and pdvli , A A A A A as well as agepdviA i , agepdvri , agepdvli and potentially age2pdvii , age2pdvri , age2pdvli ,
and (implicit in this model) upon the individual’s own discount rate.18 In this simple model with a constant marginal utility of income, increases in income Yi will increase the predicted point estimate of the V SI. The effect of income on V SIiA is given by ∂V SIiA /∂Yi = pdvliA which is non-negative. Thus the effect of an increase in income on the predicted V SI will be larger (i.) as more life-years are lost, (ii.) as the individual is older, so that life-years lost come sooner in time. The effect of income on V SI can be estimated more generally if the marginal utility of income is not constant.19 18
Subsequent work will preserve individual discount rates as systematically varying parameters, to be estimated with reference to the individual’s responses to a hypothetical “how to take your lottery winnings” question. 19 Nothing in this specification precludes negative point estimates of the V SI. A positive V SI estimate will result, however, if the estimated value of the marginal utility of income is positive and there are negative
30
The existing literature, especially the hedonic wage-risk literature, focuses most intently on society’s willingness to pay for incremental reductions in the chance of a sudden accidental death in the current period. Thus there are no age-at-health-status differences that are not captured by agei0 . In the framework of our illness profiles, such an event would be captured by zero years of morbidity and death in the current year, with the remainder of the individual’s A nominal life expectancy experienced as lost life-years. Since the terms in pdviA i and pdvri
will be zero, our analog to the conventional VSL formula will be simply: ¤ µ £ ¶ E P V (cA∗ −α30 i ) = + Yi pdvliA E[V SL] = AS β |∆Πi | where
pdvliA =
X
δ t lylitA . The summation in the formula for pdvliA is from the present
until the individual’s nominal life expectancy. This interval depends upon the individual’s current age, so even in the version of our model with homogeneous preferences, the VSI will vary with age. Our VSI estimates also depend upon the individual’s income and discount rate. We emphasize that the sampling distribution of fitted VSIs for the range of stylized illness profiles generated for the choice experiments in our survey will not necessarily match the actual distribution of VSIs in the general population that faces a real range of illness profiles.20 To be clear on what is needed to construct fitted VSIs using our present results, one would need, for the illness in question, an approximate joint distribution for the possible ages of onset, possible reductions in lifespans, and possible outcomes (i.e. recovery, sudden death, or extended morbidity). In practice, this joint distribution will be constructed using expert values for the marginal utilities of each adverse health state. 20
An appendix available from the authors nevertheless describes this sampling distribution for each of our models.
31
judgment and its validity will in part determine the validity of the eventual VSI estimates our models will produce. For the population affected by this health threat, one would need an approximate joint distribution of age, gender, and income level. This joint distribution should again be based on expert judgment combined with exposure and epidemiological data, and the validity of this distribution will also determine the validity of the VSI estimates. To simulate the predicted distribution of VSIs for a particular range of illnesses afflicting a particular population, one would make a large number of random draws (say, 10,000 or even 1,000,000) from the joint distribution of illness profile characteristics and affected population characteristics and combine the results of each draw with our estimated formulas for the value of a statistical illness to produce a marginal distribution for the range of implied VSIs. The central tendency of this distribution can be interpreted as our model’s prediction about the population average of VSIs for this type of health threat.
4
Survey Methods and Data
Adequate market data are not available to illustrate how individuals allocate risk mitigation expenditure across competing risks and across their remaining years of life.21 Therefore, we have conducted a survey of 1,619 randomly chosen adults in the United States. The centerpiece of the survey is a conjoint choice experiment that presents individuals with hypothetical specific illness profiles and programs designed to mitigate these risks. Here, we briefly describe the five modules of this survey.22 21
Most market data characterizes at best only one source of risk (e.g. hedonic wage data) and are often missing essential variables such as the baseline risk, risk reduction, the latency of the programs or the costs of programs. For example, using the Health and Retirement Survey, Picone, Sloan and Taylor (2004) expertly explored how time preferences, expected longevity and other demand shifters affect individuals’ propensity to get mammograms, pap-smears and regular breast self-exams. However missing data on program costs, baseline risks, and latency of program benefits prevented a fuller demand analysis. 22 An example of the primary survey instrument is available from the authors.
32
The first module evaluates the individual’s subjective risk assessment for each of the major illnesses considered in the study, their familiarity with each illness, and behaviors they may currently undertake to mitigate or avert these health risks.23 The second module of the survey consists of a tutorial that introduces individuals to the idea of an illness profile and programs that may manage these illness-specific risks. Each illness profile is a description of a time sequence of health states associated with a major illness that the individual is described as facing with some probability over the course of his or her lifetime. The attributes of the illness profiles are randomly varied, subject to some plausibility constraints for each illness type.24 We summarize the key attribute levels employed in the choice sets in Appendix Table A-2. Twelve specific illnesses are used in our study, and up to eleven attributes characterize each illness profile and program. These illness profiles include an illness name, age of onset, treatments, duration and level of pain and disability, and a description of the outcome of the illness.25 We next explain to individuals that they could purchase a new program that would be coming on the market that would reduce their risk of experiencing a specific illness over current and future periods of their life. These programs are described as involving annual diagnostic testing and, if needed, associated drug therapies and recommended life23
Prior to the choice experiments, we ask individuals questions about their subjective assessment of: 1) various background environmental risks, 2) their risk of each illness, 3) their personal experience with illness, and 4) the experience of friends and family with each illness. 24 We took great care to try to ensure individuals did not reject the scenario because it was implausible (e.g., one does not recover from Alzheimer’s or die suddenly from diabetes). An appendix, available from the authors, provides more details. 25 Our selection of these attributes was guided by a focus on those attributes that 1) most affected the utility of individuals and 2) spanned all the illnesses that the individuals evaluated (Moxey et al. 2003). In terms of the number and type of attributes, our design is comparable to existing state-of-the-art health valuation studies (Viscusi et al., 1991; O’Connor and Blomquist, 1997; Sloan et al., 1998; Johnson et al., 2000). We sought to estimate demand conditional on the individual’s ex ante information set. Therefore, we chose not to give individuals extensive background information on illnesses, which might make one illness risk appear more salient than others.
33
style changes.26 The effectiveness of these programs at reducing risk is described using four means: 1) graphically, with a risk grid, 2) risk probabilities, 3) measures of relative risk reduction across the two illness profiles and 4) a qualitative textual description of the risk reductions (Corso et al., 1999; Krupnick et al., 2002). The payment vehicle for each program is described as a co-payment, expressed in both monthly and annual terms, that would be necessary for the remainder of their life unless they actually experienced that illness.27 In the preamble to the five choice sets that form the core of the survey, we implement several measures to avoid potential biases.28 We include an explicit “cheap talk” reminder to ensure that respondents carefully consider their budget constraint and to discourage them from overstating their willingness to pay (Cummings and Taylor, 1999; List, 2001).29 We also carefully explain to individuals that they can choose Neither Program and offer several reasons why a reasonable person might do so.30 The third module of the survey contains the five key choice sets that respondents are asked to consider independently.
Each choice set offers two programs to reduce the risk
of two distinct illness profiles.31 In a preliminary ad hoc analysis to assess the construct 26
We selected this class of interventions because pretesting showed that individuals viewed this combination of programs (diagnostic tests, followed by drug therapies) as feasible, potentially effective and familiar for a wide range of illnesses. Depending upon their gender and age individuals were familiar with comparable diagnostic tests such as mammograms, pap smears and prostrate exams, or the new C-reactive protein tests for heart disease. 27 Most respondents’ experience with co-payments and finely designed insurance premiums for different levels of service made this annual cost assumption entirely plausible. 28 Targeted biases include hypothetical and incentive compatibility biases as well as yea-saying behavior. Other biases that we address include order and sequencing effects, Weber’s law in risk perception and various framing and anchoring concerns. 29 This screen began “In surveys like this one, people sometimes do not fully consider their future expenses. Please think about what you would have to give up to purchase one of these programs. If you choose a program with too high a price, you may not be able to afford the program when it is offered. . . ” An appendix available from the authors provides the complete context. 30 These reasons include that they 1) cannot afford either program, 2) did not believe they faced these illness risks, 3) would rather spend the money on other things, 4) believed they would be affected by another illness first. If the individual did choose neither program we ask them why they did so in a follow-up question. 31 Presenting individuals with a large array of illness risks had advantages and disadvantages. The greatest
34
validity of our study, we also explore whether individual choices are sensitive to the scope of the illness profile and risk-mitigating program (Hammitt and Graham, 1999; Yeung et al., 2003).32 The fourth module contains various debriefing questions that can be used to document the individual’s status quo health state profile and to cross check the validity of their subjective responses in the first module (Baron and Ubel, 2002). Module five was administered separately from the choice experiment. It collects a detailed medical history of the individual and household socio-economic information.33 The development of this survey instrument involved 36 cognitive interviews, three pretests (n=100 each) and large pilot study (n=1,109).34 Knowledge Networks Inc. (KNI) administered the final version of the demand survey and the health-profile survey to a sample of 2,439.35 In addition to the benefits of regular KNI panel membership, respondents were paid an additional $10 incentive for completing the survey. Our response rate for those panelists contacted was 79 percent. Although our observable sample characteristics are generally repadvantage was that individuals considered a more complete choice set, allowing us to observe how they substitute across programs associated with these competing illness risks. Second, presenting a range of major illnesses increases the representativeness of our estimates and makes the motivation of a fuller range of illness profiles plausible, and thus possible. One disadvantage is that it limits the background information that we could provide about each illness. A second potential disadvantage is the cognitive complexity associated with the choice task, which we sought to minimize through the survey design and to evaluate ex post. 32 Additional appendix tables, available from the authors, report estimates for a standard atheoretic linear additively separable conjoint analysis. Individuals are highly sensitive to changes in the scope of our central attributes: the number of years spent in a morbid condition, the number of premature lost life years, the costs of the program, and the size of the risk reduction. 33 We have a great deal of health and sociodemographic profile information on each individual that helps us to characterize his or her future health state expectations. A detailed assessment of the effects of comorbidity on willingness to pay for health risk reductions is the subject of a separate paper. 34 We thank Vic Adamowicz, Richard Carson, Maureen Cropper, Baruch Fischhoff, Jim Hammitt, Alan Krupnick, and V. Kerry Smith for their careful reviews of the second of four versions of this instrument. 35 Respondents are recruited in the Knowledge Network sample from standard RDD techniques. They are then equipped with WebTV technology that enables them to receive and answer our surveys. More information about Knowledge Networks is available from their website www.knowlegdenetworks.com.
35
resentative of the US population (Appendix Table A-1 compares the marginal distributions for our sample against those for the 2000 Census for age, income and gender), we have elsewhere explored for any problems in terms of sample selection on unobservables, relative to the entire pool of random-digit dialed original recruiting contacts for the KNI panel. Minimal sample selection distortions are present, so the results we present here do not include these further minor corrections.
5
Results and Discussion
Table 2 compiles estimation results for six different specifications estimated under an assumption of a 5% individual discount rate. Additional appendices, available from the authors, detail the results of sensitivity analyses with respect to the discounting assumptions used by considering both a 3% discount rate and a 7% discount rate. These rates were chosen based on the official range of values recommended for benefit-cost analysis by the Science Advisory Board of the US EPA. For any given discount rate, we must calculate in advance the various present discounted value terms (capturing the time profiles of morbidity and mortality) employed in construction of the eventual estimating variables.36 Before discussing the empirical results, it may be helpful to provide concrete examples of the specifications we will consider. Our baseline Model 1 allows for the level of income to affect the marginal utility of additional income, but excludes any age effects on the marginal 36
In the models presented in this paper, we lean heavily on linearities that allow us to estimate our parameters using packaged software algorithms for McFadden’s conditional logit models.
36
(dis)utilities of health states: £ ¤ £ ¤ A E ViA − E ViN = β 0 (β0termA i ) + β 1 (β1termi )
(9)
A AS A AS A +α10 ∆ΠAS i pdvii + α20 ∆Πi pdvri + α30 ∆Πi pdvli + εi
Models 2 and 3 introduce the respondent’s current age as the only age-related shifter on each of the marginal utilities of the adverse health states. In Model 2, current age enters linearly, while in Model 3 it is entered in quadratic form. Model 4 allows the marginal utility of income to be shifted by the respondent’s current age (agei0 ), and allows the marginal (dis)utility of each health state to vary linearly with both the respondent’s current age (agei0 ) and and future “age-at-health-state” (ageit ): £ ¤ £ ¤ A E ViA − E ViN = (β 00 + β 01 agei0 )(β0termA i ) + β 1 (β1termi )
(10)
A AS A +(α100 + α101 agei0 )∆ΠAS i pdvii + α11 ∆Πi agepdvii
A AS A +(α200 + α201 agei0 )∆ΠAS i pdvri + α21 ∆Πi agepdvri A AS A +(α300 + α301 agei0 )∆ΠAS i pdvli + α31 ∆Πi agepdvli + εi
Model 5 allows each of the (dis)utilities of the different health states to be fully quadratic
37
in the respondent’s age now (agei0 ) and age-at-event (ageit ). £ ¤ £ ¤ E ViA − E ViN
(11)
A = (β 00 + β 01 agei0 )(β0termA i ) + β 1 (β1termi ) A +(α100 + α101 agei0 + α102 age2i0 )∆ΠAS i pdvii A AS A + (α110 + α111 agei0 ) ∆ΠAS i agepdvii + α12 ∆Πi age2pdvii A +(α200 + α201 agei0 + α202 age2i0 )∆ΠAS i pdvri A AS A + (α210 + α211 agei0 ) ∆ΠAS i agepdvri + α22 ∆Πi age2pdvri A +(α300 + α301 agei0 + α302 age2i0 )∆ΠAS i pdvli A AS A + (α310 + α311 agei0 ) ∆ΠAS i agepdvli + α32 ∆Πi age2pdvli + εi
The marginal utility of income should be positive, but is not constrained to be so in these models. Our competing specifications also involve sets of parameters that describe the marginal (dis)utility of a sick year, a recovered year, and a lost life-year. The marginal utilities of adverse health states should also be negative on average, but for tractability in estimation we do not enforce this restriction.37
5.1
Parameter Estimates
Our estimating sample is reduce in size from the 2439 respondents to 1619 by three main exclusion criteria. First, we exclude choice sets whith outright scenario rejection. This is 37
One would typically expect that the marginal utility of a lost-life-year would be negative, but there may be specific exceptions. A positive marginal utility associated with a lost life-year might be expected when the illness is question constitutes a “fate worse than death.” For certain illnesses, such as severe Alzheimers’ disease, we might expect that death would “come as a blessing.” In any situation where the pre-death state was less onerous, however, we would expect death to be unwelcome, and hence that the marginal utility of a lost life-year would be negative.
38
defined as a case where the individual selected the Neither Program alternative and indicated that their sole reason for doing so was that they did not believe the programs would work as described. Second, after the risk tutorial, respondents were asked to answer a question to verify their comprehension of the notion of risk used in the survey scenarios.
If an
individual answered this question incorrectly, we do not use any of their choices. Finally, some choice sets were excluded because a programming error in the randomized design algorithm created some illness profiles where the illness was reported to extend, rather than reduce, the individual’s life expectancy. While this outcome may be possible, we make a conservative choice and exclude these choice sets. In Table 2, Model 1 without age effects shows robust significance and the expected signs on all five basic marginal indirect utility parameters.
The marginal utility of income is
positive, but declines as expected with the level of income. The marginal utilities of sickyears, recovered years, and lost life-years are all negative. Post-illness (“recovered”) years are not interpreted by respondents to be equivalent to pre-illness years. Respondents seem to impute reduced health or reduced function to these recovered years. The similarity in the magnitude of the marginal utility of a sick-year and a recovered-year may be due to the fact that the illnesses are described as major life-threatening illnesses, including cancers, respiratory disease, and stroke, for example. Previous empirical studies of the effects of age on WTP for risk reductions have typically allowed VSL estimates to depend systematically upon the respondent’s current age, and have not considered their future age at the time they would be experiencing different adverse health states. In most cases, this is an artifact of the strategy of considering only currentperiod risk reductions, and only mortality risks. By comparing the results from Model 3 to those from Model 2, we see that the disutility of a sick-year is predicted to lessen with
39
age now, that of a post-illness year to increase with age now, and that of a lost life-year to lessen with age now. There appear to be no individually statistically significant quadraticin-current-age effects in our data. Model 4 represents a distinct departure from any model that has been estimated in the prior literature.
This model allows the marginal utility of a year in each type of
adverse health state to vary linearly with both the respondent’s age at the time of the survey and their future age when they would be experiencing each probabilistic adverse health state. The respondents present age dictates the range of future ages at which these health states can possibly be experienced, so these two age variables will be correlated in our data. The estimates for Model 4 reveal that failure to control for age-at-health-state produces a substantial bias in the apparent effects of age now on the marginal disutility of each type of health state. In these linear specifications, age now decreases the disutility from a year in each health state (although not significantly so for the post-illness state), whereas age-at-health-state significantly increases the disutility from each type of adverse health state. Model 4, with linear effects for both age variables, makes the main empirical point in this paper. Figures 1, 2, and 3 depict the implications of the estimated parameters from Model 4. We use representative individuals at each of five current ages (25, 35, 45, 55 and 65) and with income of $40,000. We then calculate the fitted marginal (dis)utilities for a year in each of the three adverse health states at each future age. The next step is to monetize these marginal disutilities into undiscounted WTP measures to avoid that health state at that future age. This is accomplished by dividing each fitted health-state marginal utility by the marginal utility of income. For each current age, the only relevant portions of these curves lie to the right of that current age. In using this information to construct the value
40
of a statistical illness (VSI) for a particular illness profile, one could compute the present value of the relevant intervals of future age for an individual of a given age now, and then add these present values across the adverse health states in question. Figures 1, 2, and 3 reveal that, controlling for the individual’s age now, WTP to avoid each adverse health state is greater, the more advanced the future age at which that health state would be experienced. However, the older the respondent is now, the less they are willing to pay to avoid adverse health states at any future age. These tendencies are very clear for avoided sick-years and avoided lost life-years, although they are less pronounced for avoided post-illness years. These findings are fully consistent with the two main hypotheses discussed in the theoretical section of this paper. One troubling feature of the Model 4 with its linear age effects is the persistence of negative undiscounted WTP estimates in early future years. These portions of the undiscounted WTP function will tend to bias downward the present value employed as an estimate of the Value of a Statistical Illness (VSI) for near-term health threats. Figures 1, 2, and 3 show, for example, that undiscounted WTP to avoid a statistical sick-year, recovered-year, and lost life-year for a currently 25-year-old respondent (the line tagged with “25”) appear to be negative for the first few years into the future. We suspect that many respondents, feeling currently rather healthy, doubt that the health risk we describe will actually affect them in the next 5-10 years, although the possibility of becoming ill in the years beyond that is more credible.38 There is no opportunity for any respondent to express a negative willingness to pay explicitly. At a minimum, respondents can imply that the value they place on a program 38
It is not clear whether this should be interpreted as a form of partial scenario rejection or scenario revision in response to our stated preference choice scenarios, or whether this is a legitimate property of people’s preferences.
41
is zero (i.e. no greater than the cost of the Neither Program alternative, available at zero net cost). To determine whether these negative fitted WTP estimates in the linear models are merely an artifact of a too-restrictive functional form, we estimate a specification that allows the marginal utilities associated with all three health states to be fully quadratic in both age now and age-at-health-state. This strategy maintains the hypothesis that survey subjects are responding to exactly the illness profile information provided in the survey, but assesses whether the intervals of age-at-health-status displaying negative fitted undiscounted WTP may be simply an artifact of functional form. In Model 5, we allow all marginal utilities from adverse health states to be fully quadratic functions of both age now and age-at-event. In terms of statistical significance, the results can only be described as underwhelming. However, we suspect that our empirical results are being somewhat confounded by a minority of individuals who rushed through the survey without devoting much careful consideration to their choices. As a check on this possibility, we limit the sample to only those individuals who spent an average of at least 20 seconds on each choice set page of the survey. We term this restriction a “minimum time-on-task” requirement. This limitation reduces the number of choice sets in the estimation from 7520 to 6627.
However, Model 5’ estimated on this sample produces a number of statistically
significant coefficients among the terms associated with the marginal disutility of a sick-year and the marginal disutility of a lost life-year, and the signs of the statistically significant coefficients match those produced with the full sample. Figures 4, 5 and 6 display the profiles of undiscounted WTP to avoid statistical years in each adverse health state, according to the fully quadratic Model 5. Despite the statistical insignificance of many of the coefficients, the fitted undiscounted WTP schedules preserve the essence of the results of the linear Model 4, but the troublesome negative portions of
42
these curves are greatly reduced. We will use Model 5 as the basis for our simulations in the next section, since the results appear to be the least contaminated by negative undiscounted WTP to avoid very near-term portions of any risk profile. For completeness, Model 6 in Table 2 backs off from the fully quadratic specification to consider a more parsimonious version of the quadratic specification. An interaction term between age now and age-at-health-state is strongly statistically significant for the marginal disutility of a sick-year. This same interaction term, plus the quadratic term in age now, are significant shifters of the marginal disutility of a lost life-year. These results suggest that the most statistically defensible model is richer than the simple linear specification in Model 4, but perhaps not as elaborate as the fully quadratic specification in Model 5. However, the further we retreat from the fully quadratic form, the more the fitted WTP estimates are plagued by the negative portions in the linear model, so we elect to avoid the downward biases that the linear models produce by sticking with the full quadratic form.39 Figures 4, 5, and 6 reveal the consequences of allowing a more general functional form. These diagrams strongly suggest that most respondents place little value on avoiding a sickyear that will occur prior to their 50s.40 Respondents who are currently younger place higher value on avoiding future sick-years at specified ages than do currently older respondents (for those same specified ages). Similar patterns, to a greater or lesser degree, are apparent for recovered-years and lost life-years.41 39
It is very likely that the statistical insignificance of individual quadratic and interaction terms in age now and age-at-health-state is, to some extent, a result of the degree of collinearity between these two age variables that is created because we ask individuals to consider only future health states. In any event, the medians for simulations we explore are not qualitatively different for Models 5 and 6, and the confidence intervals are only slightly narrower. 40 The downward-sloping portions of the curves in these figures are most likely just an artifact of fitting the best quadratic form to the mass of the data, which will tend to lie higher up in the age-at-health-state distribution. 41 Model 5 creates a strong impression that it will be desirable in future work to break away from linearin-parameters models, in spite of their extremely attractive properties for ease of estimation. A non-linear
43
5.2
Fitted VSIs
Fitted VSIs for the estimating sample of illness profiles can be computed. However, these VSI estimates reflect the artificial range of illness profiles generated for use in eliciting individual choices.
They do not reflect the true joint distribution, in the real world, of illnesses,
symptoms and treatments, and prognoses.
In particular, there are many short-term and
non-fatal illnesses among the programs we presented to respondents. Thus, we do not expect to see the usual $6.1 million VSL estimate in these distributions.42 How do the WTP implications from our statistical illness model comport with those of earlier VSL studies? Many hedonic wage estimates of “the” VSL estimate wage-risk tradeoffs for middle-aged white males in blue collar jobs.
For comparison with earlier results, we
should probably focus on the VSI for an illness profile consisting of sudden death for a 45year-old. However, in order to highlight the generality of our WTP models, compared to earlier VSL models, we will consider four classes of simulations: Simulation 1. How would a 25-, 35-, 45-, 55-, and 65-year-old value a reduction in the chance of sudden death starting now? Simulation 2. How would a 25-, 35-, 45-, 55-, and 65-year-old value a reduction in the chance of sudden death starting 10 years from now? Simulation 3: How would a 25-, 35-, 45-, 55-, and 65-year-old value a reduction in the chance of sudden death starting at age 70? Simulation 4. How would a 25-year-old value a reduction in the chance of sudden death starting 5, 15, 25, 35 and 45 years from now? model, wherein we estimate the logarithms of the marginal utilities of income and years in each health state rather than their absolute levels, may be promising. 42 Descriptive statistics of VSIs for the estimating sample are provided in an appendix, available from the authors.
44
Table 3 summarizes the results of these four classes of simulations for Models 1, 3, 4, and 5. The results for the parsimonious Model 6 are not reported, since they closely match those for Model 5. For each simulation, we make 1000 random draws from the joint distribution of the maximum likelihood conditional logit parameters. For each set of parameter values, we calculate the desired VSI. We report the median of this distribution, as well as the 5th and 95th percentiles.43 The middle row for Simulation 1 is the closest thing to a conventional wage-risk VSL that our model produces, so it is highlighted in boldface type. With no age effects, Model 1 conforms as closely as possible to the implicit risk assumptions underlying many previous studies.
This model does not differentiate the marginal
utility of a lost life-year according to the age of the respondent now or the age the respondent would have been during each life-year lost under a current-period risk of sudden death. The median predicted VSI is nevertheless still expected to differ with the respondent’s current age because our model emphasizes life-years and involves discounting. Remarkably, despite these fundamental differences from previous models, our median VSI for sudden death for a 45-year-old is $5.86 million, with simulated 90% confidence bounds of ($4.44 million to $7.45 million). This range of estimates compares very favorably to the roughly $6 million estimate used routinely by the US EPA in their major benefit-cost analyses. For Model 1, evidence for any substantial “senior death discount” is sparse in simulations 1 and 2. The medians do decline monotonically with the current age of the respondent, but the differences are small. In Simulations 3 and 4, there are larger effects from the latency differences we explore. In Simulation 3, we see marked increases in the VSI for sudden death at age 70 as the respondent is presently closer to 70 in age. In Simulation 4, where 25-year-olds are 43
Technically, the mean of this distribution is undefined, since it the mean of a ratio of normally distributed random variables. We report these selected percentiles of the sampling distribution of these different point estimates to convey the implications of the degree of precision in the parameter estimates.
45
asked to consider risks of sudden death at increasingly distant future times, the VSI falls substantially and significantly. However, our data emphatically reject Model 1 in favor of a model that acknowledges the systematic variation of WTP for risk reductions with respect to the respondent’s age now and the age at which they would experience future lost life-years. However, if only the respondents current age is used as a linear shifter on the estimated marginal disutilities, as in Model 2, the simulated distribution is implausible. We do not convert negative fitted VSI estimates to zero in summarizing these simulated estimates, and Model 2 implies that the value of the statistical illness in Simulation 1 is overwhelmingly negative. While the quadratic form in Model 3 did not yield individually statistically significant coefficients on either the linear or the quadratic current-age interaction terms that capture the marginal disutility of a lost life-year, the implied VSI distribution is nevertheless much more plausible and the results are therefore included in Table 3. Model 4’s estimates reveal the bias in Model 2’s estimates of the age-now effects when age-at-health-state is ignored, but its VSI implications are compromised somewhat by the negative undiscounted WTP estimates for very near-term risks that tend to produce a downward bias the resulting present-value WTP estimates for illness profiles that constitute the VSI. Model 5 is our preferred specification, despite the individual insignificance of its parameter estimates, because it largely undoes the negative undiscounted WTP problem. However, the imprecision in the resulting parameter estimates produces an undesirably wide 90% simulated confidence interval for the VSI. Still, its median value of $3.54 million is probably the least biased number produced by the variety of models we consider. For Model 5, for simulation 1, there is only a very limited suggestion of a “senior discount”
46
in WTP to avoid sudden death in the current period, although Model 4, with its regions of unsubstantiated negative undiscounted fitted WTP dictated by the linear form, suggests a substantial discount.44
However, simulation 2 (sudden death in 10 years) avoids some
of the problems with negative near-term undiscounted WTP and reveals a somewhat morepronounced variant of the senior discount phenomenon for each of Models 4 and 5. We reiterate that these models control both for the respondent’s age now and for their prospective age at the time the health state is to be experienced, and we have demonstrated that both of these age-related considerations are important to people’s choices concerning health risks. For simulation 3, which pertains to sudden death at age 70 for people of different ages now, the progression in median simulated VSIs is non-monotonic, and is somewhat lower for both the oldest and youngest age groups, although the differences, especially for Model 5, are not very large. In simulation 4, VSIs for 25-year-olds appear to be strongly declining for latencies greater than 15 years.
6
Conclusions
Policy analysis with respect to risk-management programs requires detailed information about consumer demand for these programs. We begin with a concise theoretical model, adapted from Ehrlich (2001), that produces two key insights. First, individuals will derive increasing marginal utility from reducing risks that they will face later in life, which implies that individuals will be willing to pay more to reduce risks that will afflict them when they are older (and correspondingly less to reduce risks that will afflict them when they are younger). The second insight is that health and other consumption goods are likely to be complements. 44
Perhaps, however, sudden death is viewed as less likely for older respondents. It is possible that respondents substitute lower risks than the survey instrument suggests, interpreting their own risk to be lower than the “average” that they assume is being quoted in the survey.
47
As individuals age, they learn more about the extent of complementarity between health and other goods—in particular, they learn that future consumption will provide less utility because of declining physical well-being. Hence they are inclined to shift more consumption forward in time and their willingness to pay for health risk reductions will fall as they are older. Which of these two countervailing effects will dominate is an empirical question, so we have set out to build a formal utility-theoretic model that captures the relevant considerations in private ex ante consumer choices about incurring ongoing expenditures to reduce risks to life and health. Most past studies have focused on current-period costs and current-period benefits. In contrast, our model recognizes the future time profiles of illnesses and injuries for which individuals may choose to act to reduce their risks. Intertemporal consumer optimization requires explicit treatment of the interaction between disease latencies and individual discount rates. Our model permits us to derive option prices for programs that reduce well-defined types of risks. Option prices are the appropriate theoretical construct for decision-making under uncertainty, where the uncertainty in this case concerns whether the individual will actually suffer the illness or injury that the proposed risk reduction measure addresses. While we believe that it is important to preserve information about the nature of the risk reduction involved (its size, and perhaps the baseline risk), we show that our option price WTP formulas lead naturally to what we have labeled as the “value of a statistical illness” (VSI). The VSI is the present discounted value of the stream of maximum annual payments that the individual would be willing to pay for the specified (typically small) risk reduction, scaled up proportionately to correspond to a risk reduction of 100%. This construct is analogous to the more familiar, but more-limited, concept of the value of a statistical life (VSL). A VSL is typically constructed by looking simply at the static single-
48
period willingness to pay for a specified risk reduction, and scaling this willingness to pay up to a 100% risk reduction. However, static VSL estimates do not typically vary with important morbidity/mortality attributes such as latency, time profiles of illness, symptoms and treatments, outcomes, or life-years lost. In the empirical analysis presented in this paper, we first consider a model wherein preferences are considered to be homogenous across all types of individuals and where the marginal (dis)utility of a sick-year or a lost life-year is independent of the respondent’s age now and his or her age at the time he or she would be experiencing that health state (or the age that they would have been, had they not died prematurely). Even these very simple models can be used to display the sensitivity of option prices to the timing of events in an illness profile. The pattern of future health states in question matters for willingness to pay to avoid different types of risks to life and health.. Our empirical analysis also demonstrates conclusively that the current age of the respondent, as well as the prospective age at which they will experience illness or premature death, will have a systematic effect on willingness to pay for programs that reduce health risks. These findings are relevant to the current debate about whether there should be a “senior death discount” in assessing the health benefits of costly risk reductions. The choices made by the individuals in our sample strongly suggest that, ceteris paribus, the older an individual is when asked to begin paying for a particular health risk reduction, the less he or she will be willing to pay. However, this tendency can be confounded by the fact that for individuals of a given age, willingness to pay for health risk reductions increases with the age at which these health risks would be experienced.
Any given individual, looking
forward, may feel that they would be willing to pay more to reduce risks to their health that materialize when they are older. This tendency may feed the intuition that the benefits of
49
risk reductions should be, if anything, higher for older persons. However, across individuals of different ages, older individuals seem willing to pay less to reduce risks to their health.
50
7
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54
Table 1 Descriptive statistics for Risk Reduction Programs (5% discount rate); (7520 choice sets; 22560 alternatives; 15040 profiles; 1619 individuals) Variable
Mean
Std. Dev.
Min
Max
2.21 0.474 2.57 $ 29.87 -.00341
2.51 1.36 2.93 28.71 .00167
0 0 0 2 -0.006
16.3 15.9 17.8 140 -0.001
$ 50,771 50.30 0.513
33,966 15.21
5,000 25
150,000 93
Risk Reduction Programs Present discounted sick-years Present discounted recovered-years Present discounted lost life-years Monthly cost Risk change Respondents income age (years) female (1=female, 0=male)
55
Table 2 Parameter Estimates for Competing Specifications
Parameter and description of variable(s)
β 00 *10-5 (linear net Y term)
Model 1 No Age Effects
Model 2 Age Now Effects (linear)
Model 3 Age Now Effects (quadratic)
Model 4 Both Age Effects (linear)
Model 5 Both Age Effects (quadratic)
Model 5’ Both Age Effects (Min. time on task)
Model 6 Parsimonious Variant
4.87503 (8.60)***
4.022 (2.39)**
4.020 (2.34)**
6.987 (3.92)***
8.07941 (4.24)***
8.186 (4.01)***
7.542 (4.13)***
0.287 (0.11)
0.292 (0.11)
-3.345 (-1.25)
-5.259 (-1.78)*
-5.034 (-1.61)
-4.656 (-1.66)*
-0.2199 (-4.71)***
-0.2033 (4.30)***
-0.2033 (4.30)***
-0.2015 (-4.25)***
-0.2017 (-4.25)***
-.233 (-4.59)***
-.2031 (-4.29)***
-8.390 (-5.00)***
-32.47 (5.00)***
-33.05 (1.62)
1.316 (0.11)
-10.7925 (-0.19)
-79.87 (-1.27)
47.80 (2.65)***
0.489 (3.91)***
0.513 (0.64)
1.1783 (4.95)***
-1.1702 (-0.59)
-3.547 (-1.65)*
-0.0133 (-0.59)
-0.0292 (-1.21)
1.1535 (0.42)
5.106 (1.70)*
-1.561 (-3.82)***
0.0504 (1.00)
0.1083 (1.98)**
0.0142 (4.58)***
-0.0333 (-0.97)
-0.0842 (-2.25)**
β 01 *10-7 ( agei 0 interaction) β 3 *10-9 (DMU(Y) term)
Sick years
δ100 ∆Π iAS pdvii
δ101∆Π iAS pdvii × agei 0 δ102 ∆Π iAS pdvii × agei20
-0.0002 (-0.03)
δ110 ∆Π iAS agepdvii
-1.0029 (-3.43)***
δ111∆Π iAS agepdvii × agei 0 δ12 ∆Π iAS age2 pdvii
Continued… 56
Table 2, continued:
Parameter and description of variable(s)
Model 1 No Age Effects
Model 2 Age Now Effects (linear)
Model 3 Age Now Effects (quadratic)
Model 4 Both Age Effects (linear)
Model 5 Both Age Effects (quadratic)
Model 5’ Both Age Effects (Min. time on task)
Model 6 Parsimonious Variant
-8.0219 (-2.48)**
14.62 (1.27)
19.97 (0.54)
53.26 (2.27)**
-49.15 (-0.51)
-31.26 (-0.30)
42.38 (2.13)**
-0.4615 (-2.00)**
-0.6913 (-0.45)
0.3965 (0.85)
-4.449 (-1.04)
-5.869 (-1.26)
-0.0132 (-0.27)
-0.0218 (-0.42)
5.6919 (1.1)
6.523 (1.15)
0.0881 (0.81)
0.1235 (1.03)
-0.0835 (-1.21)
-0.1055 (-1.38)
Recovered years
δ 200 ∆Π iAS pdvri δ 201∆Π iAS pdvri × agei 0 δ 202 ∆Π iAS pdvri × agei20
0.0023 (0.15)
δ 210 ∆Π iAS agepdvri
-1.2055 (-2.08)**
δ 211∆Π iAS agepdvri × agei 0 δ 22 ∆Π iAS age2 pdvri
57
-0.7653 (-2.67)***
Table 2, continued:
Parameter and description of variable(s)
Model 1 No Age Effects
Model 2 Age Now Effects (linear)
Model 3 Age Now Effects (quadratic)
Model 4 Both Age Effects (linear)
Model 5 Both Age Effects (quadratic)
Model 5’ Both Age Effects (Min time on task)
Model 6 Parsimonious Variant
-8.0829 (-6.04)***
-25.18 (4.92)***
-26.12 (-1.62)
8.288 (0.73)
79.36 (1.29)
45.01 (0.70)
95.08 (2.25)**
0.3571 (3.50)***
0.3965 (0.61)
1.047 (5.07)***
-0.3568 (-0.17)
-2.346 (-1.04)
0.5216 (0.73)
-0.0402 (-1.95)*
-0.0550 (-2.51)**
-0.0381 (2.27)**
-2.1085 (-0.7)
0.6322 (0.20)
-3.364 (3.28)***
0.0737 (1.50)
0.1263 (2.40)**
0.0573 (2.41)**
-0.0163 (-0.46)
-0.0579 (-1.53)
22560 -11697.01
19881 -10295.367
Lost life-years
δ 300 ∆Π iAS pdvli δ 301∆Π iAS pdvli × agei 0 δ 302 ∆Π iAS pdvli × agei20
-0.0004 (-0.06)
δ 310 ∆Π iAS agepdvli
-1.0032 (-3.69)***
δ 311∆Π iAS agepdvli × agei 0 δ 32 ∆Π iAS age2 pdvli
Alternatives Log L
22560 -11733.469
22560 -11714.541
22560 -11714.527
58
22560 -11700.9
22560 -11699.009
Table 3 VSI for Four Classes of Sudden Death Scenarios (US $ million), based on 22560 alternatives, selected Models Model 1 Model 3 Model 4 Model 5 No “Age Now” Both Age Both Age Age Effects Effects Effects Effects (quadratic) (linear) (quadratic) Age Now
Age at LatenDeath cy
50% ( 5%,95%)
50% ( 5%,95%)
50% ( 5%,95%)
50% ( 5%,95%)
1. Simulation: Sudden death this year
25 35 45 55 65
25 35 45 55 65
0 0 0 0 0
6.36 (4.82, 8.09) 6.18 (4.68, 7.86) 5.86 (4.44, 7.45) 5.46 (4.14, 6.95) 4.88 (3.70, 6.21)
15.10 ( 7.58, 38.64) 11.29 ( 7.48, 21.35) 7.85 ( 5.52, 11.33) 4.68 ( 2.68, 7.18) 1.97 (-0.15, 4.16)
4.30 ( 1.31, 8.80) 3.68 ( 1.20, 7.02) 2.69 ( 0.56, 5.17) 1.61 (-0.43, 3.69) 0.24 (-2.10, 2.46)
0.35 (-7.53, 8.06) 2.48 (-2.53, 7.53) 3.54 (-0.01, 7.36) 3.98 ( 0.66, 7.48) 3.38 (-0.14, 7.38)
2. Simulation: Sudden death in 10 years
25 35 45 55 65
35 45 55 65 75
10 10 10 10 10
3.77 (2.89, 4.88) 3.59 (2.75, 4.64) 3.27 (2.51, 4.23) 2.87 (2.20, 3.72) 2.29 (1.76, 2.97)
8.73 (4.35, 24.57) 6.62 (4.31, 12.3) 4.35 (2.98, 6.76) 2.48 (1.39, 3.87) 0.91 (-0.16, 1.91)
4.88 (3.33, 7.77) 4.43 (3.29, 6.36) 3.66 (2.87, 4.80) 2.78 (2.11, 3.60) 1.70 (0.80, 2.57)
3.48 (1.64, 6.01) 3.87 (2.80, 5.56) 3.45 (2.61, 4.58) 2.60 (1.80, 3.51) 1.31 (0.22, 2.19)
3. Simulation: Sudden death @ fixed age (70)
25 35 45 55 65
70 70 70 70 70
45 35 25 15 5
0.42 (0.32, 0.53) 0.70 (0.53, 0.90) 1.15 (0.87, 1.46) 1.99 (1.51, 2.54) 3.43 (2.60, 4.37)
0.99 ( 0.50, 2.54) 1.29 ( 0.85, 2.44) 1.54 ( 1.08, 2.22) 1.71 ( 0.97, 2.62) 1.38 (-0.11, 2.93)
1.31 ( 0.93, 1.97) 1.86 ( 1.37, 2.56) 2.28 ( 1.78, 2.95) 2.54 ( 1.97, 3.17) 1.40 ( 0.01, 2.70)
1.93 ( 1.20, 3.04) 2.26 ( 1.61, 3.17) 2.21 ( 1.69, 2.94) 2.12 ( 1.31, 2.92) 2.12 ( 0.60, 3.70)
4. Simulation: Sudden death varying latency
25 25 25 25 25
30 40 50 60 70
5 15 25 35 45
4.91 (3.72, 6.25) 2.89 (2.19, 3.67) 1.65 (1.25, 2.09) 0.88 (0.67, 1.13) 0.42 (0.32, 0.53)
11.66 ( 5.86, 29.85) 6.86 ( 3.45, 17.56) 3.91 ( 1.96, 10.02) 2.10 ( 1.06, 5.39) 0.99 ( 0.50, 2.54)
4.78 ( 2.79, 8.16) 4.58 ( 3.31, 7.11) 3.55 ( 2.61, 5.34) 2.37 ( 1.70, 3.54) 1.31 ( 0.93, 1.97)
2.33 (-1.39, 6.38) 4.08 ( 2.52, 6.34) 4.04 ( 2.73, 6.12) 3.13 ( 2.13, 4.78) 1.93 ( 1.20, 3.04)
59
Figure 1 – Linear Model: Fitted future current values for Statistical Sick-years, for individual now 25, 35, 45, 55 and 65.
Figure 2 – Linear Model: Fitted future current values for Statistical Recovered-years, for individual now 25, 35, 45, 55 and 65. 60
Figure 3 – Linear Model: Fitted future current values for Statistical Lost Life-years, for individual now 25, 35, 45, 55 and 65.
Figure 4 – Quadratic Model: Fitted future current values for Statistical Sick-years, for individual now 25, 35, 45, 55 and 65. 61
Figure 5 – Quadratic Model: Fitted future current values for Statistical Sick-years, for individual now 25, 35, 45, 55 and 65.
Figure 6 – Quadratic Model: Fitted future current values for Statistical Lost Life-years, for individual now 25, 35, 45, 55 and 65.
62
Mortality and Morbidity Risk Reduction: An Empirical Life-cycle Model of Demand with Two Types of Age Effects 1
Appendix I - Abbreviations for Present Value terms
We abbreviate each of the numerous present value terms in the paper as follows: X X X = δt agepdvcA δ t ageit age2pdvcA δt age2it pdvcA i i = i = X X X t t A A A A pdveA = δ pre agepdve = δ age pre age2pdve = δ t age2it preA it i it i it i it X X X pdviA = δ t illitA agepdviA δ t ageit illitA age2pdviA δt age2it illitA i i = i = X X X pdvriA = δ t rcvitA agepdvriA = δ t ageit rcvitA age2pdvriA = δ t age2it rcvitA X X X pdvliA = δ t lylitA agepdvliA = δ t ageit lylitA age2pdvliA = δ t age2it lylitA
Notice that the following relationships hold, since the indicator variables for each health status are mutually exclusive and exhaustive: A A A = pdveA pdvcA i i + pdvii + pdvri + pdvli A A A agepdvcA = agepdveA i i + agepdvii + agepdvri + agepdvli A A A age2pdvcA = age2pdveA i i + age2pdvii + age2pdvri + age2pdvli
To accommodate the different time profiles of income and program costs over the individual’s remaining lifespan, we must also define five additional terms X ¡ ¢ A A A A A A A δt preA pdvyiA = it + γ 1 illit + rcvit + γ 2 lylit = pdvei + γ 1 pdvii + pdvri + γ 2 pdvli X ¡ ¢ A A A A A A A pdvpA = δt preA i it + γ 3 illit + rcvit + γ 4 lylit = pdvei + γ 3 pdvii + pdvri + γ 4 pdvli X ¡ ¢ A A A 2 2 A 2 A pdvyyiA = δt preA = pdveA it + γ 1 illit + rcvit + γ 2 lylit i + γ 1 pdvii + pdvri + γ 2 pdvli X ¡ ¢ A A A 2 2 A A 2 A pdvppA = δt preA = pdveA i it + γ 3 illit + rcvit + γ 4 lylit i + γ 3 pdvii + pdvri + γ 4 pdvli X ¡ ¢¡ A ¢ t A A A A A A A pdvypA = δ + γ ill + rcv + γ lyl + γ ill + rcv + γ lyl pre pre 1 2 3 4 i it it it it it it it it A A A = pdveA i + γ 1 γ 3 pdvii + pdvri + γ 2 γ 4 pdvli
1
APPENDIX TABLES Table A-1 – Sample versus Population Characteristics (percent) Sample n=1619 individuals
Sample n=1407 individuals (Min time on task)
2000 U.S. Census
18 23 21 17 14 7
16 22 21 19 15 8
% of 25+ pop 22 25 21 7 6 10
Income Less than $10,000 $10,000 to $15,000 $15,000 to $20,000 $20,000 to $25,000 $25,000 to $30,000 $30,000 to $40,000 $40,000 to $50,000 $50,000 to $60,000 $60,000 to $75,000 $75,000 to $100,000 $100,000 to $125,000 More than $125,000
5.7 6.1 4.9 6.1 6.6 7.4 8.6 13.3 11.1 11.1 10.4 4.2
5.3 6.1 5.0 6.3 6.5 16.2 13.3 11.0 11.3 10.5 4.1 4.3
% of hhlds 9.5 6.3 6.3 6.6 6.4 6.4 5.9 10.7 9.0 10.4 10.2 5.2
Female
0.51
0.53
0.51
Age 25 to 34 35 to 44 45 to 54 55 to 64 65 to 74 75 and older
64
Table A-2 Distribution of Program Characteristics within Illness Types (1619 individuals, 7520 choice sets, 15040 illness profiles, 22560 alternatives) Breast Prostate Lung Cancer Cancer Cancer # profiles Monthly cost Risk change Latency Illness years Lost life-years
Colon Cancer
Skin Heart Heart Stroke Cancer Attack Disease
Respir. Traffic Diabetes Alz. Disease Accident Disease
697
676
1357
1368
1353
1406
1423
1424
1337
1295
1357
1347
30.78 (30.09) -0.0033 (0.0016) 16.97 (10.95) 4.861 (3.481) 11.54 (11.4)
28.12 (26.09) -0.0034 (0.0017) 18.52 (11.2) 4.917 (3.853) 12.03 (11.5)
29.35 (28.37) -0.0034 (0.0017) 18.37 (11.57) 8.546 (8.295) 8.88 (9.71)
30.4 (28.7) -0.0034 (0.0017) 19.35 (11.46) 8.294 (7.681) 10.32 (9.75)
30.19 (28.81) -0.0035 (0.0017) 17.6 (11.68) 7.478 (7.322) 10.33 (10.79)
29.85 (29.62) -0.0035 (0.0017) 20.48 (12.54) 3.421 (6.649) 13.54 (11.26)
29.87 (28.63) -0.0034 (0.0017) 19.42 (11.94) 10.239 (8.84) 7.41 (8.42)
30.85 (29.43) -0.0034 (0.0017) 21.79 (12.67) 3.593 (6.429) 12 (10.07)
29.77 (29.41) -0.0034 (0.0017) 21.39 (12.18) 7.37 (6.529) 7.99 (7.81)
29.72 (27.92) -0.0034 (0.0017) 18.21 (12.32) 4.036 (7.596) 14.49 (12.51)
29.17 (28.07) -0.0033 (0.0016 18.23 (10.82) 6.798 (5.817) 13.44 (10.72)
29.84 (28.54) -0.0033 (0.0016) 22.63 (12.51) 6.805 (4.661) 8.8 (6.42)
0
0
0
0
0
0.52
0
0.51
0
0.51
0
0
0.40
0.36
0.22
0.36
0.30
0.08
0.11
0.07
0.21
0.07
0.85
0.84
0
0
0.37
0.41
0.30
0.21
0.63
0.24
0.41
0.23
0.15
0.16
0.60
0.64
0.39
0.23
0.40
0.19
0.26
0.19
0.38
0.19
0
0
Die suddenly Die sick Chronic condition Recover
65
Mortality and Morbidity Risk Reduction: An Empirical Life-cycle Model of Demand with Two Types of Age Effects 1
Reviewer Appendix - Survey Methods and Sample
We developed the initial version of the survey after an extensive review of the existing literature in April 2002. Over the course of nine months the survey went through four significant revisions. It was fielded to a Canadian sample of 1109 respondents in November 2002, and to a US sample of 2439 respondents in December 2002. Throughout the design process, it was necessary to work within the cognitive limits of respondents and the time constraints of the on-line survey format. Very early in the design process it became clear that respondents were able to make trade-offs across complex illness profiles, but only if these profiles were as concrete and complete as possible. Respondents wanted to know with specificity the symptoms, end result, timing and duration of an illness. We spent the early portion of the design phase determining which of many possible attributes of future illness individuals cared about. Since we anticipated using a conjoint approach, we sought to identify the ten to twelve most important attributes that were common to the top ten to twelve causes of death or chronic disease. We then searched for ways to consistently and clearly define these attributes, and to communicate them to respondents. This process proved to be a great challenge that required the field testing of many survey questions, graphics and formats. Because of the personal nature of many of the questions in the survey, we choose to evaluate prototypes of the instrument in one-on-one cognitive interviews, rather than in a focus group setting. A principal investigator conducted each cognitive interview. These interviews began with the respondent taking the online survey as they would in the respondent’s home, using a TV screen and keyboard. A PI remained present to answer and record questions of clarification and observe the respondent’s behavior and attitude while taking the survey. Once the respondent had completed the survey, a PI carefully debriefed the respondent by reviewing the survey modules and important questions, graphics or pre-designed response categories. Each interview lasted approximately an hour. We conducted over 36 cognitive interviews over the study period. We also pre-tested the penultimate version of the instrument on 142 respondents in our sample before administering it to the rest of the Canadian sample. We had a couple of weeks to look at the Canadian returns and make minor modifications to the survey instrument before launching the survey for the US sample. During the development of the instrument, we drew on the expertise of scholars in fields including the psychology of risk communication, health economics, environmental valuation and survey design. In particular, we benefited from a peer review panel that evaluated the 1
second of the four versions of the instrument. Each of these six advisors provided extensive verbal and written feedback on the survey instrument.1 The design of the final survey format was guided by several goals and constraints. One goal was to evaluate the individual’s ex ante demand for reducing the risk of a particular illness based on each individual’s current information set. To accomplish this, Module 1 contained questions that carefully documented the individual’s current knowledge of, and beliefs about, specific health risks. Yet, in contrast with many earlier studies, we deliberately chose not to provide respondents with augmenting information. A second goal was to identify sources of heterogeneity in individuals’ demands for risk reductions. This required not only that we define heterogeneous illness profiles, but also that we ensure respondents understood the metric in which each attribute was defined, and the range of values along which each attribute could vary. As part of this tutorial process, Module 2 sequentially introduced the attributes of the illness profile, its baseline risk, the risk-reducing intervention, and the payment vehicle that would achieve the risk reduction. After this preparation, Module 3 presented individuals with six choices sets, each containing two illness profiles and associated risk-reducing interventions. Module 4 contained debriefing questions. Meeting the time constraint of 30 minutes for survey completion posed a great challenge. We, and our technical advisory board members, had many more questions that would have been useful, but could not be fit into this timeframe. Ultimately, it was necessary to decide what information we would have to live without, and we chose to devote over forty percent of survey time to the tutorial module. The conjoint choice questions consumed about thirtyfive percent of the survey time. The introductory and debriefing questions consumed the remaining twenty-five percent of survey time.
1.1
Modules of the Survey
The survey is structured around four modules: 1) the introductory module, 2) the illness profile and risk tutorial, 3) the presentation of the choice sets, and 4) a debriefing and followup module. One example of the randomized survey instrument is presented as an Appendix. In the following subsections we refer to the form numbers that have been inserted in braces at the bottom center of each page. 1.1.1
Risk Beliefs and Attitudes
The survey opens with some questions that encourage respondents to think about the environmental and illness-specific threats that they face (Forms 2- 8-Private). Importantly, these questions introduce the choice set of risks that respondents will be asked to consider in 1
This panel included Victor Adamowicz, Richard Carson, Baruch Fischhoff, James Hammitt, Alan Krupnick and Kerry Smith. We are thankful for the invaluable experience, constructive criticism and advice that each of these people has shared with us.
2
the remainder of the survey. These early questions also document respondents’ experience with, and information on, each illness (Form 3-Private). We also introduce and document respondents’ knowledge of various states of morbidity (Form 4-Private). Next we directly solicit a rating score describing how “at risk” respondents feel they are of experiencing each of these illnesses over the course of their lifetime (Form 5-Private). In our empirical analysis, we allow each respondent’s answers to these questions to shift their marginal willingness-topay for a risk reduction. We interpret this ”at risk” variable as a subjective attribute of the risks that respondents will consider in the following choice exercises. The questions on Form 6-Private, are also relevant to our formal analysis. These questions explore the extent to which respondents feel they could mitigate or control illness-specific threats to their health through changes in their lifestyle and behaviour. We use respondents’ answers to these questions in our statistical analysis as additional subjective risk attributes. This form is followed by the questions of how hard or personally costly it would be to undertake these lifestyle or behavioural changes (Form 7-Private). These questions help us to distinguish between the respondent’s understanding of the opportunity to control risks and their own inclination or ability to do so. 1.1.2
Illness Profile Tutorial
Sequencing the elements of the illness profile was probably the most challenging aspect of survey design we faced. This module began by establishing the respondent’s intertemporal frame of reference. We reminded them of their current age and told them their expected age of death (Form 9-Private) based on their personal characteristics.2 We also informed the respondent that the rest of the survey would focus on health programs that would reduce their risk of getting sick and dying between now and their expected time of death. (We conditioned the presentation of information on the respondent’s age and gender throughout the survey as we discuss below.) Below we describe how we characterize and present respondents’ risk of getting sick. We adapted the risk-grid approach used by Krupnick et al. (2001). Both visually and numerically we represented a risk of 1 in 1,000 over the individual’s remaining years of life expectancy. In our example, on Form 6-Private this is 34 years. To further illustrate and make the risk personal, we then described a general mortality risk of 30 in 1,000 over forty years (Form 11-Private). (We further refine respondents’ understanding of risk on Forms 19and 2-Private, and within each choice set.) We then describe to respondents what we ultimately want to know from them, so they understand why the information on the ensuing pages is relevant. Specifically, we tell them 2
In our one-on-one cognitive interviews and pretesting, we found that the typical respondent overestimated his or her life expectancy by 5 to 8 years, compared to standard age-dependent actuarial tables. Individuals frequently referred to their longest living relatives when answering our longevity question (Form 44). To prevent scenario rejection, we added eight years to our calculation of each respondent’s life expectancy. This is a particularly important adaptation for rrespondents over the age of sixty.
3
we will want to know which of the following two illnesses they most want to avoid and that we will describe how each illness might affect him or her (Form 12-Private). They are told about two illnesses they might face, and at what age these illnesses might strike (Form 12Private). If they have already experienced one of the illnesses, they are asked to view the described onset as a recurrence. Next we define the measure of morbidity that we use to describe these two illnesses (Forms 14- and 15-Private). Adapting the types of pain and disability scales from several QALY indexes, we define for respondents what we mean by “moderate” and “severe” pain and disability. This provides respondents with a more concrete interpretation of these attributes and an ex ante understanding of the possible range of variation in each. We also introduce some types of treatments that are associated with morbidity. These include major surgery and minor surgery as well as the duration of hospitalization (measured in weeks) (Form 15-Private). We then describe for respondents the eventual outcome of each illness (Form 16-Private). For the two illnesses under consideration in each choice set, we note there are four possible outcomes: 1) full recovery, 2) sudden death, 3) morbidity for less than six years with no recovery, followed by death, 3) chronic morbidity for more than six years, followed by death. For each of the two illnesses under consideration we describe the conclusion of the profile in terms of the extent to which death is premature. We follow up this comparative information with a comprehension assessment to evaluate respondents’ understanding of the information. This completes the introduction of the elements of the illness profiles. Next we introduce the interventions that could reduce the risk of experiencing these profiles. The vast majority of interventions took the form of medically driven risk management programs that centered upon an annual diagnostic test (Form 17-Private). We chose this class of interventions because respondents viewed them as technically feasible and potentially effective. Respondents were familiar with comparable and pre-existing diagnostic tests such as mammograms, pap smears and prostrate exams, or the new C-reactive protein tests for heart disease.) Important from our perspective was the fact that this class of interventions could plausibly be applied to all of the illnesses upon which we focused. A second type of intervention (Form 18-Private) involved the installation of new safety equipment in the respondent’s vehicle to prevent the risk of injury in the event of an auto accident. The effectiveness of these interventions was described using a risk grid (Form 19-Private). All colored squares represented the baseline risk, from which reductions would take place as a result of the intervention program. Although not visible from the attached black-and-white copy, the graphic represents the risk reduction by blue squares and the remaining risk in red squares. Risk reductions are thus represented in three metrics: a numeric representation, a graphical representation and finally as a percentage of the baseline. We included the percentage reduction for two reasons. First, it allowed us to directly address a common reasoning error described in the risk literature in which individuals only focus on the relative size of risk reduction (Fetherstonhaugh, et al,1997; Baron, 1997). We took pains to point 4
out cases where overall risk reduction was in fact very small even if the percentage (or relative) risk reduction looked large. A second reason for expressing the reduction as a percentage is that it may be used to directly compare two illnesses with the same baseline risk. Therefore, it saves the respondent several cognitive operations that they would otherwise need to undertake. (Note the presence and use of the comprehension question about the size of the risk reduction (Form 20-Private).) The payment vehicle for each intervention was presented as a co-payment that would have to be paid by the respondent for as long as the diagnostic testing and medication was needed. For the sake of concreteness we asked the respondents to assume the payments would be needed for the remainder of his or her lifespan unless they actually experienced that illness. Costs were expressed in both monthly and annual terms. To ensure that respondents carefully considered their budget constraint, we included a “cheap talk” reminder as well as language to discourage overstating their willingness to pay (Form 22-Private). The survey reminded respondents, “In surveys like this one, people sometimes do not fully consider their future expenses. Please think about what you would have to give up to purchase one of these programs. If you choose a program with too high a price, you may not be able to afford the program when it is offered.” We then focused respondents’ attention on their option to choose neither program. In an effort to mitigate yea-saying, by dispelling the respondent’s assumption that they were “supposed” to choose one or the other program, we listed four plausible and legitimate reasons why a reasonable person might reject both programs, choosing instead the “neither” option (Form 22-Private). We sought to ensure that respondents clearly understood the intertemporal range of program benefits. The survey described (for two illnesses) the time of onset, time of death relative to their expected lifespan, the baseline risk and risk reduction (Form 23-Private). On the same page, the survey said, “We want to be clear about when the benefits of each program begin. For example, the benefits of Program A are that your risk of is reduced from in 1,000 to in 1,000, starting when you are years old and continuing for the rest of your life.” We also focused the respondent’s attention on the status quo option, if they choose neither program. Recall that we have already elicited the respondent’s beliefs about what illnesses he or she is most likely to experience over his or her lifetime in the absence of these risk reducing programs (Form 5-Private). The survey stated, “If you DO NOT choose Program A, your risk of will remain at X in 1,000 over this time period” (Form 23-Private). Prior to the choice questions, the survey stated, “If you choose neither program, remember that you could die early from a number of causes (of death), including the one described below” (Form 25-Private).
5
1.1.3
Choice Set Design
We presented each respondent with a sequence of five choice sets, each of which contained the option to choose neither program. Prior to presenting these choice sets, the survey states, “Given the costs, choose the program that reduces the illness you most want to avoid. Would you prefer Program A, Program B, or neither?” We wanted respondents to treat each choice set as independent of the others in order to minimize order effects that may be induced by cumulative expenditure effects or prior-purchase substitution effects. After the presentation of the first choice set, but before the second one, the survey states, “please evaluate each new pair of programs independently of the ones you saw earlier” (Form 29-Private). Following the same chronological sequence used in the tutorial, each choice set presents the attributes of two illness profiles: the illness name, the age of onset, medical treatments, duration and level of pain and disability and a description of the outcome of the illness. This is followed by a description of the cost and effectiveness of the risk reducing program (Form 29-Private). Both the age of onset and the final stage of each illness are determined by the respondent’s current age. Gender specific illnesses (e.g., breast and prostate cancer) are chosen to comport with the respondent’s gender. Based on the cognitive interviews, we took several steps to ensure that respondents accepted illness profiles as realistic. Whether a respondent accepted an illness profile as “realistic” depended upon both whether he or she could construct a plausible narrative and whether that narrative was consistent with the respondent’s prior beliefs about the illness in question. In cognitive interviews, a single unrealistic profile caused respondents to reject the entire choice set, and often caused respondents to disengage from the remainder of the survey, taking it much less seriously. The full inventory of restrictions we placed on the combinations, and ranges, of particular attributes are too numerous to review here. There were many disease specific restrictions we imposed to ensure that the general description of the illness comported with respondents’ general expectations. Examples of unacceptable illness profiles include cases in which the respondent was told that he or she would recover from Alzheimer’s disease or die suddenly at the onset of diabetes. There were also many attribute combinations that respondents viewed as unrealistic. Examples include profiles in which the respondent has major surgery, but spends no time in the hospital or he or she survives the onset of a major illness, but experiences not even moderate pain and disability. As a result, statistically optimal choice set designs were not possible. Rather, the attributes of each profile were drawn randomly subject to several restrictions. Some of these restrictions did enhance the statistical efficiency of random draws, such as a restriction on strict dominance across illness profile pairs. We summarize the results of this choice set design process in Table XX. The first row in this table presents the frequency with which each of the twelve illnesses appeared in the choice sets. In the subsequent rows, the table presents the mean levels of each of the risk, morbidity, and mortality attributes associated with that illness. While the mean levels of
6
the costs, baseline risk, and risk change are very comparable across all of the illnesses, the average levels of the other attributes vary greatly across illnesses. For example, heart attacks are associated with much shorter periods of pain, hospitalization, and death after a chronic condition than is lung cancer.3 For each illness profile, the choice set presents both individual and comparative risk reduction information. The individual baseline risk and risk reduction is presented as “Risk reduction from X0 in 1,000 to X1 in 1,000.” To ease the cognitive burden associated with comparing the risk reductions across the two programs, we do the following. First, we randomly choose a common baseline for each of the two illness profiles within a choice set. (While the baseline risks vary across choice sets, we hold them constant across illness profiles within each choice set.) Then we describe the randomly chosen risk reduction for each program in terms of its percent reduction from the baseline risk. Because the baseline risk is common to both illness profiles, these percentage reductions describe the relative effectiveness of the two risk reducing programs. The benefits of this approach are that it eliminates the need for the respondent to undertake two cognitive operations (e.g., subtraction and division) that would normally be required for a careful comparison the two programs. The only potential cost of this approach arises if the respondent rejects the conjecture that the hypothetical baseline risk for the two illness profiles would be the same. Not once did respondents raise this concern in cognitive interviews, while many said that the availability of the percentages facilitated their comparisons of the programs. If the respondent chose neither program, he or she was asked, “Which reasons best describe why you did not want to pay?” (E.g. Form 28-Private.) Based on our cognitive interviews, we identified six possible response categories.4 Five of these categories included reasons that would be considered valid economic grounds for rejecting the programs. Either the program’s price was too high, the respondent’s income is too low, there exist cheaper substitute actions for the program, they did not personally face the ”typical” baseline risk as described for each illness. One reason involved the respondent’s rejection of some critical portion of the scenario: that the interventions would not reduce the risks as described.5 3
Table 4 may also help the reader more fully understand the concept of a VSI and how it varies across particular illnesses. Notice that each illness differs in its distribution of “final outcomes.” Some illnesses predominantly result in sudden death, some in recovery, while others result in death only after a short period of illness or a chronic period of illness. (We conservatively define a chronic illness as one lasting six years or more.) For example, in our choice set the distribution of final outcomes for colon cancer is: zero percent die suddenly, 37 percent recover, 23 percent die within six years, and 40 percent after six years. The VSI for colon cancer will depend upon this distribution of fatal and non-fatal outcomes, as well as all the other attributes. In contrast, a traditional VSL for colon cancer would be based on fatal outcomes, and theoretically recover only the value of reducing the risk of mortality, not the value of reducing associated sick years. 4 The Canadian version contained a seventh reason, that the program should be covered by the respondent’s provincial health plan. 5 In the Canadian version, an additional scenario rejection answer was deemed to be the belief that program costs should be covered under the respondent’s health plan.
7
1.1.4
Debriefing Questions
While most of the debriefing questions focus on better characterizing the respondent’s status quo health profile, we also address several other issues. One question documents whether respondents invested more in their health relative to ten years ago (Form 45-Private). Several questions explore whether the respondent expects to experience one of the illnesses discussed within the next 20 years (Form 46-Private); if they do, whether they think they will be given adequate diagnosis and treatment (Form 48-Private); and until what age he or she will ultimately live (Form 49-Private).6 Finally, because of its importance as an attribute in the choice set, the survey also asked respondents directly if they considered whether they could actually afford to pay for these programs. All of these questions may play an important role in future extensions of our analysis. 6
In the Canadian version, another question asked respondents whether they had ever gone outside of their health plan to consume a list of possible medical services including diagnostic tests. (Other services included cosmetic surgery, immunization, flu shots, physical exams and major surgery.)
8
2 2.1
Reviewer Appendix - Additional or Expanded Formulas Elaboration of expected indirect utility terms
In the body of the paper, we do not spell out the general formulas for the present discounted values of expected utility, saving space by leaving them implicit in the terms such as P DV (ViAS ). Expected utility if the individual buys program A is: ¡ ¢ £ ¤ × P DV (ViAS ) + 1 − ΠAS (1) × P DV (ViAH ) E ViA S,H = ΠAS i i X X ⎡ ⎤ ∗ A∗ β0 δ t f (Yit∗ − cA∗ δ t (Yit∗ − cA∗ it ) + β 1 it )f (Yit − cit ) ⎢ ⎥ ⎢ ⎥ X X X ⎢ ⎥ t A t t A 2 A +α ill + α age ill + α age ill δ δ δ ⎢ ⎥ 10 11 it it 12 it it it ⎢ ⎥ ⎢ ⎥ AS X X X = Πi ⎢ ⎥ t t t A A 2 A ⎢ +α20 ⎥ rcv + α age rcv + α age rcv δ δ δ 21 it 22 it it it it ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ X X X ⎣ +α30 δ t lylA + α31 δ t ageit lylA + α32 δ t age2 lylA + εAS ⎦ it
it
it
it
i
i X X ¡ ¢h t t A A A AH + 1 − ΠAS f (Y − c ) δ + β (Y − c )f (Y − c ) δ + ε β i i 0 1 i i i i i i
Expected utility if the program is not purchased (i.e. ”no program”, N), with the expectation taken over uncertainty about whether the individual will suffer the illness, is: ¡ ¢ £ ¤ × P DV (ViNS ) + 1 − ΠNS (2) × P DV (ViNH ) E ViN S,H = ΠNS i i X X ⎡ ⎤ β0 δ t f (Yit∗ ) + β 1 δ t (Yit∗ )f (Yit∗ ) ⎢ ⎥ ⎢ ⎥ X X X ⎢ ⎥ δ t illitA + α11 δt ageit illitA + α12 δ t age2it illitA ⎢ +α10 ⎥ ⎢ ⎥ ⎥ NS ⎢ X X X = Πi ⎢ ⎥ t t t A A 2 A ⎢ +α20 ⎥ δ rcvit + α21 δ ageit rcvit + α22 δ ageit rcvit ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ X X X t t t ⎣ +α30 δ lylA + α31 δ ageit lylA + α32 δ age2 lylA + εNS ⎦ it
it
i X X ¡ ¢h t t NS NH + 1 − Πi δ + β 1 (Yi ) f (Yi ) δ + εi β 0 f (Yi )
9
it
it
i
2.2
Extensive formulas for expected utility differences
For a simplified variant of our model with β 1 = 0 and β 0 = β, it may be helpful to show in much more detail the calculations involved in constructing the estimating variables for the conditional logit models used in this study. We show these calculations below for a version of the model without age-at-health-state heterogeneity (i.e. no ageit shifters on the underlying undiscounted marginal (dis)utilities of adverse health states). The difference in expected present discounted utility between the program and noprogram cases will be: £ ¤ £ ¤ E ViA − E ViN (3) h X i X X X β δ t f (Yit∗ − cA∗ δ t illitA + α2 δ t rcvitA + α3 δ t lylitA + εAS = ΠAS i it ) + α1 i i h X ¡ ¢ + 1 − ΠAS δ t + εAH βf (Yi − cA i i ) i h X i X X X t t A t t NS ∗ A A NS −Πi β δ f (Yit ) + α1 δ illit + α2 δ rcvit + α3 δ lylit + εi i h X ¡ ¢ − 1 − ΠNS δ t + εNH βf (Yi ) i i Distributing terms, this expected utility difference can be written as: £ ¤ £ ¤ E ViA − E ViN h X i t ∗ A∗ = ΠAS β δ f (Y − c ) i it it h X i X X £ AS ¤ t A t t A A AS +ΠAS α + Π ill + α rcv + α lyl δ δ δ εi 1 2 3 i it it it i h X i + βf (Yi − cA δt i ) h X i ¡ ¢ £ AH ¤ AS A −Πi βf (Yi − ci ) δ t + 1 − ΠAS εi i i h X t ∗ −ΠNS f (Y ) β δ i it h X i X X £ NS ¤ t A t t NS A A −Πi εi α1 δ illit + α2 δ rcvit + α3 δ lylit − ΠNS i h X i − βf (Yi ) δt h X i ¡ ¢ £ NH ¤ S +ΠNS ) δ t − 1 − ΠN εi βf (Y i i i
(4)
In the process of simplifying this expression, there are four components involving error k are independent terms. We will define the compound error term as ε. If the error terms εN i and identically distributed according to an extreme value distribution, and if the εNk are i 10
similarly independent and identically distributed extreme value, then the resulting error term can be assumed to be logistic, so that a logit model is appropriate. £ AS ¤ ¡ ¢ £ AH ¤ £ NS ¤ ¡ ¢ £ NH¤ AS NS NS ε + 1 − Π ε − Π ε − 1 − Π εi (5) εi = ΠAS i i i i i i i
In equation (4), pairs of terms can be combined in three cases.First, the two term containing no probability term can be combined: h X i h X i t − βf (Y ) δ ) δt βf (Yi − cA i i £ ¤X t = β f (Yi − cA δ i ) − f (Yi ) Next, the two terms involving ΠAS and the income expression can be combined: i h X h i X i t ∗ A∗ AS A ΠAS β δ βf (Y f (Y − c ) − Π − c ) δt i i it it i i nhX i h X io t AS ∗ A∗ A = βΠi δ f (Yit − cit ) − f (Yi − ci ) δt
and the income expression can be collected: Finally, the two terms involving ΠNS i nh io X i h X t t ∗ ΠNS βf (Y − β δ ) δ f (Y ) i i it nh io X i hX t t NS ∗ = βΠi f (Yi ) δ f (Yit ) δ −
Taking advantage of these simplifications, and collecting terms, the expected indirect utility difference can be re-written as: £ ¤ £ ¤ (6) E ViA − E ViN X ¤ £ δt = β f (Yi − cA i ) − f (Yi ) nhX i h X io t ∗ A∗ A δ +βΠAS f (Y − c ) − f (Y − c ) δt i i it it i nhX i h X io t NS ∗ −βΠi δ f (Yit ) − f (Yi ) δt i X X ¡ ¢h X t A t t NS A A + ΠAS δ δ δ − Π ill + α rcv + α lyl α 1 2 3 i i it it it + εi
In this expression, the second and third lines to the right of the equals sign are present because of the different patterns of income and program costs implied by a choice of the program. It is probably appropriate to assume that there is zero income (meaning no consumption of other goods and services) after death and that program costs will not be paid if the individual is ill or dead. In what follows, we will also maintain the hypotheses that 11
(γ 1 , γ 2 , γ 3 , γ 4 ) = (1, 0, 0, 0). In words, usual income is sustained through illness by insurance, but not after death (there are no bequests), and program costs are only paid while alive and healthy.
2.3
Generalizing to the model with age-at-health-state effects
Restoring the age-at-health-state effects that are central to this paper, and also allowing for systematically varying marginal utilities of income by including all four β parameters, the expected indirect utility difference can be expressed as: £ ¤ £ ¤ E ViA − E ViN ¤X t £ δ = β 0 f (Yi − cA i ) − f (Yi ) ¤X t £ A +β 1 (Yi − cA δ i )f (Yi − ci ) − (Yi )f (Yi ) nhX i h X io t AS ∗ A∗ A δ f (Yit − cit ) − f (Yi − ci ) +β 0 Πi δt nhX i h X io t ∗ A∗ ∗ A∗ A A +β 1 ΠAS δ (Y − c )f (Y − c ) − (Y − c )f (Y − c ) δt i i i it it it it i i nhX i h X io t ∗ −β 0 ΠNS δ f (Y ) − f (Y ) δt i i it i h nhX X io t NS ∗ ∗ −β 1 Πi δt δ (Yit )f (Yit ) − (Yi )f (Yi ) X X X ⎡ α10 δ t illitA + α11 δ t ageit illitA + α12 δ t age2it illitA ⎢ X X X ¢⎢ ¡ AS t t A A N S ⎢ +α20 rcv + α age rcv + α δ δ δt age2it rcvitA 21 it 22 it it + Πi − Πi ⎢ X X X ⎣ +α30 δt lylitA + α31 δ t ageit lylitA + α32 δ t age2it lylitA
(7)
⎤
⎥ ⎥ ⎥ + εi ⎥ ⎦
The expected indirect utility difference can be simplified if we assume f (Y ) = Y. Under
12
this assumption, equation (7) can be simplified to: £ ¤ £ ¤ E ViA − E ViN ¤X t £ ¤X t £ A δ δ + β 1 (Yi − cA = β 0 −cA i i )(Yi − ci ) − (Yi )(Yi ) nhX ¡ io i h ¢ ¡ ¢X t δ t Yit∗ − cA∗ +β 0 ΠAS − Yi − cA δ i it i nhX ¡ i h ¢¡ ∗ ¢ ¡ ¢¡ ¢ X t io t ∗ A∗ A∗ A A +β 1 ΠAS δ − c − c − c − c Y Y − Y Y δ i i i it it it it i i nhX i h X io −β 0 ΠNS δ t Yit∗ − (Yi ) δt i nhX i h X io t ∗ ∗ NS −β 1 Πi δ Yit Yit − (Yi ) (Yi ) δt X X X ⎡ α10 δ t illitA + α11 δ t ageit illitA + α12 δ t age2it illitA ⎢ X X X ⎢ ¢ ¡ AS t t A A N S ⎢ +α20 rcv + α age rcv + α δ δ δt age2it rcvitA 21 it 22 it it + Πi − Πi ⎢ X X X ⎣ +α30 δt lylitA + α31 δ t ageit lylitA + α32 δ t age2it lylitA
(8)
⎤
⎥ ⎥ ⎥ + εi ⎥ ⎦
¡ ¢ Now define ∆ΠAS = ΠAS − ΠNS i i i £ ¤ £ ¤ (9) E ViA − E ViN £ A¤ X t = β 0 −ci δ nhX ¡ ¢i h¡ ¢ X t io t ∗ A∗ A δ +β 0 ΠAS − c − c Y − Y δ i i it it i nhX i h io X −β 0 ΠNS δ t Yit∗ − (Yi ) δt i ¤X t £ A +β 1 (Yi − cA )(Y − c ) − (Y )(Y ) δ i i i i i nhX ¡ i h¡ ¢ ¡ ¢ ¢¡ ¢ X t io t ∗ A∗ ∗ A∗ A A δ +β 1 ΠAS − c − c − c − c Y Y − Y Y δ i i i it it it it i i nhX i h X io −β 1 ΠNS δt δ t Yit∗ Yit∗ − (Yi ) (Yi ) i X X X +α10 ∆ΠAS δt illitA + α11 ∆ΠAS δ t ageit illitA + α12 ∆ΠAS δt age2it illitA i i i X X X t t A AS A AS +α20 ∆ΠAS rcv + α ∆Π age rcv + α ∆Π δ δ δ t age2it rcvitA 21 it 22 i it i it i X X X t t A AS A AS δ δ δ t age2it lylitA + εi +α30 ∆ΠAS lyl + α ∆Π age lyl + α ∆Π 31 it 32 i it i it i
As described in the body of the paper, solving this model requires that we isolate an expression for the common certain paymenr cA it that makes this utility difference exactly zero. 13
The previous equation therefore needs to be re-written to isolate this variable. Maintaining generality, let the present discounted profiles for income (pdvy) and program costs (pdvp) be abbreviated as follows. = pdvcA i pdveA = i pdviA = i pdvriA = pdvliA =
X X X X X
δt
agepdvcA i =
δ t preA it
agepdveA i =
δ t illitA δ t rcvitA δ t lylitA
X
X
X
δ t ageit
δ t ageit preA it
δ t ageit illitA X agepdvriA = δ t ageit rcvitA X agepdvliA = δ t ageit lylitA agepdviA i =
X age2pdvcA = δt age2it i X age2pdveA = δ t age2it preA i it X t A 2 A age2pdvii = δ ageit illit X age2pdvriA = δ t age2it rcvitA X age2pdvliA = δ t age2it lylitA
Notice that the following relationships hold, since the indicator variables for each health status are mutually exclusive and exhaustive: A A A = pdveA pdvcA i i + pdvii + pdvri + pdvli A A A agepdvcA = agepdveA i i + agepdvii + agepdvri + agepdvli A A A age2pdvcA = age2pdveA i i + age2pdvii + age2pdvri + age2pdvli
To accommodate the different time profiles of income and program costs over the individual’s remaining lifespan, we must also define two additional terms X ¡ ¢ A A A A A A A δ t preA pdvyiA = it + γ 1 illit + rcvit + γ 2 lylit = pdvei + γ 1 pdvii + pdvri + γ 2 pdvli X ¡ ¢ A A A A A A A pdvpA = δ t preA i it + γ 3 illit + rcvit + γ 4 lylit = pdvei + γ 3 pdvii + pdvri + γ 4 pdvli
To isolate cA it , we now focus separately on each of the terms involving income and the different β coefficients in equation (8). We re-write each of the terms, then collect them to highlight the resulting quadratic form in cA it . The terms in β 0 can be re-written in a couple
14
of steps as follows: £ ¤X t δ β 0 −cA i nhX ¡ ¢i h¡ ¢ X t io t AS ∗ A∗ A +β 0 Πi δ Yit − cit − Yi − ci δ nhX i h X io −β 0 ΠNS δt δ t Yit∗ − (Yi ) i £ A¤ £ A¤ £ ¤ A AS A AS A = −cA i β 0 pdvci − −ci β 0 Πi pdvci + −ci β 0 Πi pdvpi © ª © ª +Yi β 0 ΠAS pdvyiA − pdvcA − Yi β 0 ΠNS pdvyiA − pdvcA i i i i ¤ £¡ ¢ ¤ £ AS A 1 − ΠAS pdvcA = −cA i β0 i i + Πi pdvpi © ª +β 0 Yi ∆ΠAS pdvyiA − pdvcA i i
A 2 The term involving β 1 can be written to isolate terms in cA i and (ci ) if we define the following additional abbrevations for new terms to be introduced. Again, the fact that our indicator variables for each distinct health status are mutually exclusive and exhaustive allows some important simplifications: X ¡ ¢ A A A 2 2 A A 2 A δt preA = pdveA pdvyyiA = it + γ 1 illit + rcvit + γ 2 lylit i + γ 1 pdvii + pdvri + γ 2 pdvli X ¡ ¢ A A A 2 2 A A 2 A pdvppA = δt preA = pdveA i it + γ 3 illit + rcvit + γ 4 lylit i + γ 3 pdvii + pdvri + γ 4 pdvli X ¡ ¢¡ A ¢ A A A pdvypA = δt preA preit + γ 3 illitA + rcvitA + γ 4 lylitA i it + γ 1 illit + rcvit + γ 2 lylit A A A = pdveA i + γ 1 γ 3 pdvii + pdvri + γ 2 γ 4 pdvli
15
These terms allow us to streamline our notation via the following steps: ¤X t £ A δ β 1 (Yi − cA i )(Yi − ci ) − (Yi )(Yi ) nhX ¡ ¢ ¡ ¢i h¡ ¢¡ ¢ X t io t AS ∗ A∗ ∗ A∗ A A +β 1 Πi δ Yit − cit Yit − cit − Yi − ci Yi − ci δ nhX i h X io −β 1 ΠNS δt δ t Yit∗ Yit∗ − (Yi ) (Yi ) i A A 2 A 2 A 2 A = [−cA i ]2β 1 (Yi )pdvci + [−ci ] β 1 pdvci + β 1 (Yi ) pdvci − β 1 (Yi ) pdvci 2 A AS A 2 AS A +β 1 ΠAS i (Yi ) pdvyyi + [−ci ]2β 1 Πi (Yi ) pdvypi + [−ci ] β 1 Πi pdvppi 2 A AS A 2 AS A −β 1 ΠAS i (Yi ) pdvci − [−ci ]2β 1 Πi (Yi ) pdvci − [−ci ] β 1 Πi pdvci S −β 1 ΠNS (Yi )2 pdvyyiA + β 1 ΠN (Yi )2 pdvcA i i i A A AS AS A = [−ci ]2β 1 (Yi )pdvci + [−ci ]2β 1 Πi (Yi ) pdvypA i − [−ci ]2β 1 Πi (Yi ) pdvci 2 A 2 AS A 2 AS A +[−cA i ] β 1 pdvci + [−ci ] β 1 Πi pdvppi − [−ci ] β 1 Πi pdvci 2 A NS +β 1 ΠAS (Yi )2 pdvyyiA i (Yi ) pdvyyi − β 1 Πi 2 A NS −β 1 ΠAS (Yi )2 pdvcA i (Yi ) pdvci + β 1 Πi i £¡ ¢ ¤ A AS A AS = [−ci ]β 1 2Yi 1 − Πi pdvci + Πi pdvypA i £¡ ¢ ¤ 2 AS A +[−cA 1 − ΠAS pdvcA i ] β1 i i + Πi pdvppi ¡ ¢ A A − pdvc +β 1 Yi2 ∆ΠAS pdvyy i i i
With a little more incidental factoring of the shared β coefficients, these final simplifications match the constructed variables outlined in the body of the paper, to be used in estimation of the underlying marginal utility parameters originating in the undiscounted per-period indirect utility function for arbitrary health states.
16
Reviewer Table A-3 Sensitivity to Scope, Key Choice Attributes Variable Monthly Cost of Program Risk Reduction Sick-Years
Estimate -0.007491 (-9.48)*** 57.64 (5.77)*** 0.0879 (3.85)***
Lost Life-Years
0.01138 (7.13)***
Alternatives Log-likelihood
22560 -11706.11
17
Reviewer Table A-4 Fitted VSIs for Illness Profiles used in Estimating Sample ($ million)
Model 1 No Age Effects
Model 2 Age Now Effects (linear)
Model 3 Age Now Effects (quadratic)
Model 4 Both Age Effects (linear)
Model 5 Both Age Effects (quadratic)
Model 5’ Both Age Effects (Min time on task)
Model 6 Parsimonious Variant
Sample mean VSI
1.63
0.69
2.75
7.09
2.54
2.62
4.85
Sample 5th % Sample 25th % Sample 50th % Sample 75th % Sample 95th %
0a 0.41 1.08 2.16 5.09
0 0 0 0 3.03
0 0.2 1.22 3.26 10.44
0a 0.81 2.04 3.77 10.45
0a 0.17 1.25 2.59 5.46
0a 0.13 1.60 3.29 7.46
0 0.61 1.97 3.63 9.78
22560
22560
22560
22560
22560
19881
22560
Descriptive Statistic
Alternatives used
a
In the choice scenarios presented to respondents, there was no opportunity for any individual to express a negative willingness to pay for a program. At most, they could choose the other alternative, or “Neither Program.” As a consequence, for these descriptive statistics, we interpret negative fitted point values of the VSI for a particular program as zero values, both in computing the marginal mean and in describing the percentiles of the marginal distribution. Means are sensitive to large positive outliers
18
Reviewer Table A-5 Sensitivity of Parameter Estimates to Alternative Discount Rate Assumptions (Sensitivity Analysis uses 19788 alternative sample) 3% discount rate ___________________________ Parameter and description of variable(s)
β 00 *10-5
(linear net income term)
β 01 *10-7 ( agei 0 β3
No Linear Quadratic Age Age Age Effects Effects Effects ___________________________ 3.83 (8.60)***
6.11 (4.20)***
7.26 (4.54)***
-3.65 (-1.64)
-5.44 (2.15)**
-0.1350 (4.40)***
-0.1514 (4.08)***
-0.1468 (3.95)***
-7.460 (6.01)***
0.0323 (0.00)
-69.03 (-1.40)
0.6526 (4.22)***
-2.101 (-1.52)
interaction)
*10-9 (DMU(Y) term)
5% discount rate ___________________________ No Linear Quadratic Age Age Age Effects Effects Effects ____________________________ 4.62 (8.31)***
8.29 (4.37)***
10.09 (4.81)***
-5.61 (1.98)**
-8.34 (2.58)***
-0.2130 (4.62)***
-0.195 (4.17)***
-0.1917 (4.09)***
-9.6248 (5.41)***
2.096 (-0.17)
-85.16 (-1.35)
1.314 (5.04)***
-3.6357 (1.68)*
7% discount rate ___________________________ No Linear Quadratic Age Age Age Effects Effects Effects ___________________________ 5.336 (7.97)***
9.658 (4.14)***
12.93 (4.94)***
-6.296 (1.84)*
-11.19 (2.83)***
-0.2670 (4.78)***
-0.2398 (4.22)***
-0.2378 (4.17)***
-11.582 (4.79)***
3.653 (-0.23)
-105.6 (-1.33)
2.2562 (5.43)***
-6.095 (1.86)*
Sick years
δ100 ∆Π iAS pdvii δ101∆Π iAS pdvii × agei 0 δ102 ∆Π iAS pdvii × agei20 δ110 ∆Π iAS agepdvii δ111∆Π iAS agepdvii × agei 0 δ12 ∆Π iAS age2 pdvii
0.0018 (-0.13) -0.5568 (2.76)***
-0.0931 (2.22)**
-0.0243 (-1.01)
3.600 (1.72)*
-1.136 (3.55)***
5.352 (1.77)*
-1.9604 (4.04)***
8.043 (1.88)*
0.0352 (-1.13)
0.1025 (1.86)*
0.2536 (2.66)***
-0.0428 (1.86)*
-0.0842 (2.23)**
-0.167 (2.73)***
19
Table A-2, continued: Recovered years
δ 200 ∆Π iAS pdvri
-6.423 (2.86)***
δ 201∆Π iAS pdvri × agei 0
47.78 (2.59)***
-41.11 (-0.47)
0.2011 (-0.67)
-3.570 (-1.16)
δ 202 ∆Π iAS pdvri × agei20
-9.329 (2.70)***
65.91 (2.49)**
-82.44 (-0.69)
0.5672 (-1.11)
-7.320 (-1.47)
-0.0048 (-0.16)
δ 210 ∆Π iAS agepdvri
-0.9283 (2.31)**
δ 211∆Π iAS agepdvri × agei 0 δ 22 ∆Π iAS age2 pdvri
-12.69 (2.51)**
86.57 (2.37)**
-143.8 -0.92
1.139 (-1.36)
-13.75 (1.76)* -0.0807 (-0.87)
-0.0238 (-0.45)
4.635 (-1.12)
-1.523 (2.34)**
-2.345 (2.32)**
9.132 (-1.43)
16.53 (1.74)*
0.0592 (-0.85)
0.1453 (-1.17)
0.3319 (-1.55)
-0.062 (-1.28)
-0.1318 (-1.59)
-0.2643 (1.92)*
Lost life-years
δ 300 ∆Π iAS pdvli
-6.929 (7.70)***
δ 301∆Π iAS pdvli × agei 0
11.06 (-1.29)
16.38 (-0.34)
0.6312 (5.13)***
-1.796 (-1.25)
δ 302 ∆Π iAS pdvli × agei20
17.66 (-1.45)
44.20 (-0.68)
1.289 (5.80)***
-2.321 (-1.02)
-0.0177 (-1.49)
δ 310 ∆Π iAS agepdvli
-0.6921 (3.88)***
δ 311∆Π iAS agepdvli × agei 0 δ 32 ∆Π iAS age2 pdvli
Alternatives Log L
-9.454 (6.70)***
19788 -7175.195
19788 -7146.39
-11.93 (5.75)***
24.70 (-1.5)
81.99 (-0.97)
2.255 (5.96)***
-2.609 (-0.75) -0.1224 (3.13)***
-0.0525 (2.41)**
1.010 (-0.47)
-1.337 (4.56)***
-2.265 (4.89)***
0.7263 (-0.22)
0.0092 0
0.0563 (1.87)*
0.1225 (2.31)**
0.2461 (2.68)***
-0.0315 (-1.35)
-0.0579 (-1.51)
-0.1064 (1.71)*
19788 -7141.535
20
19788 -7186.375
19788 -7149.029
19788 -7142.45
19788 -7196.076
19788 -7154.093
19788 -7143.988
Reviewer Table A-6 Sensitivity of Fitted VSIs in Estimating Sample to Alternative Discount Rate Assumptions
7% discount rate 3% discount rate 5% discount rate ___________________________ ___________________________ ___________________________
Descriptive Statistic
No Linear Quadratic No Linear Quadratic No Linear Quadratic Age Age Age Age Age Age Age Age Age Effects Effects Effects Effects Effects Effects Effects Effects Effects ___________________________ ____________________________ ___________________________
Sample mean VSI ($ million)
4.17
4.09
4.65
2.2
3.65
8.92
1.96
2.69
2.11
Sample 5th % Sample 25th % Sample 50th % Sample 75th % Sample 95th %
0.13 1.21 2.4 4.16 11.74
0.03 1.26 2.62 4.28 8.51
0a 1.45 2.78 4.13 8.25
0 0.61 1.54 2.88 6.76
0 0.96 2.01 3.29 6.65
0 1.09 2.11 3.11 6.16
0 0.36 1.07 2.29 6.4
0 0.71 1.59 2.68 5.46
0 0.82 1.65 2.46 4.72
a
In the choice scenarios presented to respondents, there was no opportunity for any individual to express a negative willingness to pay for a program. At most, they could choose the other alternative, or “Neither Program.” As a consequence, for these descriptive statistics, we interpret negative fitted point values of the VSI for a particular program as zero values, both in computing the marginal mean and in describing the percentiles of the marginal distribution.
21
Expanded Reviewer Version of Text Table 3 VSI for Four Classes of Sudden Death Scenarios (US $ million), based on 22560 alternatives Model 1 Model 2 Model 3 No “Age Now” “Age Now” Age Effects Effects Effects (linear) (quadratic) Age Now
Age at Latency Death
50% ( 5%,95%)
50% ( 5%,95%)
50% ( 5%,95%)
1. Simulation: Sudden death this year
25 35 45 55 65
25 35 45 55 65
0 0 0 0 0
6.36 (4.82, 8.09) 6.18 (4.68, 7.86) 5.86 (4.44, 7.45) 5.46 (4.14, 6.95) 4.88 (3.70, 6.21)
2.03 (-1.69, 7.50) -3.33 (-11.26, 2.75) -7.94 (-18.51, -0.30) -11.79 (-24.23, -2.51) -14.47 (-28.93, -3.94)
15.10 ( 7.58, 38.64) 11.29 ( 7.48, 21.35) 7.85 ( 5.52, 11.33) 4.68 ( 2.68, 7.18) 1.97 (-0.15, 4.16)
2. Simulation: Sudden death in 10 years
25 35 45 55 65
35 45 55 65 75
10 10 10 10 10
3.77 (2.89, 4.88) 3.59 (2.75, 4.64) 3.27 (2.51, 4.23) 2.87 (2.20, 3.72) 2.29 (1.76, 2.97)
-0.25 (-4.45, 2.65) -3.36 (-9.77, 0.42) -5.70 (-12.43, -1.01) -7.33 (-13.81, -2.01) -7.68 (-14.63, -2.5)
8.73 (4.35, 24.57) 6.62 (4.31, 12.3) 4.35 (2.98, 6.76) 2.48 (1.39, 3.87) 0.91 (-0.16, 1.91)
3. Simulation: Sudden death @ fixed age (70)
25 35 45 55 65
70 70 70 70 70
45 35 25 15 5
0.42 (0.32, 0.53) 0.70 (0.53, 0.90) 1.15 (0.87, 1.46) 1.99 (1.51, 2.54) 3.43 (2.60, 4.37)
-0.54 (-1.86, -0.01) -1.25 (-3.06, -0.19) -2.51 (-5.39, -0.55) -5.29 (-10.47, -1.41) -10.75 (-21.37, -3.04)
0.99 ( 0.50, 2.54) 1.29 ( 0.85, 2.44) 1.54 ( 1.08, 2.22) 1.71 ( 0.97, 2.62) 1.38 (-0.11, 2.93)
4. Simulation: Sudden death varying latency
25 25 25 25 25
30 40 50 60 70
5 15 25 35 45
4.91 (3.72, 6.25) 2.89 (2.19, 3.67) 1.65 (1.25, 2.09) 0.88 (0.67, 1.13) 0.42 (0.32, 0.53)
0.64 (-3.35, 4.72) -0.79 (-4.44, 1.84) -1.07 (-4.16, 0.60) -0.88 (-3.14, 0.14) -0.54 (-1.86, -0.01)
11.66 ( 5.86, 29.85) 6.86 ( 3.45, 17.56) 3.91 ( 1.96, 10.02) 2.10 ( 1.06, 5.39) 0.99 ( 0.50, 2.54)
22
Reviewer Version: Table 3, continued VSI for Four Classes of Sudden Death Scenarios (US $ million), based on 22560 alternatives Model 4 Model 5 Model 6 Both Age Both Age Parsimonious Effects Effects Variant (linear) (quadratic) Age Now
Age at Latency Death
50% ( 5%,95%)
50% ( 5%,95%)
50% ( 5%,95%)
1. Simulation: Sudden death this year
25 35 45 55 65
25 35 45 55 65
0 0 0 0 0
4.30 ( 1.31, 8.80) 3.68 ( 1.20, 7.02) 2.69 ( 0.56, 5.17) 1.61 (-0.43, 3.69) 0.24 (-2.10, 2.46)
0.35 (-7.53, 8.06) 2.48 (-2.53, 7.53) 3.54 (-0.01, 7.36) 3.98 ( 0.66, 7.48) 3.38 (-0.14, 7.38)
-0.46 (-5.84, 5.49) 1.89 (-0.84, 5.34) 3.25 ( 0.95, 6.12) 3.88 ( 1.02, 7.28) 3.47 ( 0.21, 7.61)
2. Simulation: Sudden death in 10 years
25 35 45 55 65
35 45 55 65 75
10 10 10 10 10
4.88 (3.33, 7.77) 4.43 (3.29, 6.36) 3.66 (2.87, 4.80) 2.78 (2.11, 3.60) 1.70 (0.80, 2.57)
3.48 (1.64, 6.01) 3.87 (2.80, 5.56) 3.45 (2.61, 4.58) 2.60 (1.80, 3.51) 1.31 (0.22, 2.19)
4.03 (2.00, 6.68) 4.23 (3.07, 5.88) 3.58 (2.73, 4.70) 2.50 (1.76, 3.47) 1.01 (0.00, 2.02)
3. Simulation: Sudden death @ fixed age (70)
25 35 45 55 65
70 70 70 70 70
45 35 25 15 5
1.31 ( 0.93, 1.97) 1.86 ( 1.37, 2.56) 2.28 ( 1.78, 2.95) 2.54 ( 1.97, 3.17) 1.40 ( 0.01, 2.70)
1.93 ( 1.20, 3.04) 2.26 ( 1.61, 3.17) 2.21 ( 1.69, 2.94) 2.12 ( 1.31, 2.92) 2.12 ( 0.60, 3.70)
1.86 ( 1.24, 3.00) 2.14 ( 1.56, 3.04) 2.05 ( 1.51, 2.72) 1.87 ( 1.13, 2.71) 2.02 ( 0.54, 3.63)
4. Simulation: Sudden death varying latency
25 25 25 25 25
30 40 50 60 70
5 15 25 35 45
4.78 ( 2.79, 8.16) 4.58 ( 3.31, 7.11) 3.55 ( 2.61, 5.34) 2.37 ( 1.70, 3.54) 1.31 ( 0.93, 1.97)
2.33 (-1.39, 6.38) 4.08 ( 2.52, 6.34) 4.04 ( 2.73, 6.12) 3.13 ( 2.13, 4.78) 1.93 ( 1.20, 3.04)
2.47 (-0.79, 6.43) 4.61 ( 2.94, 7.52) 4.33 ( 3.03, 6.88) 3.18 ( 2.17, 5.09) 1.86 ( 1.24, 3.00)
23
