2.1. Standard cost-effectiveness and WTP for health gains

The formal version of cost-effectiveness analysis (CEA), consistent with microeconomic principles, was first set forth by Garber and Phelps (Reference Garber and Phelps1997). They assumed that health is produced with a two-input process, $ H\hskip0.35em =\hskip0.35em H\left(a,b\right) $ , where $ {p}_a\;\mathrm{and}\;{p}_b $ are the prices of $ a\hskip0.24em \mathrm{and}\;b, $ and $ {H}_a^{\prime } $ and $ {H}_b^{\prime } $ are the associated marginal products. Define utility as a function of consumption, C, equal to income less medical spending, and health, H. In a two-period model, $ {H}_0 $ is baseline period zero health, measured on a scale of $ 0\hskip0.35em \le \hskip0.35em H\hskip0.35em \le \hskip0.35em 1, $ and typically set at $ {H}_0\hskip0.35em =\hskip0.35em 1 $ , perfect health. Overall utility is given by $ V\left(C,H\right)\hskip0.35em =\hskip0.35em U(C)H $ , where $ U(C) $ is assumed to have diminishing returns, but utility is assumed to have constant returns to $ H. $ Defining the elasticity of utility with respect to C as $ {\omega}_C\hskip0.35em \equiv \hskip0.35em {U}^{\prime }(C)\left[\frac{C}{U(C)}\right], $ the WTP for a quality-adjusted life-year (QALY) is the marginal utility of a QALY scaled by the marginal utility of consumption, or:

(1) $$ K\hskip0.35em \equiv \hskip0.35em \frac{U(C)}{U^{\prime }(C){H}_0}\hskip0.35em =\hskip0.35em \left[\frac{C}{H_0}\right]\left[\frac{1}{\omega_C}\right]. $$

Insurance coverage choices are made in period 0, before individual illness states are known. These choices affect access to care in period 1, after illness states are realized. Expected utility is maximized by adjusting the use of inputs so that $ \frac{p_a}{H_a^{\mathrm{\prime}}}\hskip0.35em =\hskip0.35em \frac{p_b}{H_b^{\mathrm{\prime}}}\hskip0.35em \le \hskip0.35em K $ . WTP for one QALY exceeds the value of a year’s consumption, C, only because of diminishing returns to consumption, that is, $ 0<{\omega}_C<1. $ Current estimates suggest that $ 2C\hskip0.35em \le \hskip0.35em K\hskip0.35em \le \hskip0.35em 3C $ in industrialized countries, lower in developing countries (Phelps, Reference Phelps2019; Phelps & Cinatl, Reference Phelps and Cinatl2021; Phelps & Lakdawalla, Reference Phelps and Lakdawalla2023).

Once K is determined, CEA and BCA are equivalent under certain conditions (Bleichrodt & Quiggin, Reference Bleichrodt and Quiggin1999). The CEA rule says that the incremental cost-effectiveness ratio (ICER) of any medical intervention – marginal cost divided by marginal product – should not exceed WTP, normally stated as $ \frac{p_a}{H_a^{\prime }}\hskip0.35em \le \hskip0.35em K $ . Once a specific value of K is chosen, CEA value measures convert directly into net monetary benefit (NMB) measures and hence to standard BCA (Phelps & Mushlin, Reference Phelps and Mushlin1991).

Since CEA presumes constant returns to H, it does not matter to whom health improvements are given, what their untreated illness severity is, what their level of preexisting disability is, or what degree of uncertainty about treatment benefits exists. Constant returns to $ H $ also require that the marginal rate of substitution (MRS) between gains in health-related quality of life (HRQoL) and LE be the same in all situations. The mantra of standard CEA says that “… a QALY is a QALY is a QALY…” (Williams, Reference Williams1992). Mathematically, the total value of a medical intervention in CEA is

(2) $$ \mathrm{TVM}{\mathrm{I}}_{\mathrm{CEA}}\hskip0.35em =\hskip0.35em \mathrm{LE}\times \Delta \mathrm{HRQoL}+\mathrm{HRQoL}\times \Delta \mathrm{LE}. $$

This formula values gains in LE at the existing HRQoL, and thus values LE gains for disabled people less than the value for otherwise-similar nondisabled people. Similarly, if disability reduces LE, then gains in HRQoL are similarly lower for disabled than for nondisabled people. This may have contributed to the U.S. Affordable Care Act’s banning of cost-effectiveness measures that discriminate against disabled people for federally related health programs.

2.2. How GRACE differs from CEA

GRACE retains CEA’s assumption of separable utility in consumption, C, and health, H Footnote ¹:

(3) $$ V\left(C,H\right)\hskip0.35em =\hskip0.35em U(C)W(H). $$

However, in GRACE, both $ U(C)\;\mathrm{and}\;W(H) $ exhibit positive but diminishing returns to health.Footnote ² As proven in Lakdawalla and Phelps (Reference Lakdawalla Darius and Phelps2020) and Lakdawalla & Phelps (Reference Lakdawalla and Phelps2021, Reference Lakdawalla and Phelps2022), this simple generalization alters standard CEA methods in six important ways:

1. (i) GRACE demonstrates greater WTP for HRQoL gains as untreated illness severity increases.

2. (ii) GRACE reveals greater WTP for both HRQoL and LE gains as disability increases.

3. (iii) GRACE shows that with diminishing returns to health, standard CEA over-values gains in health, with the magnitude of over-valuation depending on how rapidly marginal utility declines as health increases.

4. (iv) Combining these three insights, standard CEA methods overvalue treatments for low-severity illnesses and undervalue treatments for high-severity illnesses, and they undervalue treatments for persons with preexisting disabilities.

5. (v) The relative value of LE and HRQoL varies by initial values of those measures, contrary to the CEA implication that it is identical in all situations.

6. (vi) When consumers exhibit relative risk aversion and prudence (Kimball, Reference Kimball1990), uncertainty in treatment outcomes for HRQoL lowers the value of treatment gains, but increases in the probability of unusually good outcomes (positive skewness in HRQoL outcome distributions) increase the value of treatments.

GRACE defines five new parameters that can be computed using knowledge of underlying preference parameters, along with measures of treated and untreated illness and disability, as well as the distributions of health outcomes in the treated and untreated states.

We also note that, like traditional CEA and almost the entire health economics literature, GRACE presumes time-separable utility, which in turn implies consumer risk-neutrality over changes in mortality risk. Therefore, GRACE primarily affects the value of HRQoL, rather than of LE gains.

2.2.1. Nonstochastic parameters

Three GRACE parameters depend only on levels of health outcomes. These parameters combine to alter the traditional CEA value per unit of health gain, K, to reflect the consequences of diminishing return to health. We discuss these next.

Diminishing returns to H

From Equation (3),

(4a) $$ \frac{\partial V\left(C,H\right)}{\partial C}\hskip0.35em =\hskip0.35em {U}^{\prime }(C)W(H), $$

(4b) $$ \frac{\partial V\left(C,H\right)}{\partial H}\hskip0.35em =\hskip0.35em {W}^{\prime }(H)U(C). $$

Therefore, the MRS between C and H is

(4c) $$ \mathrm{MRS}\hskip0.35em \equiv \hskip0.35em \frac{W^{\prime}\left({H}_0\right)U\left({C}_0\right)}{U^{\prime}\left({C}_0\right)W\left({H}_0\right)}. $$

Define $ {\omega}_H\hskip0.35em =\hskip0.35em {W}^{\mathrm{\prime}}({H}_0)[\frac{H_0}{W({H}_0)}], $ the elasticity of utility with respect to health. Treat baseline health as “perfect” so that $ {H}_0\hskip0.35em =\hskip0.35em 1, $ and define baseline consumption, $ {C}_0 $ . Then,

(4d) $$ \mathrm{MRS}\hskip0.35em =\hskip0.35em \left[\frac{C_0}{H_0}\right]\left[\frac{\omega_H}{\omega_C}\right]. $$

This alters the WTP value K from Equation (1) in several distinct ways. Using Equation (4d), we define the first adjustment to $ K\hskip0.35em =\hskip0.35em \left[\frac{C_0}{\omega_C}\right]\left[\frac{1}{H_0}\right] $ as

(5a) $$ {K}^{\ast}\hskip0.35em =\hskip0.35em K{\omega}_H. $$

Since $ 0<{\omega}_H\hskip0.35em \le \hskip0.35em 1 $ , $ {K}^{\ast}\hskip0.35em \le \hskip0.35em K. $ Therefore, before adjusting for illness severity or preexisting disability, CEA weakly overvalues gains in HRQoL compared with GRACE, with equality obtaining only under the standard CEA restriction that $ {\omega}_H\hskip0.35em =\hskip0.35em 1 $ .

Illness severity

The second GRACE change introduces illness severity. Define $ {\ell}^{\ast } $ as the proportional loss in untreated health in period 1. Further, define $ {H}_{1S}\hskip0.35em =\hskip0.35em {H}_0\left(1-{\ell}^{\ast}\right) $ as the untreated health level in period 1 after the illness occurs and $ {\mu}_H\hskip0.35em \equiv \hskip0.35em E\left({H}_{1S}\right) $ as its mean. For ease of intuition, GRACE uses mean QoL gain as the unit of health improvement; later, we explain how GRACE accounts for variance and higher-order moments of QoL gain. As with standard CEA, GRACE assumes full annuitization and constant consumption, so that $ {C}_0\hskip0.35em =\hskip0.35em {C}_1\hskip0.35em =\hskip0.35em C $ . Therefore, $ \mathrm{MRS}\hskip0.35em =\hskip0.35em \frac{W^{\prime}\left({\mu}_H\right)U(C)}{U^{\prime }(C)W\left({H}_0\right)} $ . In effect, the MRS in Equation (4c) is multiplied by $ R\hskip0.35em =\hskip0.35em \frac{W^{\prime}\left({\mu}_H\right)}{W^{\prime}\left({H}_0\right)} $ , the ratio of marginal utilities in the average untreated sick state to the healthy state. This adjustment shifts the location at which MRS is measured on indifference curves $ \mathrm{from}\;{H}_0\;\mathrm{to}\;{\mu}_H. $ Lakdawalla and Phelps (Reference Lakdawalla Darius and Phelps2020) prove that the WTP measure K in standard CEA then becomes

(5b) $$ {K}_{\mathrm{GRACE}}\hskip0.35em =\hskip0.35em {RK}^{\ast}\hskip0.35em =\hskip0.35em K{\omega}_HR. $$

For low-severity illnesses, $ R\approx 1 $ , and $ R $ rises exponentially with severity. Since $ 0<{\omega}_H<1 $ under diminishing returns, it follows that standard CEA overvalues treatments for low-severity illnesses and undervalues them for high-severity illnesses.

Permanent disability

Lakdawalla and Phelps (Reference Lakdawalla and Phelps2021, Reference Lakdawalla and Phelps2022) further generalized this approach by introducing the possibility of preexisting permanent disability. Just as the MRS between well and sick states changes with untreated illness severity, so also the MRS changes with permanent disability, rising as the degree of disability increases. In perfect health, before any illness has presented itself, the period zero MRS $ \hskip0.35em =\hskip0.35em \left[\frac{C}{H_0}\right]\left[\frac{\omega_H}{\omega_C}\right]. $ Define $ {d}^{\ast } $ as the proportional loss in HRQoL created by disability and $ {H}_D\hskip0.35em =\hskip0.35em {H}_0\left(1-{d}^{\ast}\right) $ . To properly represent the ratio of marginal utilities when preexisting disability is present (see Equation (4c)), the MRS must be adjusted by the factor

(5c) $$ D\hskip0.35em =\hskip0.35em \frac{W\left({H}_0\right)}{W\left({H}_D\right)}. $$

$ D\hskip0.35em =\hskip0.35em 1 $ with no disability, and $ D $ rises exponentially as $ {d}^{\ast } $ rises. This adjustment is necessary to correct for the new base from which ex ante resource allocation decisions are made, accounting for the way the utility of health, $ W(H) $ , affects the marginal utility of consumption in Equation (4a) and hence in the MRS in Equation (4c).

The GRACE-adjusted WTP

These changes lead to the final difference in WTP compared with standard CEA:

(6) $$ {K}_{\mathrm{GRACE}}^D\hskip0.35em =\hskip0.35em D{K}_{\mathrm{GRACE}}\hskip0.35em =\hskip0.35em DR{K}^{\ast}\hskip0.35em =\hskip0.35em DR{\omega}_HK. $$

CEA is a special case of GRACE, characterized by the restrictions that $ {d}^{\ast}\hskip0.35em =\hskip0.35em 0\hskip0.35em \mathrm{and}\hskip0.35em {\omega}_H\hskip0.35em =\hskip0.35em 1 $ so that R and D and $ {H}_0 $ all equal 1.0, and WTP collapses to the standard CEA measure of K. GRACE’s new parameters rely only on parameters of utility functions themselves and upon levels of HRQoL in treated and untreated states. Therefore, estimating them does not require information about the distribution of treatment outcomes.

A graphical presentation of these changes may further assist in understanding how GRACE differs from standard CEA. These figures show indifference curves between consumption, C and health, H. The MRS is the negated slope of the indifference curve at any point along it, and it equals WTP for health gains in each condition.

Setting aside, for now, any randomness in $ {H}_{1S} $ , Figure 1 shows how the MRS increases when moving from perfect health, $ {H}_0 $ , to untreated health with an illness in period 1, $ {H}_{1S}\hskip0.35em =\hskip0.35em {H}_0\left(1-{\ell}^{\ast}\right). $ The second panel in Figure 1 shows the additional increase in the MRS for a person who begins with a permanent disability and then incurs the same illness. In this case, the untreated level of illness is $ {H}_{1S}\hskip0.35em =\hskip0.35em {H}_0\left(1-{\ell}^{\ast}\right)\left(1-{d}^{\ast}\right). $ This demonstrates items (i) and (ii) shown above.

Figure 1. MRS between consumption and HRQoL increases with acute illness severity and/or permanent disability.

Figure 2 shows how the MRS changes as the elasticity of utility with respect to consumption, $ {\omega}_C $ changes. In general, $ MRS\hskip0.35em =\hskip0.35em \frac{\omega_H}{\omega_C}\frac{W\left({H}_{1S}\right)}{W\left({H}_0\right)}\frac{C}{H_{1S}} $ . Under the assumptions of standard CEA, this collapses to $ K\hskip0.35em =\hskip0.35em \frac{C}{\omega_C} $ . As $ {\omega}_C $ declines from $ {\omega}_C^1\hskip0.35em \mathrm{to}\hskip0.35em {\omega}_C^2\hskip0.35em \mathrm{and}\hskip0.35em {\omega}_C^3, $ the opportunity cost of consumption falls, and the WTP, the MRS in this figure, grows as the indifference curves steepen.

Figure 2. The willingness to pay for health improvement at any given illness severity rises as the utility elasticity of consumption falls.

In each of these curves, the elasticity of utility with respect to health, $ {\omega}_H, $ is kept at 1.0, the standard CEA assumption. Thus, Figure 2 demonstrates how the CEA measure of WTP, K, changes as $ {\omega}_C $ changes.

Figure 3 shows the effect of introducing the fundamental GRACE change in assumptions about the nature of utility. For any one of the curves shown in Figure 2, if the elasticity of utility with respect to health, $ {\omega}_H, $ embeds positive but strictly diminishing returns, that is, $ 0<{\omega}_H<1, $ then the indifference curves flatten out, so the MRS falls at every point on the curve. Figure 3 shows the primary effect of allowing diminishing returns, item (iii) shown above.

Figure 3. The willingness to pay for health improvement at any given illness severity rises as the utility elasticity of health falls.

In combination, these results demonstrate how GRACE changes WTP from the standard CEA value of K to $ {K}_{\mathrm{GRACE}}^D\hskip0.35em =\hskip0.35em DR{\omega}_HK $ .

2.2.2. Stochastic parameters

The previous discussion dealt with three GRACE parameters that do not involve the distribution of health outcomes. Next, we discuss two additional parameters that require information about those distributions.

Uncertainty of treatment outcome

Lakdawalla and Phelps (Reference Lakdawalla Darius and Phelps2020, Equation (5)) define the expected period utility gain of a stochastic treatment B for a patient known to have the disease in question as

(7a) $$ EW(T)\hskip0.35em =\hskip0.35em E\left[W\left({H}_{1S}+B\right)-W\left({H}_{1s}\right)\right] $$

and the expected monetary value of that treatment gain asFootnote ³

(7b) $$ EV(B)\hskip0.35em =\hskip0.35em \left[\frac{U(C)}{U^{\prime }(C)}\right]\left[\frac{EW(T)}{W\left({H}_0\right)}\right]. $$

Next, define

(7c) $$ ETG\hskip0.35em =\hskip0.35em \frac{EW(T)}{W^{\prime}\left({\mu}_H\right)}. $$

Normalizing $ EW(T) $ by $ W^{\prime}\left({\mu}_H\right) $ converts it into units of average HRQoL improvement, ETG.

Recalling that $ R\hskip0.35em \equiv \hskip0.35em \frac{W^{\prime}\left({\mu}_H\right)}{W^{\prime}\left({H}_0\right)} $ , simple algebraic manipulation now gives

(7d) $$ EV(B)\hskip0.35em =\hskip0.35em K{\omega}_HR\left\{ ETG\right\}. $$

By computing a Taylor series expansion of $ EW(T) $ around the level of untreated health, Lakdawalla and Phelps (Reference Lakdawalla Darius and Phelps2020) show that $ {\mu}_B\epsilon \approx ETG $ , where $ {\mu}_B $ is the average HRQoL improvement, and $ \epsilon $ is the “certainty-equivalence” ratio measuring the number of riskless QoL units the individual would give up in exchange for the treatment’s stochastic HRQoL gains. The certainty-equivalence ratio, $ \epsilon $ , allows for the approximation of health gains in terms of nonstochastic QoL improvement. It can be computed using the information on higher-order parameters of the statistical distributions of treated and untreated outcomes, including mean, variance, skewness, and kurtosis. We shortly use this Taylor Series approximation in a comparison of GRACE and standard CEA methodologies.

The MRS between LE and HRQoL

To convert longevity and HRQoL gains into a single index of health improvement, GRACE, like traditional CEA, converts longevity gains into units of HRQoL improvement. This requires use of the MRS between LE gains and average HRQoL gains. Furthermore, GRACE, like traditional CEA and almost all the health economics literature, assumes that utility is time-separable and thus linear in the probability of survival, holding consumption, and HRQoL constant. Therefore, the marginal utility of gains in LE is the expected utility of the level of health experienced in after the effects of treatment on HRQoL are received: $ E\left(W\left({H}_T\right)\right) $ . The marginal utility of gains in average HRQoL is simply $ W^{\prime}\left({\mu}_H\right) $ , the marginal utility of the average value of untreated health. The ratio of these forms the MRS between LE and HRQoL, $ \delta $ , and has units of measurement of H

(8a) $$ \delta \hskip0.35em =\hskip0.35em \left\{\frac{E\left[W\left({H}_T\right)\right]}{W^{\prime}\left({\mu}_H\right)}\right\}. $$

Lakdawalla and Phelps (Reference Lakdawalla and Phelps2022) define a new variable, the ratio of expected utility in the treated state to utility of perfect health,

(8b) $$ \rho \hskip0.35em =\hskip0.35em \frac{E\left[W\left({H}_T\right)\right]}{W\left({H}_0\right)}. $$

They then prove that

(8c) $$ {\omega}_H R\delta \hskip0.35em =\hskip0.35em \rho {H}_0. $$

Therefore, $ K{\omega}_H R\delta {\mu}_P\hskip0.35em =\hskip0.35em K{\rho H}_0{\mu}_P. $ We will use this alternative expression below.

Under traditional CEA, $ E\left(W\left({H}_T\right)\right)\hskip0.35em =\hskip0.35em {H}_T $ , $ {W}^{\prime}\left({\mu}_H\right)\hskip0.35em =\hskip0.35em 1 $ , and the MRS becomes simply $ {H}_T $ . However, with diminishing returns to H, people should be more willing to trade LE for HRQoL gains when HRQoL is low, and conversely. Put slightly differently, $ \frac{1}{\delta } $ represents the WTP for HRQoL gains expressed in terms of LE, not consumption.

2.2.3. The total value of medical interventions in GRACE

Combining these parameters, we are now in a position to specify the total value of medical interventions (TVMI) for GRACE. Define the mean gain in survival probability (LE gain) from treatment as $ {\mu}_P $ , and define the ex ante probability of illness as $ \phi $ . Now, from the ex ante period 0 perspective,

(9a) $$ TVM{I}_{GRACE}\hskip0.35em =\hskip0.35em KD\phi \left\{{\mu}_P\rho {H}_0+{\mu}_B\epsilon {\omega}_HR\right\}. $$

In contrast and relying on Equation (2), traditional CEA computes TVMI as

(9b) $$ TVM{I}_{CEA}\hskip0.35em =\hskip0.35em K\phi [{\mu}_P{H}_T+{\mu}_B]. $$

This highlights the differences between GRACE and CEA. In CEA where $ {\omega}_H\hskip0.35em =\hskip0.35em 1 $ , then so also $ R\hskip0.35em =\hskip0.35em 1 $ , $ \epsilon \hskip0.35em =\hskip0.35em 1 $ , $ \rho \hskip0.35em =\hskip0.35em {H}_T $ and $ D\hskip0.35em =\hskip0.35em 1. $ Compared with GRACE, CEA overlooks the effects of the effect of diminishing returns to health, $ {\omega}_H $ , the extra value of treating disabled people, $ D $ , the effect of illness severity on treatment, $ R $ , the effect of QoL levels on the MRS between LE and HRQoL $ , $ $ \rho $ , and the effect of treatment uncertainty on value, $ \epsilon . $

The terms in Equation (9a) can be recovered without knowledge of a specific utility function, provided the analyst can estimate relative risk aversion, relative prudence, and other higher-order risk preferences (Lakdawalla & Phelps, Reference Lakdawalla Darius and Phelps2020; Lakdawalla & Phelps, Reference Lakdawalla and Phelps2022). However, this process introduces approximation error into the TVMI expression, because of the Taylor series expansions used to estimate the parameters in the equation. As an alternative, we can unravel the definitions of $ D,\rho, \epsilon, {\omega}_H, $ and $ R $ to express an “exact” form of Equation (9a):

(9c) $$ TVM{I}_{GRACE}\hskip0.35em =\hskip0.35em \frac{K\phi {H}_0}{W({H}_0(1-{d}^{\ast }))}\{{\mu}_PE[W({H}_T)]+E[W(T)]\}. $$

The first term in curly braces describes the expected utility from gains in LE and the second term describes the expected utility from gains in HRQoL, as defined in Equation (7a). When the function $ W $ is known, along with the distributions of $ {H}_{1s} $ and $ B $ , all terms in Equation (9c) can be calculated exactly, without reliance on approximations.

3.1. Constant relative risk aversion

The simplest case to understand is to specify that utility has CRRA $ . $ This formulation lends itself to intuitive understanding of the way GRACE functions. With CRRA, utility and marginal utility are given by

(10a) $$ W(H)\hskip0.35em =\hskip0.35em \left[\frac{1-\gamma }{\gamma}\right]{H}^{\gamma } $$

and

(10b) $$ {W}^{\prime }(H)\hskip0.35em =\hskip0.35em \left(1-\gamma \right){H}^{\gamma -1}. $$

With CRRA utility, relative risk aversion is given by $ {r}_H^{\ast}\hskip0.35em =\hskip0.35em \left(1-\gamma \right) $ , and the elasticity of utility with respect to health is $ {\omega}_H\hskip0.35em =\hskip0.35em \gamma . $ Further, $ {r}_H^{\ast }+{\omega}_H\hskip0.35em =\hskip0.35em 1. $ This simplified structure allows us to easily calculate exact forms for the key GRACE parameters. Positive but diminishing marginal utility requires that $ 0<\gamma <1. $

3.1.1. Nonstochastic parameters

First, as noted, with CRRA, $ {\omega}_H\hskip0.35em =\hskip0.35em \gamma $ . Now, consider $ R $ , the severity of illness multiplier, and D, the adjustment to WTP for permanent disability. Since $ {H}_{1S}\hskip0.35em =\hskip0.35em {H}_0\left(1-{\ell}^{\ast}\right) $ , using Equation (10a), we can readily write R, the ratio of two marginal utilities, as

(11a) $$ R\hskip0.35em =\hskip0.35em {\left[\frac{H_{1S}}{H_0}\right]}^{\gamma -1}\hskip0.35em =\hskip0.35em {\left[\frac{H_0}{H_0\left(1-{\ell}^{\ast}\right)}\right]}^{1-\gamma}\hskip0.35em =\hskip0.35em {\left[\frac{1}{1-{\ell}^{\ast }}\right]}^{1-\gamma}\hskip0.35em =\hskip0.35em {\left[\frac{1}{1-{\ell}^{\ast }}\right]}^{r_H^{\ast }}. $$

When $ {r}_H^{\ast}\hskip0.35em =\hskip0.35em 0, $ the standard CEA assumption, then $ R\hskip0.35em =\hskip0.35em 1 $ .

Next, $ D $ is a ratio of utilities, not marginal utilities, so where $ {H}_{1d}\hskip0.35em =\hskip0.35em {H}_0\left(1-{d}^{\ast}\right) $ , using Equation (10a)

(11b) $$ D\hskip0.35em =\hskip0.35em {\left[\frac{H_0}{H_0\left(1-{d}^{\ast}\right)}\right]}^{\gamma}\hskip0.35em =\hskip0.35em {\left(\frac{1}{1-{d}^{\ast }}\right)}^{\gamma}\hskip0.35em =\hskip0.35em {\left(\frac{1}{1-{d}^{\ast }}\right)}^{\omega_H}. $$

Therefore, $ D\ge 1 $ and increases exponentially with the severity of disability, d*. When $ {d}^{\ast}\hskip0.35em =\hskip0.35em 0, $ the standard CEA assumption, then $ D\hskip0.35em =\hskip0.35em 1 $ .

Since $ {\omega}_H\hskip0.35em =\hskip0.35em \gamma $ with CRRA utility, we can now directly compute the multiplier that adjusts for diminishing returns to H. Combining these, the GRACE value per unit of health gain is

(12) $$ {K}_{GRACE}^D\hskip0.35em =\hskip0.35em K{\omega}_H DR\hskip0.35em =\hskip0.35em K\gamma {\left[\frac{1}{1-{d}^{\ast }}\right]}^{\gamma }{\left[\frac{1}{1-{\ell}^{\ast }}\right]}^{1-\gamma }. $$

Asymptotically, CRRA utility is linear in H when $ \gamma \hskip0.35em =\hskip0.35em 1 $ , that is, standard CEA. In this case, adopting the usual CEA perspective that $ {d}^{\ast}\hskip0.35em =\hskip0.35em 0, $ the value per unit of health gain collapses to K.

3.1.2 Stochastic parameters

The MRS between LE and HRQoL

The general equation for this tradeoff is given in Equation (8a). To fully specify this, we must first define the probability distribution of outcomes for individuals who are sick but treated. Define each possible treated outcome as $ {H}_n^T $ for n = 1…. N, each with an associated probability $ {\pi}_n^T $ . Then, using Equation (8b),

(13a) $$ \rho {H}_0\hskip0.35em =\hskip0.35em {H}_0\left[{\sum}_n{\pi}_n^T\left(\frac{W\left({H}_n^T\right)}{W\left({H}_0\right)}\right)\right]. $$

When utility is CRRA

(13b) $$ \rho {H}_0\hskip0.35em =\hskip0.35em {H}_0{\sum}_n{\pi}_n^T\left(\frac{W\left({H}_n^T\right)}{W\left({H}_0\right)}\right)\hskip0.35em =\hskip0.35em {H}_0{\sum}_n{\pi}_n^T{\left(\frac{H_n^T}{H_0}\right)}^{\gamma }. $$

Defining $ {H}_n^T\hskip0.35em =\hskip0.35em {H}_0\left(1-{t}_n^{\ast}\right), $ then

(13c) $$ \rho {H}_0\hskip0.35em =\hskip0.35em {H}_0{\sum}_n{\pi}_n^T{(\frac{H_0(1-{t}_n^{\ast })}{H_0})}^{\gamma}\hskip0.35em =\hskip0.35em {H}_0{\sum}_n{\pi}_n^T{((1-{t}_n^{\ast })}^{\gamma }). $$

Note that when constant returns to H are specified, then, asymptotically, $ \gamma \hskip0.35em =\hskip0.35em 1 $ and $ {H}_0{\sum}_n{\pi}_n^T{\left(\frac{H_n^T}{H_0}\right)}^{\gamma } $ collapses to $ {\sum}_n{\pi}_n^T\left({H}_n^T\right), $ so $ \rho {H}_0 $ equals expected treated health in period 1, identified in Garber and Phelps as $ {H}_1. $ GRACE differs from CEA in this calculation, using CRRA utility, by the effect of the power exponent $ \gamma . $

Uncertain treatment outcomes

Adding the disability adjustment to Equation (7b), and recalling that $ K\hskip0.35em =\hskip0.35em \left[\frac{C}{\omega_C}\right]\left[\frac{1}{H_0}\right], $ the value of the medical intervention (in units of consumption, e.g., dollars) is

(14a) $$ EV(B)\hskip0.35em =\hskip0.35em K{\omega}_H RD\left\{ ETG\right\}. $$

To complete Equation (14a) using CRRA, define the possible treatment outcomes as $ {H}_{T_i} $ and their associated probabilities, $ {\pi}_i^T. $ Similarly, define the possible outcomes in the untreated state as $ {H}_{1{S}_j} $ and their associated probabilities as $ {\pi}_j^U. $ Then, ETG, measured in units of HRQoL gains, becomes

(14b) $$ ETG\hskip0.35em =\hskip0.35em \left\{{\sum}_i{\pi}_i^T\frac{W\left({H}_{T_i}\right)}{W^{\prime}\left({H}_0\right)}-{\sum}_j{\pi}_j^U\frac{W\left({H}_{1{S}_j}\right)}{W^{\prime}\left({H}_0\right)}\right\}. $$

Finally, applying the CRRA definitions of $ W(H)\;\mathrm{and}\;{W}^{\prime }(H) $ from Equations (10a) and (10b), we have

(14c) $$ EV(B)\hskip0.35em =\hskip0.35em K{\omega}_H RD\left\{\sum {\pi}_i^T\left[\frac{H_0\;}{\gamma}\right]{\left(\frac{{H_T}_i}{H_0}\right)}^{\gamma }-\left[\frac{H_0\;}{\gamma}\right]\sum {\pi}_j^T{\left(\frac{H_{1{S}_j}}{H_0}\right)}^{\gamma}\right\}. $$

When utility is CRRA, the term in curly braces, ETG, is an exact measure of the Taylor Series approximation $ {\mu}_B\epsilon $ , where the term ϵ is defined below in Equation (15).

Both $ \left(\frac{{H_T}_i}{H_0}\right) $ and $ \left(\frac{H_{1{S}_j}}{H_0}\right) $ are ratios between 0 and 1, and are magnified as $ \gamma $ becomes smaller, as also is $ \left[\frac{H_0\;}{\gamma}\right]. $ In CRRA, $ \gamma \hskip0.35em =\hskip0.35em {\omega}_H $ , so treatment gains are larger, the faster utility of health declines as H increases, that is, the lower the value of $ {\omega}_H $ . This matches intuition well.

If $ \gamma \hskip0.35em =\hskip0.35em 1, $ which asymptotically equates to linear utility in health in CRRA, the health gain is simply $ \left\{\sum {\pi}_i^T{H}_i^T-\sum {\pi}_j^T{H}_{1{S}_j}\right\} $ , the difference in expected health levels between the treated and untreated states. Once again, this shows that CEA is a restricted version of GRACE.

For this specific GRACE parameter, the Taylor Series itself provides good intuition about how GRACE works. In the Taylor Series, the mean gain in HRQoL is $ {\mu}_B $ and it is adjusted by the risk term $ \epsilon $ , so the “certainty equivalent” health gain is approximately equal to $ {\mu}_B\epsilon . $ Lakdawalla and Phelps (Reference Lakdawalla Darius and Phelps2020) prove that

(15) $$ \epsilon \hskip0.35em =\hskip0.35em 1+[\frac{\mu_H}{\mu_B}][-\frac{1}{2}{r}_H^{\ast}\Delta [{\sigma}_H^2][\frac{1}{\mu_H^2}]+\frac{1}{6}{r}_H^{\ast }{\pi}_H^{\ast}\Delta [{\gamma}_1{\sigma}_H^3][\frac{1}{\mu_H^3}]-\frac{1}{24}{r}_H^{\ast }{\pi}_H^{\ast }{\unicode{x03C4}}^{\ast}\Delta [{\gamma}_2{\sigma}_H^4][\frac{1}{\mu_H^4}]+\dots . $$

Since the risk adjustment terms are all multiplied by $ \left[\frac{\mu_H}{\mu_B}\right], $ for any level of untreated illness, $ {\mu}_H, $ they become magnified as the mean gain in treatment, $ {\mu}_B, $ shrinks. For relatively large average treatment gains, $ \epsilon \approx 1 $ in many cases, but for relatively small treatment gains, $ \epsilon $ can become quite important in assessing overall treatment value.

This illuminates the consequences of uncertain treatment outcomes in terms familiar to the world of finance, but converted into risk about health outcomes rather than risk about income or wealth. If utility is linear in H, so that all risk terms equal zero, then $ \epsilon \hskip0.35em =\hskip0.35em 1 $ . Similarly, if the distributions of health outcomes are identical except for a shift in the mean, so that $ \Delta {\sigma}_H^2 $ and all higher terms show differences in the distributions of outcomes equal 0, then similarly, $ \epsilon \hskip0.35em =\hskip0.35em 1. $

The term $ \frac{1}{2}{r}_H^{\ast}\Delta \left[{\sigma}_H^2\right] $ is exactly parallel to the standard “risk premium” in mean-variance tradeoffs in the world of finance. It incorporates the changes in variance between treated and untreated conditions, $ \Delta {\sigma}_H^2, $ weighted by the relative risk aversion term $ \frac{1}{2}{r}_H^{\ast } $ . This term can be labeled as the “value of insurance” or the value of health risk-reduction; it is conceptually similar to the value of financial insurance that reduces risks to financial assets or income streams. If treatments reduce outcome uncertainty, $ \Delta {\sigma}_H^2<0 $ and value rises.

In parallel, if the distribution of treated patient outcomes has an unusually high proportion of very positive outcomes, this positive skewness in treatment outcomes also increases treated patients’ expected utility. This is captured in the term $ \frac{1}{6}{r}_H^{\ast }{\pi}_H^{\ast}\Delta \left[{\gamma}_1{\sigma}_H^3\right], $ which measures the changes in positive skewness of the outcome distribution, valued by the product $ \frac{1}{6}{r}_H^{\ast }{\pi}_H^{\ast } $ Footnote ⁴. This formulation shows that positive skewness is an economic “good” for any given degree of variance. This term has been described as measuring “the value of hope,” a phenomenon that has been observed in actual cancer patient preferences (Lakdawalla et al., Reference Lakdawalla Darius, Romley, Sanchez, Ross Maclean, Penrod and Phillipson2012).

The next Taylor Series term encompasses changes in kurtosis of the outcome distributions. Generally, kurtosis magnifies the effect of variance in the expected utility measure.

3.1.3 TVMI in CRRA

We can now summarize TVMI as a function of $ {d}^{\ast },{t}^{\ast },{\ell}^{\ast } $ and the CRRA power parameter $ \gamma $ , which combine, along with the distributions of health outcomes, to give the GRACE parameters $ \rho, R,\mathrm{and}\;D. $ Starting with Equation (9a) and recalling that $ {\mu}_B\epsilon $ approximates $ ETG $ , we can write

(16) $$ TVM{I}_{\mathrm{GRACE}}\hskip0.35em =\hskip0.35em KD\phi \left[\rho {H}_0{\mu}_p+ ETG{\omega}_HR\right]. $$

All parameters in Equation (16) are exactly specified in CRRA, as developed within this section in Equations (11–14). The supplemental materials to this manuscript further explore how alternative types of therapies and health states would affect GRACE valuations under CRRA utility.

3.2. HARA utility

HARA utility generalizes CRRA utility functions, which are a special case of HARA. With HARA utility

(17a) $$ W(H)\hskip0.35em =\hskip0.35em [\frac{1-\gamma }{\gamma }]{[\frac{aH}{1-\gamma }+b]}^{\gamma }. $$

Since $ 0\hskip0.35em \le \hskip0.35em H\hskip0.35em \le \hskip0.35em 1 $ , we have no need to scale the values of H, and therefore we can set $ a\hskip0.35em =\hskip0.35em \left(1-\gamma \right)\;\mathrm{so} $ :

(17b) $$ W(H)\hskip0.35em =\hskip0.35em \left[\frac{1-\gamma }{\gamma}\right]{\left(H+b\right)}^{\gamma } $$

and

(17c) $$ {W}^{\prime }(H)\hskip0.35em =\hskip0.35em \left(1-\gamma \right){\left(H+b\right)}^{\gamma -1}. $$

Further, it is easy to prove that in HARA

(17d) $$ {r}_H^{\ast}\hskip0.35em =\hskip0.35em \left(1-\gamma \right)\left[\frac{H}{H+b}\right] $$

and

(17e) $$ {\omega}_H\hskip0.35em =\hskip0.35em \gamma \left[\frac{H}{\left(H+b\right)}\right]. $$

When $ b>0 $ , utility has increasing relative risk aversion (IRRA) and when $ b<0, $ utility has decreasing relative risk aversion (DRRA). Obviously, when $ b\hskip0.35em =\hskip0.35em 0 $ , then utility has CRRA.

With CRRA utility, $ {r}_H^{\ast }+{\omega}_H\hskip0.35em =\hskip0.35em 1 $ . Introducing nonzero values of b allows this total to differ from 1.0. As we demonstrate below, this introduces a wider range of possible GRACE valuations for any value of $ \gamma $ , which will in turn affect all GRACE parameters. HARA generalizes CRRA by adding one extra parameter, b, increasing the formula’s complexity but at the same time usefully increasing generalizability.

Figure 4(a,b) demonstrates the effects of positive and negative values of b. The intuition of these effects is reasonably obvious. When $ b>0, $ as health grows, the ratios $ \left[\frac{H}{H+b}\right] $ in the expressions for $ {r}_H^{\ast } $ and $ {\omega}_H $ shrink for any given value of $ \gamma, $ but the effect diminishes as H grows. For example, if $ b\hskip0.35em =\hskip0.35em 0.05, $ then as H approaches 1.0, the ratio $ \left[\frac{H}{H+b}\right]\hskip0.5em \mathrm{approaches}\hskip0.5em 1.0\hskip0.5em \mathrm{and}\ \mathrm{the} $ values of $ {r}_H^{\ast}\;\mathrm{and}\;{\omega}_H $ increase from smaller values to those near comparable CRRA values.

Figure 4. Relative risk-aversion and the health elasticity of utility rise with health for IRRA utility but fall with health for DRRA utility.

When $ b>0, $ utility is IRRA. When the utility is IRRA, $ {r}_H^{\ast }+{\unicode{x03C9}}_{\mathrm{H}}<1 $ , as in Figure 4(a). As we show below, compared with CRRA utility, this generally reduces $ TVM{I}_{GRACE} $ for any given $ \gamma $ and severity of acute illness or disability.

The reverse is true if $ b<0, $ as in Figure 4(b). As b grows in absolute value, the ratio $ \left[\frac{H}{H+b}\right] $ grows, so $ {r}_H^{\ast}\;\mathrm{and}\;{\omega}_H $ become larger, but as H grows toward 1.0 from lower values, $ {r}^{\ast } $ and $ {\omega}_H $ approach the CRRA values from above.

When utility is DRRA, $ {r}_H^{\ast }+{\omega}_H>1. $ As we demonstrate shortly, this combination of parameters, when compared with CRRA utility, generally makes $ TVM{I}_{GRACE} $ larger for any given level of $ \gamma $ and degree of illness or disability.

The expression for R is slightly more complex, but easy to follow as an extension of Equation (11a). The levels of health in the numerator and denominator are replaced with $ H+b, $ so $ R\hskip0.35em =\hskip0.35em {\left[\frac{\left({H}_0+b\right)}{H_0\left(1-{\ell}^{\ast}\right)+b}\right]}^{1-\gamma } $ . When $ b>0 $ , the ratio in square brackets becomes smaller as b rises, that is, as the extent of IRRA increases. This occurs because b is a larger proportion of the denominator than of the numerator.

Similarly, since $ {\omega}_H\hskip0.35em =\hskip0.35em \gamma \left[\frac{H}{\left(H+b\right)}\right] $ , when $ b>0, $ the larger the value of b, the smaller is $ {\omega}_H $ for any given value of $ \gamma . $ Therefore, when $ b>0, $ compared with CRRA utility, two of the key parameters in $ {K}_{GRACE} $ , $ {\omega}_H $ and R, shrink as b grows. Thus, before considering the disability multiplier D, the value of $ {K}_{GRACE} $ falls as the degree of IRRA increases.

The same is true symmetrically when $ b<0 $ , that is, when the utility is DRRA. Then both R and $ {\omega}_H $ rises as the degree of DRRA increases, before considering the disability multiplier D, the value of $ {K}_{GRACE} $ rises as the degree of DRRA increases.

We now turn to the effects of IRRA and DRRA on the disability multiplier D. In CRRA, $ D\hskip0.35em =\hskip0.35em {\left[\frac{H_0}{H_0\left(1-{d}^{\ast}\right)}\right]}^{\gamma } $ , which is similar to R except that the exponent changes from $ \left(1-\gamma \right) $ to $ \gamma $ and $ {\ell}^{\ast } $ is replaced by $ {d}^{\ast }, $ so $ D\hskip0.35em =\hskip0.35em {\left[\frac{H_0+b}{H_0\left(1-{d}^{\ast}\right)+b}\right]}^{\gamma }. $ Therefore, the sign and magnitude of b have similar effects on D as they do on R. Increasing IRRA dampens the D multiplier and DRRA increases it, relative to CRRA.

Since $ \rho \hskip0.35em =\hskip0.35em {\sum}_n{\pi}_n^T{\left(\frac{H_n^T}{H_0}\right)}^{\gamma} $ when utility is CRRA, $ {H}_n^T $ and $ {H}_0 $ are replaced by $ \left({H}_n^T+b\right) $ and $ \left({H}_0+b\right), $ so $ \rho \hskip0.35em =\hskip0.35em {\sum}_n{\pi}_n^T{\left(\frac{H_n^T+b}{H_0+b}\right)}^{\gamma } $ . For $ b>0,\mathrm{when}\ \mathrm{utility}\ \mathrm{is}\ \mathrm{IRRA},\left({H}_n^T+b\right)<\left({H}_0+b\right), $ so $ \rho $ increases as b grows. The reverse holds and $ \rho $ diminishes as $ <0 $ , when utility is DRRA.

3.3. Exponential and EP utility

Many previous health economics studies (e.g., Zeckhauser, Reference Zeckhauser1970; Feldstein, Reference Feldstein1973; Feldstein & Friedman, Reference Feldstein and Friedman1977; Keeler et al., Reference Keeler, Buchanan, Rolph, Hanley and Reboussin1988; Manning & Marquis, Reference Manning and Marquis1996; Marquis & Holmer, Reference Marquis and Holmer1996; Garber & Phelps, Reference Garber and Phelps1997) assumed CARA. We first discuss this simple form and then a generalization of it, EP utility (Saha, Reference Saha1993).

3.3.1. Exponential utility

In exponential utility, which has CARA

(18a) $$ U(x)\hskip0.35em =\hskip0.35em 1-{e}^{-\beta x}, $$

(18b) $$ {U}^{\prime }(x)\hskip0.35em =\hskip0.35em \beta x{e}^{-\beta x}, $$

(18c) $$ {U}^{\prime \prime }(x)\hskip0.35em =\hskip0.35em -{\beta}^2x{e}^{-\beta x}, $$

(18d) $$ r\hskip0.35em =\hskip0.35em -\frac{U^{\prime \prime }(x)}{U(x)}\hskip0.35em =\hskip0.35em \beta, $$

(18e) $$ {r}^{\ast}\hskip0.35em =\hskip0.35em rx\hskip0.35em =\hskip0.35em \beta x. $$

Although CARA has seen widespread use for its analytic simplicity, it has been widely rejected as appropriate in financial economics. In our setting, using CARA utility would substantially shrink $ TVM{I}_{GRACE} $ compared with using CRRA, since CARA has strong IRRA, and the discussion of HARA with IRRA, that is, when $ b>0 $ , shows that all component parts of $ {K}_{GRACE} $ decline as IRRA becomes stronger.Footnote ⁵

3.3.2. EP utility

EP utility replaces $ {e}^{-\beta x} $ with $ {e}^{-\beta {x}^{\gamma }}, $ so that, in some sense, it is a mixture of CARA and CRRA utility. In EP utility,

(19a) $$ U(x)\hskip0.35em =\hskip0.35em 1-{e}^{-\beta {x}^{\gamma }}, $$

(19b) $$ {U}^{\prime (x)}\hskip0.35em =\hskip0.35em \gamma \beta {x}^{\gamma -1}{e}^{-\beta {x}^{\gamma }}, $$

(19c) $$ {U}^{\prime \prime }(x)\hskip0.35em =\hskip0.35em -\gamma \beta {x}^{\gamma -1}{e}^{\gamma {x}^{\gamma -1}}\left[\gamma \beta {x}^{\gamma -1}\left(1-\gamma \right){x}^{-1}\right), $$

(19d) $$ r\hskip0.35em =\hskip0.35em -\frac{U^{\prime \prime }(x)}{U^{\prime }(x)}\hskip0.35em =\hskip0.35em \frac{\left[\gamma \beta {x}^{\gamma }+\left(1-\gamma \right)\right]}{x}, $$

(19e) $$ {r}^{\ast}\hskip0.35em =\hskip0.35em rx\hskip0.35em =\hskip0.35em \gamma \beta {x}^{\gamma }+\left(1-\gamma \right). $$

When $ \gamma \hskip0.35em =\hskip0.35em 1 $ , asymptotically, relative risk aversion collapses to exponential utility, where $ {r}^{\ast}\hskip0.35em =\hskip0.35em \beta x. $ As $ \gamma \to 0 $ asymptotically, $ {r}^{\ast}\to 1, $ the relative risk aversion when $ U(x)\hskip0.35em =\hskip0.35em \ln (x) $ Footnote ⁶.

Unlike the CRRA and HARA functions, no simple measures exist for the key GRACE parameters such as R, D, $ \hskip0.24em \mathrm{and} $ $ \rho . $ Nevertheless, it may be a very useful function to employ to characterize GRACE value measures. Just as HARA generalizes CRRA by adding one extra parameter, b, EP generalizes exponential utility by adding one parameter, $ \gamma $ , beyond the simpler CARA model.

6.1. Discrete choice experiments

We begin with the simplest case, that of CRRA utility. Holt and Laury (Reference Holt and Laury2002) demonstrate the key methodology, as shown in Table 1.

Table 1. Discrete choice experiment structure using financial gambles.

Notes: From Holt and Laury (Reference Holt and Laury2002, Table 1). Option A has very low risk. Option B has higher risk. The final column shows the expected difference between choosing Option A over Option B.

In this approach, people are presented with pairs of gambles that alter the risk/reward tradeoffs. This can be done either by altering the payoffs or the probabilities of the gambles; Holt and Laury modified the probabilities, while Noussair et al. (Reference Noussair, Trautmann and Van de Kuilen2014) modified the gambles with constant probabilities.

In Table 1, Option A, the “safe” choice, is less-risky than Option B. Moving down the rows, as the probabilities shift, the expected payoff steadily increases, so that choosing the riskier gamble, Option B, becomes increasingly attractive. At some point, respondents switch from making “safe” choices to the riskier choices. Table 2 uses CRRA to determine the expected utility of each gamble, and the switching point identifies the CRRA power parameter $ \gamma . $ Once this is known, and CRRA is assumed, the exact utility function expressions for TVMI are fully defined. For example, in Table 2, if the switch occurs at the fifth gamble, then relative risk aversion is somewhere between 0.15 and 0.41.

Table 2. Summary of risk aversion for CRRA utility.

Note: From Holt and Laury (Reference Holt and Laury2002, Table 3).

Only 8 per cent of the subjects exhibited risk-loving behavior (the top three rows), and two-thirds were clearly risk averse. The median number of safe choices was 5, indicating relative risk aversion in the range of $ .15<{r}^{\ast }<.41, $ with a midpoint of $ {r}^{\ast}\approx .28. $

Holt and Laury (Reference Holt and Laury2002) also estimated an EP model using their data, where, at mean values of the data, the estimated value of $ {r}^{\ast}\approx .28 $ and climbs slowly with the size of the gamble.

The same MLE approach can obviously be used for any specified utility function, although the complexity will increase with the number of parameters necessary to define utility. Just as with EP utility, HARA utility requires two parameters, the power parameter $ \gamma $ and the risk aversion adjustment b.

6.2. Happiness economics models

While DCE experiments focus on gambles involving health states, Happiness Economics models offer the ability to simultaneously estimate not only the parameters for the “health” components of GRACE, but also the elasticity of utility with respect to consumption, $ {\omega}_C $ in the same data set as is used to estimate the functional form and parameters of $ W(H). $

Easterlin (Reference Easterlin2003) pioneered the Happiness economics approach, now widely used by the United Nations using data from international Gallup polls. This approach requires (in our case) three data elements from each respondent: (i) Their level of happiness on some fixed scale, for example, 0–10 or 0–100; (ii) their income, and (iii) their level of health on a fixed interval, for example, 0–10 or 0–100. These can be gathered using direct questions, visual analog scales (“thermometer” scales) or other methods. This approach uses reported happiness as a proxy for the economists’ concept of “utility.” With such data, one can estimate a generic translog utility function (Christiansen et al., Reference Christiansen, Jorgensen and Lau1973) of the form

(24) $$ \ln \left( Happ{y}_i\right)\hskip0.35em =\hskip0.35em {\beta}_1\ln \left({H}_i\right)+\frac{1}{2}{\beta}_2{\left(\ln \left({H}_i\right)\right)}^2+{\beta}_3\ln \left({C}_i\right)+\frac{1}{2}{\beta}_4{\left(\ln \left({C}_i\right)\right)}^2+{\epsilon}_i\dots . $$

Of course, analysts can include other covariates that might affect happiness such as age, sex, ethnicity, geographic region, and others. These would reduce residual variance, and would avoid omitted variable bias if the omitted variables were correlated with H or C.

From such data, one can readily infer the key GRACE parameters such as $ {\unicode{x03C9}}_{\mathrm{H}}, $ $ {\mathrm{r}}_H^{\ast } $ (and how these change with levels of H), and $ {\unicode{x03C9}}_{\mathrm{C}} $ and how it might change with C. Phelps and Lakdawalla (Reference Phelps and Lakdawalla2023, Chapter 8) provide details for this approach.

Assuming a specific utility function offers a different approach, however, namely to fit the data from the “happiness” survey to a specific functional form. We use here the example of HARA utility, although the approach generalizes to any specific form of the utility function. Here again, the assumption of separability in utility simplifies the discussion.

Begin with the standard utility function in Equation (3), but with the specific assumption that both $ U(C) $ and $ W(H) $ have HARA form. In what follow, we will ignore the possible inclusion of covariates, which can generically be added as $ {X}_j $ values, each with their own appropriate functional form and parameterization. We add a generic error term, the details of which would emerge in econometric analysis. For individual j,

(25a) $$ \mathrm{Happiness}\hskip0.35em =\hskip0.35em U\left({C}_j\right)W\left({H}_j\right){\epsilon}_j. $$

Applying CRRA utility from Equation (17b) gives

(25b) $$ Happines{s}_j\hskip0.35em =\hskip0.35em \left[\frac{1-\gamma }{\gamma}\right]{\left[{C}_j+b\right]}^{\gamma}\left[\frac{1-\delta }{\delta}\right]{\left[{H}_j+\beta \right]}^{\delta }{\epsilon}_{\mathrm{j}}. $$

In logarithmic form,

(25c) $$ \ln \left( Happines{s}_j\right)\hskip0.35em =\hskip0.35em \ln \left(\frac{1-\gamma }{\gamma}\right)+\ln \left(\frac{1-\delta }{\delta}\right)+\gamma \ln \left[{C}_j+b\right]+\delta \ln \left[{H}_j+\beta \right]+\ln \left({\epsilon}_j\right). $$

From this, the log-likelihood function is readily composed, and maximized over the four parameters $ \gamma, b,\delta, \mathrm{and}\hskip0.35em \beta $ , plus those associated with any other covariates included in the model. One could readily test the hypotheses that the functions $ U(C) $ and $ W(H) $ were CRRA by testing whether (respectively) $ b $ and $ \beta $ equal zero. Econometric analysis would determine the best possible transformation for $ {\epsilon}_j $ , possibly using Box and Cox (Reference Box and Cox1964) transformations to normalize the distribution of the residuals. We leave these details to others.

One could obviously insert the EP function into Equation (25b) as an alternative to the HARA function, which would have a more complicated appearance, but would still only require maximization over two parameters each for $ U(C) $ and $ W(H). $

Adding covariates such as age, sex, ethnicity, and others could improve precision by reducing the residual variance. One could compare across structural models by comparing log-likelihood ratios from equations using HARA and EP. Unfortunately, HARA and EP are not nested, so they cannot be compared in a single maximum likelihood estimation.