1. Introduction
1.1. Forks in the road in health technology valuation
A fundamental policy question in the economic evaluation of health technologies and policies is whether to evaluate them narrowly or broadly (or equivalently, whether to evaluate them from a health payer’s or societal perspective) (Neumann et al., Reference Neumann, Sanders, Russell, Siegel and Ganiats2016). Narrow evaluation focuses on health-centric impacts (i.e. on mortality, morbidity, and health resource utilization), while broad evaluation in principle incorporates all impacts on the economy (e.g. worker productivity, household financial security, macroeconomic impacts) and society (e.g. sociopolitical stability). Assuming broad valuation, there are further forks in the road regarding which type of analysis to perform: cost-effectiveness- or cost–utility analysis (CEA or CUA) or one of their variants such as extended or distributional CUA (see, e.g. Neumann et al., Reference Neumann, Sanders, Russell, Siegel and Ganiats2016; Verguet et al., Reference Verguet2016; Cookson, Reference Cookson2023), cost-benefit analysis (CBA, see, e.g. Robinson et al., Reference Robinson, Hammitt, Jamison and Walker2019), multicriteria decision-analysis (MCDA, see, e.g. Ilyas et al., Reference Khan, Pintelon and Martin2022), or analyses centered on social welfare functions (SWFs, see, e.g. Adler, Reference Adler2019), equivalent incomes (EI, see, e.g. Fleurbaey et al., Reference Fleurbaey, Luchini, Muller and Schokkaert2013), wellbeing-adjusted life years (WELLBYs, see, e.g. Frijters, Reference Frijters2021), poverty-free life expectancy (PFLE, see, e.g. Riumallo-Herl et al., Reference Riumallo-Herl, Canning and Salomon2018), and others.
The fundamental normative premise underlying this article is the principle of the sovereignty of an individual’s idealized preferences in resolving trade-offs among various goods such as health and consumption within that individual’s own life (“intrapersonal trade-offs”). Call this principle “preferencism.” Idealized preferences are simply those an individual would have under ideal circumstances for making value judgments, such as having sufficient empirical information and time to deliberate, and being sufficiently free from cognitive and motivational impairments, such as biases in belief formation and addictions. Preferencism is subjectivist as opposed to objectivist with respect to value in that it holds that values ultimately flow from the subject (i.e. from the individual whose life is affected by policy) rather than from the object (e.g. from the intrinsic nature of the goods such as health or consumption conferred by policy). It agrees with Hamlet that there is nothing good or bad but thinking makes it so. And it rejects the hard paternalist notion that policymakers understand individuals’ well-being better than those individuals themselves would under ideal circumstances.
Preferencist value frameworks satisfy the preference-based version of the Pareto principle (PP), which combines Pareto Indifference (if all individuals are indifferent between two policies, then a value framework should value these policies equally) and Strong Pareto (if all but one individual weakly prefers a first policy to a second, and the remaining individual strongly prefers the first to the second, then a value framework should value the first policy strictly more than the second) (Adler, Reference Adler2019).
Preferencism traces a path through the abovementioned forks in the road. It supports broad valuation since individuals under ideal circumstances value not just the health-related impacts of health technologies and policies but their socio-economic impacts as well. It rejects types of analyses like CEA, CUA, and MCDA that shift at least partial and often significant weight away from individual preferences toward policymaker preferences (see, e.g. Brouwer et al., Reference Brouwer, Culyer, Job, van Exel and Rutten2008). It also rejects types of analyses like CUA that try to accommodate individual preferences but impose drastic simplifying assumptions on them (Bleichrodt & Quiggin, Reference Bleichrodt and Quiggin1999; Hammitt, Reference Hammitt2013), thus limiting their sensitivity to more general preferences not satisfying those assumptions.
Preferencism supports valuing policy impacts on an individual using that individual’s willingness-to-pay (WTP) for those impacts since such WTP reflects that individual’s preferences. Of the value frameworks enumerated earlier, CBA, SWFs, and EI all rely on using WTP to resolve intrapersonal trade-offs, thus satisfying preferencism. These frameworks differ with respect to how to resolve distributional issues, that is, how to trade off the well-being or preference satisfaction of different individuals (“interpersonal trade-offs”). These differences can be understood in terms of whether or how to weigh different individuals’ WTP (see discussion in Fleurbaey et al., Reference Fleurbaey, Luchini, Muller and Schokkaert2013). CBA resolves interpersonal trade-offs by adding up unweighted WTP across individuals, which risks being disproportionately sensitive to the interests of the wealthy who have higher ability-to-pay (ATP) for policy benefits. SWFs and EI address interpersonal trade-offs by differentially weighing different individuals’ WTP: utilitarian SWFs weigh individuals’ WTP by their respective marginal utilities of income, prioritarian SWFs weigh them by the product of these individuals’ marginal utilities of income and the marginal social value of these individuals’ utilities, and EI weighs them by the product of the partial derivative of an individual’s equivalent income with respect to their actual income and the marginal social value of these individuals’ equivalent incomes (see Fleurbaey et al., Reference Fleurbaey, Luchini, Muller and Schokkaert2013 for details).
In the next section, I discuss a promising microeconomic and utility-theoretic approach to the broad valuation and quantification of WTP, the health-augmented lifecycle model (HALM). The lifecycle model is a model of lifetime utility maximization subject to constraint and is the workhorse model used by economists to explain or predict economic choices or behaviors over an individual’s lifetime including those involving consumption, work, leisure, and savings. (Browning & Crossley, Reference Browning and Crossley2001). The HALM is a health-augmented version of the lifecycle model in that it incorporates lifetime mortality and morbidity prospects into the utility function and budget constraint, allowing the derivation of expressions for individual WTP for improvements in these prospects. The HALM is preferencist, satisfies the PP, and usable unweighted in CBA and weighted in SWF and EI analyses. I calibrate the HALM to the US setting and demonstrate its use in the valuation of a pediatric vaccine.
2. The health-augmented lifecycle model
2.1. Specification of lifetime utility and the budget constraint
The HALM is a model of lifetime utility maximization subject to a budget constraint, augmented to include health-related measures of mortality and morbidity prospects. Following the specification of Murphy and Topel (Reference Murphy and Topel2006), the individual solves:
$$ \underset{c(a),l(a)}{\max }U={\sum}_{a=0}^{99}\frac{s(a)\times q(a)\times u\left(c(a),l(a)\right)}{{\left(1+\rho \right)}^a}, $$
subject to the budget constraint:
$$ {\displaystyle \begin{array}{c}{\sum}_{a=0}^{99}\frac{s(a)\times o(a)+}{{\left(1+r\right)}^a}=A+{\sum}_{a=0}^{99}\frac{s(a)\times i(a)\;}{{\left(1+r\right)}^a}.\end{array}} $$
Lifetime utility
$ U $
is the sum across all ages
$ a $
of life of period utility
$ s(a)\times q(a)\times u\left(c(a),l(a)\right) $
discounted at the rate of time preference
$ \rho $
. I set the maximum lifespan at 100 years, so
$ a=0,\dots, 99 $
. Age-specific consumption
$ c(a) $
and non-market time
$ l(a) $
are the standard goods of microeconomic theory. Non-market time consists of unpaid work and leisure time.
$ u\left(\right) $
is the standard utility function of microeconomic theory. I shall refer to it as the “economic utility function” and to
$ u\left(c(a),l(a)\right) $
as “economic utility.” The age-specific survival probability
$ s(a) $
is the probability, conditional on being born, that an individual reaches age
$ a $
(known in the context of demographic life tables as the survival function). Health utility
$ q(a) $
is a scalar measure of health-related quality of life (HRQoL) taking on values from zero to one, representing death and perfect health respectively (known in CUA as the quality-adjusted life year (QALY) weight).
$ s(a) $
and
$ q(a) $
represent mortality and morbidity prospects respectively. Period utility has multiplicative form, implying that survival, health utility, and economic utility are natural complements: the higher the level of one, the larger the marginal impact of the others on period utility.
The budget constraint requires the expected present discounted value (EPDV) of lifetime resource outflows
$ o(a) $
to equal initial wealth
$ A $
and the EPDV of lifetime resource inflows
$ i(a) $
. To facilitate empirical quantification of these outflows and inflows over the lifecycle, I follow the National Transfer Accounts (NTA) project in disaggregating the components of such flows (see UNPDDESA, 2013, equation 2.1 and what follows). Outflows
$ o(a) $
equal the sum of consumption (excluding health and education)
$ c(a) $
, consumption of health and education
$ {c}^{he}(a) $
, and transfer outflows
$ {\tau}^o(a) $
. Inflows
$ i(a) $
are the sum of labor income
$ y(a) $
, transfer inflows
$ {\tau}^i(a) $
, and net asset-related inflows
$ {k}^i(a) $
. Labor income
$ y(a)=w(a)\left(T-l(a)\right) $
is the product of the hourly wage
$ w(a) $
and paid work time
$ T-l(a) $
, where
$ T $
is the time endowment. Transfer outflows
$ {\tau}^o(a) $
equal private transfer outflows and public transfer outflows, while transfer inflows
$ {\tau}^i(a) $
equal private transfer inflows and public transfer inflows. Private transfers include intra- and inter-household transfers. Public transfers include those related to education, health, pensions, social protection other than pensions, and other in-kind and cash transfers. Taxes do not explicitly appear in the budget constraint, but they determine the magnitude of public transfer outflows. Net asset-related inflows
$ {k}^i(a) $
are the sum of private asset income and public asset income. Private asset income comprises private capital income and property income, while public asset income comprises public operating surpluses and public property income. The difference between inflows and outflows is savings
$ v(a)=i(a)-o(a) $
, which is also the sum of private savings and public savings.
I take the economic utility function
$ u\left(\right) $
to be a constant relative risk aversion (CRRA) function of a composite commodity
$ z $
, which is in turn a Cobb-Douglas function of consumption and non-market time:
$$ {\displaystyle \begin{array}{c}u\left(z(a)\right)=\frac{z{(a)}^{1-\frac{1}{\sigma }}-{z}_0^{1-\frac{1}{\sigma }}}{1-\frac{1}{\sigma }},\end{array}} $$
$$ {\displaystyle \begin{array}{c}\sigma \ge 0,\sigma \ne 1,\end{array}} $$
$$ {\displaystyle \begin{array}{c}{z}_0>0,\end{array}} $$
$$ {\displaystyle \begin{array}{c}z(a)=c{(a)}^{\alpha }l{(a)}^{1-\alpha },\end{array}} $$
$$ {\displaystyle \begin{array}{c}0<\alpha <1.\end{array}} $$
The CRRA function, also called the isoelastic or power function, has two parameters, the elasticity of intertemporal substitution
$ \sigma $
(whose reciprocal
$ 1/\sigma $
is the coefficient of relative risk aversion and the income elasticity of marginal utility) and subsistence composite consumption
$ {z}_0 $
. The advantage of the CRRA form is that it incorporates the elasticity of intertemporal substitution into the economic utility function in a simple way. This elasticity represents a person’s tolerance for volatility/uncertainty. As
$ \sigma \to +\infty $
, tolerance increases, and as
$ \sigma \to 0 $
, tolerance decreases. We will find that WTP for mortality and morbidity risk reduction is sensitive to this tolerance.
Subsistence composite commodity consumption
$ {z}_0 $
is the level of composite consumption at which individuals would be indifferent between life and death. All economic models that value mortality risk reductions must specify the utility of death and we adopt the common approach of assuming that this utility is constant and that some positive level of consumption is necessary to make life worth living. The economic utility function above implies that
$ u\left({z}_0\right)=0 $
, or that the economic utility of death is normalized to zero. The Cobb–Douglas function is homogeneous of degree one and has parameter
$ \alpha $
reflecting relative preferences for consumption and non-market time.
The budget constraint reflects the assumption of perfect capital markets, including perfect credit markets for saving and borrowing, and perfect annuity markets for insuring consumption against longevity risks.
2.2. Solution and value formulas
The individual takes as given
$ s(a),q(a),w(a),A,T,{c}^{he}(a),{\tau}^o(a),{\tau}^i(a),{k}^i(a) $
and maximizes
$ U $
with respect to
$ c(a),l(a) $
subject to the budget constraint. (See the Appendix for the Lagrangian and first-order conditions [FOCs].)
2.3. Value of a statistical life year and the value of a statistical health utility
Given a health technology applied to the individual at birth and whose lifetime survival and health utility impacts are
$ \delta s(a),\delta q(a) $
for all
$ a\ge 0 $
, the individual’s WTP for this technology can be approximated, using the envelope theorem, by (see the Appendix for derivations):
$$ {\displaystyle \begin{array}{c}\mathrm{W}\mathrm{T}\mathrm{P}=\sum \limits_{a=0}^{99}\frac{\mathrm{VSLY}(a)\times \delta s(a)+s(a)\times \mathrm{V}\mathrm{S}\mathrm{H}\mathrm{U}(a)\times \delta q(a)}{(1+r)^a},\end{array}} $$
where:
$$ {\displaystyle \begin{array}{c}\mathrm{VSLY}(a)\equiv {y}^f+{c}^f\Phi, \end{array}} $$
$$ {\displaystyle \begin{array}{c}\mathrm{VSHU}(a)\equiv \left(\frac{1}{q}\right)\times {c}^f\left(\Phi +1\right).\end{array}} $$
Full consumption
$ {c}^f $
is the sum of the value of consumption and of non-market time, where each hour of non-market time is valued at the hourly wage:
$$ {\displaystyle \begin{array}{c}{c}^f=c+ wl.\end{array}} $$
Full income
$ {y}^f $
is the sum of net earned and unearned income
$ {y}^n $
and the value of non-market time:
$$ {\displaystyle \begin{array}{c}{y}^f={y}^n+ wl.\end{array}} $$
$$ {\displaystyle \begin{array}{c}{y}^n=y+{\tau}^i+{k}^i-{c}^{he}-{\tau}^o.\end{array}} $$
$ \Phi $
is consumer surplus per unit of
$ z $
or per dollar of
$ {c}^f $
, or equivalently, the monetized value of the excess of the average utility of
$ z $
over its marginal utility, and is given by:
$$ {\displaystyle \begin{array}{c}\Phi \equiv \frac{\frac{u}{z}-{u}_z}{u_z}=\frac{\frac{u}{z}}{u_z}-1=\frac{1-\sigma {\left(\frac{z_0}{z(a)}\right)}^{1-\frac{1}{\sigma }}}{\sigma -1}.\end{array}} $$
Thus, WTP for a technology depends on its impacts on survival
$ \delta s(a) $
and on health utility
$ \delta q(a) $
, as well as on the value of unit improvements in survival probability (summarized by the age-specific Value of a Statistical Life Year or
$ \mathrm{VSLY}(a) $
) and the value of unit improvements in health utility (summarized by the age-specific Value of a Statistical Health Utility or
$ \mathrm{VSHU}(a) $
).
$ \mathrm{VSLY}(a) $
and
$ \mathrm{VSHU}(a) $
are the fundamental value formulas produced by the HALM. They show how the value of health is affected by a broad range of economic variables related to consumption, non-market time, and earned and unearned income. Since these economic quantities vary with age over the lifecycle, so does the value of health.
These value formulas have two broad elements. The
$ {y}^f $
term reflects the resources (in terms of income and time) that are made available by health and that are transformable into the goods (consumption and non-market time) from which economic utility is directly derived. This term can therefore be interpreted as reflecting the instrumental value of health. The terms involving
$ {c}^f $
and
$ \Phi $
reflect the value of health in virtue of its being a direct component of period utility (see Equation (1)) and so can be interpreted as reflecting the intrinsic value of health. This intrinsic value reflects the value of
$ {c}^f $
given the natural complementarity between health and
$ {c}^f $
, so that the value of health is higher the higher is the value of
$ {c}^f $
. This intrinsic value multiplies
$ {c}^f $
by the consumer surplus term
$ \Phi $
to reflect the fact that the monetary value of the utility derived from full consumption generally exceeds the price of full consumption itself.
The
$ \Phi $
term in
$ \mathrm{VSHU}(a) $
has a “+1” while the corresponding term in
$ \mathrm{VSLY}(a) $
does not. This reflects the fact that improvements in survival impose a cost on the budget constraint that improvements in health utility do not: a living person must consume, so being alive has resource costs equal to the value of full consumption. Such cost must be offset from the value of improvements in survival but not from the value of improvements in health utility.
The fact that
$ \mathrm{VSLY}(a) $
contains a
$ {y}^f $
term while
$ \mathrm{VSHU}(a) $
does not is an artifact of my specification of the budget constraint, which gives a role to the survival probability in affecting lifetime resources (being alive can both relax the budget constraint by facilitating income and tighten it by requiring consumption) but not to health utility. A more general budget constraint, which could be pursued in future work, could allow for health utility to affect the budget constraint through positive impacts (e.g. higher earnings and lower health expenditures) and negative impacts (e.g. lower transfer inflows). Such budget impacts would introduce a resource-related term into
$ \mathrm{VSHU}(a) $
.
The
$ \mathrm{VSHU}(a) $
term has a
$ 1/q $
term while the
$ \mathrm{VSLY}(a) $
term does not. This is because the calculation of the intrinsic value of health requires computing the partial derivative of period utility
$ s(a)\times q(a)\times u\left(c(a),l(a)\right) $
with respect to health, and monetizing this period utility gain by dividing by the marginal utility of composite commodity consumption
$ q(a)\times {u}_z $
(see Appendix for details). In the case of improvements in survival, this partial derivative is
$ q(a)\times u\left(c(a),l(a)\right) $
, so that health utility cancels out when dividing by marginal utility. However, in the case of improvements in health utility, this partial derivative is
$ s(a)\times u\left(c(a),l(a)\right) $
so that dividing by marginal utility does not cancel out health utility in the denominator. Thus, holding fixed the economic aspects of life, the value of improvement in survival does not depend on baseline health utility, while the value of improvement in health utility does.
Equation (8) for
$ \mathrm{WTP} $
discounts health gains
$ \delta s(a),\delta q(a) $
at the interest rate
$ r $
. It is an unresolved and controversial question in health economic evaluation whether to discount such health gains, and if so, whether to do so at the same rate
$ r $
with which other monetary values are discounted (see, e.g. the discussion in Robinson et al., Reference Robinson, Hammitt and O’Keefe2019). I do not make any contribution to this debate. I have adopted the mathematically simplest assumption of applying the rate of time preference
$ \rho $
to all of period utility including its health and consumption elements, which results in a uniform discount rate for health and monetary values. Such uniform discounting conforms to widespread practice and influential reference cases (see, e.g. Neumann et al., Reference Neumann, Sanders, Russell, Siegel and Ganiats2016; iDSI, 2023).
Following Murphy and Topel (Reference Murphy and Topel2006), I assume the survival probability
$ s(a) $
is a function of the mortality hazard
$ \lambda (j) $
for all ages from birth to age
$ a $
:
$$ {\displaystyle \begin{array}{c}s(a)=\exp \left[-\sum \limits_{j=0}^a\lambda (j)\right].\end{array}} $$
The Value of a Statistical Life at age
$ \overline{a} $
, denoted
$ VSL\left(\overline{a}\right) $
, is the WTP for a small reduction in the mortality hazard
$ \lambda \left(\overline{a}\right) $
, or equivalently, the marginal rate of substitution between
$ \lambda \left(\overline{a}\right) $
and
$ A $
. For a small enough reduction in the hazard, the envelope theorem provides the following approximation to VSL (see Appendix for details):
$$ {\displaystyle \begin{array}{c}\mathrm{VSL}\left(\overline{a}\right)=\sum \limits_{a=\overline{a}}^{99}\frac{s(a)\times \mathrm{VSLY}(a)}{{\left(1+r\right)}^{a-\overline{a}}}.\end{array}} $$
2.4. Calibration
The fundamental value formulas of the HALM are the age-specific expressions for VSLY and VSHU. Computing these formulas therefore requires expressions for
$ {y}^f,{c}^f,\Phi $
. Consumer surplus
$ \Phi $
, in turn, requires estimates of
$ \sigma $
and
$ {z}_0 $
. In what follows, I convert all monetary variables into 2023 US dollars (USD).
2.4.1.
Model-based versus data-based values for
$ c(a),l(a) $
The value formulas for VSLY and VSHU ultimately depend on the values of
$ c(a),l(a) $
. A first way to derive
$ c(a),l(a) $
is to solve the FOCs for the optimal values for these variables, which will be functions of the exogenous variables and parameters of the model. Given values for these exogenous variables and parameters, we obtain values for optimal consumption and non-market time implied by the theory. A second and alternative way is to estimate these from data and to assume these data have been generated by individuals behaving according to the model. For realism, to facilitate widespread use, and to generate value formulas that reflect real-world patterns of consumption, income, and time use, I follow the second approach.
2.4.2.
Consumption
$ c $
and earned and unearned income
$ {y}^n $
Equations (11) and (12) show full consumption
$ {c}^f $
and full consumption
$ {y}^f $
to be functions of consumption
$ c $
and earned and unearned income
$ {y}^n, $
respectively. I estimate
$ c $
and
$ {y}^n $
using variables from the National Transfer Accounts Project database (NTA, 2023), which uses microdata to compute how various consumption-, income-, and transfer-related quantities vary by age over the lifecycle. Table A1 shows how I map
$ c $
and the components of
$ {y}^n $
onto NTA variables, as well as the definitions of the NTA variables I rely on. Since NTA values stop at age 90, I extrapolate values to all older ages by assuming that labor income is zero, and that all other values are set to their age 90 values. I convert all NTA data, which is given in 2011 international dollars, to 2023 USD by multiplying by 1.35 (US BLS, 2023a)). Table A2 shows the values (in 2023 USD) of the NTA variables I use. The age-specific values of consumption
$ c $
and earned and unearned income
$ {y}^n $
are given in Table A6.
2.4.3.
Hourly wages
$ w $
and non-market time
$ l $
Both full consumption and full income require the value of non-market time
$ w\times l $
. For hourly wages
$ w $
, the US Bureau of Labor Statistics (Table 20 in US BLS, 2023b) provides hourly earnings by age (column 1 in Table A3), but these are limited to workers paid hourly rates who may not be representative of workers as a whole. I thus compute the ratio of the hourly earnings of a particular age group to the hourly earnings across all age groups. For example, this ratio equals 12.06/17.02 for those aged 16–19 (column 2 in Table A3). I then estimate the hourly earnings for this age group across all workers (including those not paid hourly rates) as the product of this ratio and the hourly earnings across all workers, which I take as $33.74 in 2023 USD (Table B-3 in US BLS, 2023c). Thus, for example, for those aged 16–19, the hourly earnings across all workers equals (12.06/17.02) × 33.74 (column 3 in Table A3). Hourly wage data are unavailable for ages 0–15 so I backfill age 16 values to these earlier ages.
I obtain age-specific non-market time from the 2022 results of the American Time Use Survey (US BLS, 2023d, summarized in Table A4). I take unpaid work to consist of the following ATUS activity categories: household activities, purchasing goods and services, caring for and helping household members, caring for and helping non-household members, organizational, civic, and religious activities, telephone calls, mail, and e-mail. I take leisure to consist of the ATUS category “leisure and sports.” Thus, age-specific non-market time per year is 365.25 times the sum of the columns in Table A4. Time use data are unavailable for those aged 0–14, so I backfill age 15 values of non-market time to these earlier ages.
The age-specific values of non-market time (combining the value of hourly earnings and non-market time)
$ w\times l $
, full consumption
$ {c}^f $
, and full income
$ {y}^f $
are given in Table A6.
2.4.4. Health utility
The expression for
$ VSHU(a) $
depends on health utility in the general population
$ q(a) $
. I obtain these from Szende et al. (Reference Szende, Janssen and Cabases2014) and report them in Table A5. Health utility values are unavailable for those aged 0–17 so I backfill age 18 values to these earlier ages.
2.4.5. Consumer surplus
The last component of
$ \mathrm{VSLY} $
(a) and
$ \mathrm{VSHU}(a) $
is consumer surplus
$ \Phi $
, which depends on both
$ \sigma $
and
$ {z}_0 $
. I simplify by approximating
$ \frac{z_0}{z(a)} $
by
$ \frac{c_0}{c(a)} $
where
$ {c}_0 $
is subsistence consumption of goods and services so that:
$$ {\displaystyle \begin{array}{c}\Phi (a)=\frac{1-\sigma {\left(\frac{z_0}{z(a)}\right)}^{1-\frac{1}{\sigma }}}{\sigma -1}\approx \frac{1-\sigma {\left(\frac{c_0}{c(a)}\right)}^{1-\frac{1}{\sigma }}}{\sigma -1}.\end{array}} $$
I have already quantified
$ c(a) $
above, so the remaining parameters in (17) are
$ {c}_0 $
and
$ \sigma $
.
I consider two possible values for
$ {c}_0 $
: half the annual extreme poverty rate in the US, and half that of the country with the lowest annual extreme poverty rate in the world. I estimate these annual extreme poverty rates based on Allen’s (Reference Allen2017) estimate of a daily extreme poverty rate in the US and Zimbabwe of $4.28 and $1.86 respectively in 2011 international dollars. Half the annual extreme poverty rate in the US and Zimbabwe are therefore 0.5 × 365.25 × 4.28=781.64 and.5 × 365.25 × 1.86=339.68 respectively in 2011 international dollars. Converting this into 2023 USD using the CPI (US BLS, 2023a), given parity between international and US dollars, gives us the following value for
$ {c}_0 $
: 1057.62 and 459.61 based on the US and Zimbabwe rates respectively.
I estimate
$ \sigma $
by assuming the VSL of a middle-aged American is $10 million (which is conservative relative to the $11.4 million estimate for the year 2020 from the US Department of Health and Human Services (2021)) when evaluated at the age equal to half of US life expectancy at birth and backing out the value of
$ \sigma $
that makes this equality hold. Combining (9), (16), and (17):
$$ {\displaystyle \begin{array}{c}\mathrm{VSL}\left(\overline{a}\right)=\sum \limits_{a=\overline{a}}^{99}\frac{s(a)\times \left({y}^f(a)+{c}^f(a)\times \frac{1-\sigma {\left(\frac{c_0}{c(a)}\right)}^{1-\frac{1}{\sigma }}}{\sigma -1}\right)}{{\left(1+r\right)}^{a-\overline{a}}}=10M.\end{array}} $$
Given that US life expectancy is about 76 (Arias et al., Reference Arias, Tejada-Vera, Kochanek and Ahmad2022), I choose
$ \overline{a}=38 $
. I assume
$ r=\rho =0.03 $
. I derive the survival probabilities
$ s(a) $
using 2020 US lifetables from the National Vital Statistics Report (US CDC, 2022). Thus, the only remaining unknown in Equation (18)) is
$ \sigma $
. Backing out the value for
$ \sigma $
, I get
$ \sigma =2.06 $
when
$ {c}_0=1057.62 $
and
$ \sigma =2.13 $
when
$ {c}_0=459.61 $
.
(These
$ \sigma $
estimates suggest an estimated coefficient of relative risk aversion of
$ 1/\sigma <1/2 $
. There are puzzles regarding such an estimated coefficient that I do not attempt to resolve. On the one hand and reassuringly, my estimates conform to theoretical results suggesting this coefficient should be smaller than the income elasticity of VSL (Kaplow, Reference Kaplow2005) and empirical results suggesting this income elasticity is between 0.5 and 0.7 in the US. (Viscusi & Masterman, Reference Viscusi and Masterman2017). On the other hand, and problematically, my estimates also exemplify an empirical puzzle whereby VSL-based estimates of the coefficient of relative risk aversion are significantly smaller than those derived from other economic behaviors and contexts such as financial and labor markets, where the coefficient of relative risk aversion tends to be around 1 (Chetty, Reference Chetty2006; Gandelman & Hernandez-Murillo, Reference Gandelman and Hernandez-Murillo2015) or higher (e.g. between 2 and 10, Kaplow, Reference Kaplow2005.)
This completes the calibration of VSLY and VSHU. Table A6 reports values for survival
$ s $
,
$ {\Phi}^{\mathrm{US}} $
and
$ {\Phi}^Z $
which are the values for
$ \Phi $
corresponding to US and Zimbabwean utility parameter values, and the resulting values for VSLY and VSHU. Figure 1 represents the following values graphically:
$ {y}^f,{c}^f, wl,\mathrm{VSLY},\mathrm{VSHU},{\mathrm{WTP}}_{\mathrm{QALY}} $
. The left and right panels are based on the US and Zimbabwean values for
$ {c}_0 $
respectively. Note that VSLY and VSHU are very similar across panels, suggesting their robustness with respect to the choice of
$ {c}_0 $
. The jaggedness of the graphs reflects the jaggedness of the underlying data and can be smoothed away if necessary.
Figure 1. Health-augmented lifecycle model results. Value of a statistical disability year (black); Value of a statistical life year (dashed black); Full income (green); Full consumption (red); Monetized value of nonmarket time (blue).
Observe that
$ {y}^f,{c}^f, wl,\mathrm{VSLY},\mathrm{VSHU} $
share the same general inverse U-shape as a function of age, peaking at age 55. These provide evidence that the economic determinants of the value of health (e.g.
$ {c}^f $
and
$ {y}^f $
) and therefore the value of health itself (i.e. VSLY, VSHU), are generally not age-invariant. Figure 1 also shows that VSHU is everywhere higher than VSLY. The jump upward in
$ {y}^f,{c}^f,\mathrm{VSLY},\mathrm{VSHU} $
at age 90 is an artifact of how I have filled in missing NTA values in this age group, and I shall improve this feature of this analysis in future work.
2.5. Comparison with CUA
Equation (1) shows what assumptions suffice for lifetime utility
$ U $
to be multiplicatively separable in lifetime QALYs, given by
$ {\sum}_{a=0}^{99}\frac{s(a)\times q(a)}{{\left(1+\rho \right)}^a} $
, and economic utility, given by
$ u\left(c(a),l(a)\right) $
, as assumed by CUA: lifetime utility must be additively separable in period utility; period utility must be multiplicatively separable in survival probability, health utility, and economic utility; and
$ c(a) $
and
$ l(a) $
must be constant throughout the lifetime. While the first and second of these are standard in the literature, the third, the age-invariance of consumption and non-market time, is implausible in light of the evidence in Figure 1, suggesting the empirical falsity of the assumptions imposed by CUA on preferences. In Figure 1, I have also for comparison with my VSLY and VSHU estimates, graphed
$ {\mathrm{WTP}}_{\mathrm{QALY}}=\mathrm{539,083} $
, which is my estimate of CUA’s central measure of the value, the WTP per QALY, given a VSL of $10 million and given my estimate of the discounted quality-adjusted life expectancy (QALE) of a 38-year-old of 18.55 (using the
$ s $
values in Table A6, renormalized so they equal 1 for a 38-year-old; the
$ q $
values in Table A5; and a 3 % discount rate).
2.6. Application
To show how to use the VSLY and VSHU value formulas to compute WTP for a vaccine, I take an example from Sevilla et al. (Reference Sevilla2022). That paper performed a prospective evaluation of introducing a pediatric 13-valent Pneumococcal Conjugate Vaccination (PCV13) program into Egypt’s national immunization program and vaccinating 100 successive birth cohorts over the period from 2016 to 2115. Egypt is a PCV-naïve country, so the paper used an incidence rate projection model to estimate the impact of PCV13 on the incidence of various serotypes or groups of serotypes of pneumococcal disease. It used a multiple-cohort static Markov model of Streptococcus Pneumoniae to estimate PCV13’s impact (relative to no vaccination) on lifetime survival probabilities and health utility conditional for each of the 100 birth cohorts. These impacts are what are denoted in Equation (8) as
$ \delta s(a) $
and
$ \delta q(a) $
. That paper’s calculated impacts for the first birth cohort are summarized in their Figure 6 (Sevilla et al., Reference Sevilla2022), which, through visual inspection, I approximate graphically in Figure 2 and numerically in the final two columns of Table A6. Equation (8) shows how to combine
$ \delta s(a) $
and
$ \delta q(a) $
with
$ \mathrm{VSLY}(a),\mathrm{VSHU}(a) $
,
$ s(a) $
, and
$ r $
to obtain WTP for the vaccine. Sevilla et al. (Reference Sevilla2022) calculate Egyptian values for
$ \mathrm{VSLY}(a) $
and
$ \mathrm{VSHU}(a) $
, which they combine with estimates of
$ \delta s(a) $
and
$ \delta q(a) $
to derive Egyptian WTP for PCV13. I combine their estimates of
$ \delta s(a) $
and
$ \delta q(a) $
with my estimates of
$ \mathrm{VSLY}(a) $
and
$ \mathrm{VSHU}(a) $
for the US to estimate US WTP for these survival and health utility gains. Using the US and Zimbabwe values for the extreme poverty rate, the US WTP for these gains are $13,481 and $13,508 respectively.
Figure 2. Mortality and morbidity impacts of pediatric vaccination program.
(The reason the US WTP for the vaccine is relatively invariant with respect to the value of
$ {c}_0 $
is that the estimate of
$ \sigma $
adjusts in response to maintain a VSL of $10M. The change in
$ {c}_0 $
and the offsetting change in
$ \sigma $
do not fully offset each other in
$ \mathrm{VSLY}(a) $
and
$ \mathrm{VSHU}(a) $
, so WTP does vary with the value of
$ {c}_0 $
. But they offset sufficiently that the variation is slight.)
2.7. Generalization to other countries
Applying the HALM to other countries requires country-specific estimates of
$ c(a) $
and
$ {y}^n(a) $
(which can be obtained from NTA (2023)),
$ w(a) $
(which can be obtained from labor force data), and
$ l(a) $
(which can be obtained from time use surveys). Under the assumption of identical global preferences (discussed further below), the values of
$ {c}_0 $
and
$ \sigma $
derived from the Zimbabwean extreme poverty rate can be used for any country in the world. (Using these values for
$ {c}_0 $
and
$ \sigma $
for countries other than the US does not imply extrapolating the $10 million US VSL to that other country, since the VSL in that other country, as Equations (9) and (16) show, will depend not just on the values for
$ {c}_0 $
and
$ \sigma $
but also on its own levels of full consumption and full income.) An even simpler way to apply the HALM to another country is to simply take my computed VSLY and VSHU values and scale them by the ratio of per capita GDP in that other country to that of the US. Applying the HALM also requires estimates of
$ s(a),q(a),\delta s(a),\delta q(a) $
, but these are required or would be generated by any model of health technology impact.
2.8. Conservative simplification
VSLY and VSHU are positive functions of consumer surplus
$ \Phi $
, which is itself typically positive. Also, the utility function parameters
$ {z}_0 $
and
$ \sigma $
only affect VSLY and VSHU through
$ \Phi $
. These along with Equations (9) and (10) suggest an approximation that is both conservative and avoids the challenge of having to estimate utility function parameters: approximate VSLY and VSHU by
$ {y}^f $
and
$ {c}^f/q $
respectively.
3. Conclusion
The HALM is preferencist and therefore satisfies the PP. It incorporates health into economic evaluation by augmenting a workhorse of microeconomic and macroeconomic theory, the lifecycle model, which is perhaps the most foundational and widely used utility-theoretic framework for the economic evaluation of policies in general, health-related or not. This makes it an apt framework for preferencist economic evaluation involving health.
The HALM also provides a flexible preference-based framework for policy evaluation. By “flexible” I mean that it imposes relatively fewer empirically false restrictions on preferences than other preference-based approaches like CUA. (Another approach that incorporates features of lifecycle models and QALY-centric CUA is taken by Cookson et al. (Reference Cookson, Skarda, Cotton‐Barratt, Adler, Asaria and Ord2021), which proposes a measure of lifetime utility that is additive in lifetime QALYs. However, it exemplifies the problem that although it is convenient (and congenial to CUA) to specify lifetime utility as separable in lifetime QALYs and economic utility, such separability requires imposing implausible restrictions on preferences or economic behaviors. As I discussed above, multiplicative separability requires age-invariant consumption and non-market time. Cookson et al. (Reference Cookson, Skarda, Cotton‐Barratt, Adler, Asaria and Ord2021) achieve additive separability with the assumption that health and consumption are perfect substitutes/non-complementary. This is implausible given that, intuitively, the marginal utility of leisure time, books, movies, and so on depends on whether one is depressed, has good vision, has physical mobility, and conversely the marginal utility of vision and mobility depends on whether one has access to books and movies and leisure time, as they themselves acknowledge using similar examples.) The HALM’s relative flexibility allows it to reflect true preferences more accurately, which promotes its ability to satisfy the PP.
While I have focused on a benchmark specification of the HALM, other versions are possible. One important area for future work is to drop the assumption of perfect capital markets and to allow for budget constraints allowing for borrowing constraints, and imperfect insurance and annuity markets. Another area worth examining is how to extend the HALM to accommodate dimensions of well-being beyond health and economics, for example, by incorporating hedonic, eudaimonic, or social well-being.
The HALM shows that the value of health reflects not just its impact on earned income, but also on unearned income and the value of non-market time, which in turn consists of the value of unpaid work and leisure. This demonstrates the limitations of the human capital approach that values health solely in terms of its impact on (typically paid) work, ignoring leisure (and typically, unpaid work).
The HALM expresses the value of health as functions of traditional economic quantities like consumption, paid and unpaid work, leisure, health expenditures, and public and private transfers. This may give it an advantage relative to a recent approach to broad valuation centered on the WELLBY (Frijters, Reference Frijters2021), which evaluates policies through their joint impact on self-reported overall life satisfaction (OLS) as quantified over a 0 to 10 scale and on longevity. Cognitive limitations likely prevent individuals from being able to translate policy impacts on the above economic quantities into ultimate impacts on OLS, so it is useful to be able to value health directly in terms of those economic quantities.
There is some controversy regarding whether the value of health gains, especially of mortality risk reductions, should net out such economic costs such as future consumption, future health services consumed, and future pension costs (see, e.g. Basu, Reference Basu, Sanders, Russell, Siegel and Ganiats2016). Equations (9), (11)–(13) shed light on this issue and show that the value of mortality risk reduction depends on the value of savings
$ v(a) $
, which is indeed a function of consumption of goods and services, consumption of health services, transfers including pensions, along with all other savings determinants.
An issue of concern in the literature is how to address preference heterogeneity. We can distinguish two questions. The first question is intrapersonal and wholly empirical: what is the functional dependence of WTP on preferences, and how therefore does WTP vary with preferences? The second question is interpersonal and irreducibly normative: how aggregate the WTP of individuals with different preferences? There are at least three competing answers to this second question. CBA simply adds up WTP across individuals with different preferences, while SWF and EI analysis can account for these differences through differential weighing of WTP (for SWF-based approaches to preference heterogeneity, see Adler (Reference Adler2019), and for the EI-based approach, see Fleurbaey et al. (Reference Fleurbaey, Luchini, Muller and Schokkaert2013)). Both questions are essential, but they are separate questions. The HALM addresses only the first question and should not be relied on to supplant the consideration of interpersonal aggregation, distribution, and fairness required by the second question. But regardless of the answer to the second question, we need expressions for WTP as functions of preference parameters – these, after all, are what are to be aggregated given some answer to the second question--and the HALM provides such expressions.
The HALM generates expressions for WTP that can be used in either CBA, utilitarian or prioritarian SWFs, or EI, which differ from each other in whether and how to weigh the WTP of the worse off relative to the better off, or whether and how to weigh the WTP of individuals with different preferences. Thus, the HALM can be used within any of these three frameworks. However, the HALM can make an even further contribution to the implementation of SWF-based approaches. Utilitarian- and prioritarian SWFs respectively weigh individuals’ WTP wholly or partially in proportion to their marginal utility of income. The HALM facilitates the construction of such weights, and therefore the operationalization of SWF approaches: the Lagrange multiplier serves as an estimate of that marginal utility that is directly comparable among individuals with identical preferences and that can be adjusted using the methods described in Adler (Reference Adler2019) to render them comparable among individuals with different preferences.
Competing interest
JS is employed as a health economist at Data for Decisions, LLC, and as a Research Associate at the Harvard T.H. Chan School of Public Health. Through these employers or personal consulting fees, he has received financial support from vaccine- and medical device manufacturers including Pfizer, GSK, Merck, Sanofi, Bavarian-Nordic, Johnson & Johnson, Janssen, and Edwards Life Sciences. Through these employers or personal consulting fees, he has also received financial support from the Bill and Melinda Gates Foundation, the World Health Organization, and Open Philanthropy.
A. Appendix
A.1. HALM mathematical details
Denoting the Lagrangian by
$ L $
, the individual solves:
$$ {\displaystyle \begin{array}{c}\underset{c(a),l(a),\mu }{\max }L={\sum}_{a=0}^{99}\frac{s(a)\times q(a)\times u\left(c(a),l(a)\right)}{{\left(1+\rho \right)}^a}+\mu \left[A+{\sum}_{a=0}^{99}\frac{s(a)\times v(a)\;}{{\left(1+r\right)}^a}\right].\end{array}} $$
The FOCs are:
$$ {\displaystyle \begin{array}{c}\frac{\partial L}{\partial c(a)}=0\to \mu =q\times {u}_z\times {z}_c\times \left(\frac{1+r}{1+\rho}\right),\end{array}} $$
$$ {\displaystyle \begin{array}{c}\frac{\partial L}{\partial l(a)}=0\to \mu \times w=q\times {u}_z\times {z}_l\times \left(\frac{1+r}{1+\rho}\right),\end{array}} $$
$$ {\displaystyle \begin{array}{c}\frac{\partial L}{\partial \mu }=0\to A+{\sum}_{a=0}^{99}\frac{s(a)\times v(a)\;}{{\left(1+r\right)}^a}=0,\end{array}} $$
where
$ {u}_z,{z}_c,\mathrm{and}\;{z}_l $
are the partial derivatives of
$ u $
with respect to composite commodity consumption and the partial derivatives of composite commodity consumption with respect to consumption and nonmarket time, respectively.
Given a health technology applied to the individual at birth and whose lifetime survival and health utility impacts are
$ \delta s(a),\delta q(a) $
for all
$ a\ge 0 $
, the individual’s WTP for this technology can be approximated, using the envelope theorem, by
$ WTP=\frac{\delta L}{\mu } $
where:
$$ {\displaystyle \begin{array}{c}\mathrm{WTP}=\frac{\delta L}{\mu }=\frac{1}{\mu}\left(\sum \limits_{a=0}^{99}\frac{\partial L}{\partial s(a)}\delta s(a)+\frac{\partial L}{\partial q(a)}\delta q(a)\right).\end{array}} $$
Equations (A1) and (A2) imply:
$$ {\displaystyle \begin{array}{c}\frac{1}{\mu}\frac{\partial L}{\partial s(a)}=\frac{1}{\mu}\frac{q\times u}{{\left(1+\rho \right)}^a}+\frac{v}{{\left(1+r\right)}^a}=\frac{\frac{u}{u_z\times {z}_c}+v}{{\left(1+r\right)}^a},\end{array}} $$
$$ {\displaystyle \begin{array}{c}\frac{1}{\mu}\frac{\partial L}{\partial q(a)}=\frac{1}{\mu}\frac{s\times u}{{\left(1+\rho \right)}^a}=\frac{s\times \left(\frac{1}{q}\right)\times \frac{u}{u_z\times {z}_c}}{{\left(1+r\right)}^a}.\end{array}} $$
We can rewrite
$ \frac{u}{u_z\times {z}_c} $
as follows. The Cobb-Douglas function’s homogeneity of degree one implies (suppressing
$ a $
for notational compactness):
$$ {\displaystyle \begin{array}{c} kz=z\left( kc, kl\right)\to \frac{\partial (kz)}{\partial k}={z}_c\frac{\partial (kc)}{\partial k}+{z}_l\frac{\partial (kl)}{\partial k}\to \mathrm{z}={z}_cc+{z}_ll\to \frac{z}{z_c}=c+\frac{z_l}{z_c}l.\end{array}} $$
Combining (A2) and (A3) yields:
$$ {\displaystyle \begin{array}{c}w=\frac{z_l}{z_c}.\end{array}} $$
Equations (A8), (A9), and (10) imply:
$$ {\displaystyle \begin{array}{c}\frac{z}{z_c}={c}^f,\end{array}} $$
which in turn implies:
$$ {\displaystyle \begin{array}{c}\frac{u}{u_z\times {z}_c}={c}^f\left(\Phi +1\right),\end{array}} $$
where
$ \Phi $
is given by (14).
We also have:
$$ {\displaystyle \begin{array}{c}v=i-o={y}^n-c,\end{array}} $$
and:
$$ {\displaystyle \begin{array}{c}\frac{u}{u_z\times {z}_c}+v={c}^f\left(\Phi +1\right)+\left({y}^n+ wl\right)-\left(c+ wl\right)={y}^f+{c}^f\Phi, \end{array}} $$
where
$ {y}^f $
and
$ {y}^n $
are given by (12) and (13).
Substituting all the above into (A5) yields (8).
Equation (15) implies, for
$ a\ge \overline{a} $
:
$$ {\displaystyle \begin{array}{c}\frac{\partial s(a)}{\partial \lambda \left(\overline{a}\right)}=-s(a).\end{array}} $$
For a small enough reduction in the hazard, the envelope theorem provides the following approximation to VSL (relying on (A1), (A2), and (A6)):
$$ {\displaystyle \begin{array}{c}\mathrm{VSL}\left(\overline{a}\right)=\frac{\frac{\partial L}{\partial \lambda \left(\overline{a}\right)}}{\frac{\partial L}{\partial A}}=\left(\frac{1}{\mu}\right)\frac{\partial L}{\partial \lambda \left(\overline{a}\right)}=\left(\frac{1}{\mu}\right)\sum \limits_{a=\overline{a}}^{99}\frac{\partial L}{\partial s(a)}\frac{\partial s(a)}{\partial \lambda \left(\overline{a}\right)}=\sum \limits_{a=\overline{a}}^{99}\frac{s(a)\times \mathrm{VSLY}(a)}{{\left(1+r\right)}^{a-\overline{a}}}.\end{array}} $$
Table A1. HALM variables and National Transfer Accounts equivalents
Table A2. NTA variables
Table A3. Hourly wages, 2023 USD
Table A4. Nonmarket Time use (hours per day)
Table A5. US general population health utilities
Table A6. HALM quantities
Open access
