The dominant policy approach to monetizing mortality risk reductions is the VSL. The VSL represents the monetary amount people need to be paid to accept additional risk in their lives. However, the ease and regularity of the VSL usage come with the disadvantages of biases in the publication process, lack of recognition of the differences between occupation and industry as well as the ambiguity of the effect age and cohort group have on the value (Aldy and Viscusi 2008; Kip Viscusi 2004; Viscusi 2018a, b). This paper will present the theoretical research behind the VSL then the numerical analysis before introducing a new method of calculating the VSL based on Gary Becker's method found in his paper "Health as Human Capital: Synthesis and Extensions" (Becker 2007).
In a 2018 paper, Viscusi states that when publishing research, it is possible for researchers to have publication selection bias and best-estimate selection bias when choosing VSL values. These biases cover the possibility that researchers may use estimates that are most likely to be published or choose the value from their preferred model. Fortunately, Viscusi states that the biases are only moderate when studying data from the Bureau of Labor Statistics Census of Fatal Occupational Injuries (CFOI). Viscusi offers solutions such as undertaking meta-regression studies, focusing on characteristics associated with studies with the least biases, and having government agencies work with researchers to identify reliable estimates (Aldy and Viscusi 2008; Viscusi 2015).
Viscusi details in 2004 how the current ways of measuring VSL do not differentiate between occupation and industry. Viscusi argues that this disadvantage creates four deficiencies: (1) erroneous variables; (2) failure of significant full-sample estimates; (3) failure in the analysis of nonfatal job risks; and (4) grouping all workers in the same industry or occupation under the same value of job risk. These deficiencies dilute and deviate the actual job risk, dampening the credibility of the VSL. Viscusi calls for a more robust database of evidence, concluding that VSLs can differ significantly once measurements are disaggregated (Viscusi 2004).
Lastly, a 2008 study by Aldy and Viscusi argues that the effect of age and cohort is not analyzed accurately during wage-risk tradeoffs. The ambiguity behind these factors maybe for a good reason. The authors state that the relationship age and cohort have with VSL is incredibly complex; however, they end up finding the relationship to be an inverted U-shape, signaling that the VSL rises and then falls with age. Aldy and Viscusi state that age and VSL are linked through factors such as "life cycle consumption pattern" while arguing that VSLs vary per age cohort (Aldy and Viscusi 2008; Viscusi 2019).
Many studies have utilized the VSL to make significant and costly efforts to reduce mortality by comparing the anticipated economic costs to the accumulation of saved VSLs. The VSL reported in the reviewed studies was between $1 million to $10.3 million for an average American, and these values are not adjusted for publication selection effects (Bethune and Korinek 2020; Goldstein and Lee 2020; Hammitt 2019, 2020; Pindyck 2020; Robinson et al. 2021). In 2003, Viscusi and Aldy estimated the USA's VSL to be $10 million, based on estimates of the extra wages that workers received to accept increased fatality risk at work (Viscusi and Aldy, 2003). The USA government uses an average VSL for the population (Hammitt 2019; Johansson, 2019; Viscusi 2020). When the VSL was estimated based on age across the 35 cohorts (weighted for the different proportions of the age groups), the value ranged approximately between $300,000 to $37 million with an average VSL of $8,635,355 (Adler 2020).
A comparison by Robinson et al. (2021) between the VSL estimates based on three different approaches: (1) an invariant population-average VSL; (2) a constant value per statistical life-year (VSLY); and (3) a VSL that follows an inverse-U pattern, found that the average VSL estimates were $10.63 million, $4.47 million, and $8.31 million, respectively, when applied to the USA COVID-19 deaths. However, despite the importance of these estimates, the analysis does not address other factors such as income level, change in risk, or uncertainty, among many other factors (Robinson et al., 2020).
Viscusi (2020) addressed the international differences by using income elasticity to estimate an international VSL derived from the USA VSL to provide country-specific global mortality cost estimates. He pointed out the need for both comprehensive and individual-specific estimates of the VSL to reflect the measure's heterogeneity and address the personal valuations of risk besides age and income adjustments.
Therefore, deficiencies exist in the VSL when using the traditional cost–benefit analysis method. The vast majority of the cost–benefit analysis value comes from foregone earnings and does not account for leisure time and utility loss nor preferences for risk reductions. Personalizing the VSL to a particular individual, or even to specific countries, is becoming a primary focus of public health officials during the COVID-19 pandemic.
The VSL can be revisited and measured using the modifications to the theory of the demand for health done by Becker in "Health as Human Capital: Synthesis and Extensions," where Becker presents the theory of human capital and integrated various contributions previously relevant yet ignored ideas into the model. Becker uses the results of the consumer optimization analysis to their utility over time, subject to constraints in budgeting and to investments in health made to change their survivorship at different ages (Becker 2007).
Becker's contributions enable us to calculate the VSL, now definable as the tradeoff between wage and risk, while also accounting for the loss of leisure, differences between average and marginal utilities in combination with losses from foregone earnings.
A more formal representation of the model is presented in his paper by using the expected utility function that is homogeneous of degree γ, simplified to a two-period model, and maximized subject to a budget constraint, as indicated below in the maximization problem:
$${\text{Maximize}}\; u_{0} (x_{0} ,l_{0} ) + BS_{1} \left( h \right)u_{1} (x_{1} ,l_{1} )$$
$${\text{Subject}}\;{\text{to:}}\;x_{0} + \frac{{S_{1} x_{1} }}{1 + r} + g\left( h \right) = w_{0} \left( {1 - l_{0} } \right) + \frac{{S_{1} w_{1} \left( {1 - l_{1} } \right) }}{1 + r}$$
where,
\({u}_{0}:\text{Utility in the current period zero}\);
\({u}_{1}:\text{Utility in period one}\);
\({x}_{i}: \text{are consumption goods at period} i\)
; \({l}_{i}: \text{is leisure at period}\, i\); B: is the time discount rate (time preference) that depends on age. Diseases, and many other factors \({S}_{i}: \text{is the probability of survival to age} i\);
\(g(h): \text{is the expenditure function for health}\);
\(g(h): \text{is the expenditure function for health}\); \({w}_{i}:\text{ is wage rate at period} i\);
\(r:\text{is the interest rate}\).
The maximization problem above, with respect to the x's, l's, and h subject to the constraint, will yield the first-order conditions (1), (2), (3), and (4):
$${u}_{0x}=B\left(1+r\right){ u}_{1x}$$
(1)
$$\frac{{u}_{0l}}{{u}_{0x}}={w}_{0x}$$
(2)
$$\frac{{u}_{1l}}{{u}_{1x}}={w}_{1x}$$
(3)
$$\frac{1}{{1 + r}}(d\log S_{1} /dh)S_{1} u_{1} /u_{{1x}} = g^{\prime}\left( h \right) + w_{0} \left( {1 - l_{0} } \right) + \frac{1}{{1 + r}}(dS_{1} /dh)\left( {x_{{1 - }} w_{1} \left( {1 - l_{1} } \right)} \right)$$
(4)
In the above first-order condition with respect to investment in health (h) represented in Eq. (4), the willingness to pay is given by the right-hand side of the equation, and the change in the probability of survival (ds) is explicitly on the left-hand side. Therefore, the maximization problem developed by Becker concerning consumption goods, leisure, and investments in health, along with the concavity of the utility function, results in the formula (5) as below that gives an estimate of the value of a statistical life life that basically equals full wealth, adjusted upwards for the degree of concavity in the single period utility function1/γ (Eq. (14) in Becker’s paper p. 385):
$$V=\frac{1}{\left(1+r\right)}\frac{1}{\gamma }\left({x}_{1}+{l}_{1}{w}_{1}\right)= \frac{1}{\left(1+r\right)}\frac{1}{\gamma } {C}_{1}$$
(5)
where \({C}_{1}\) is full consumption in period 1.
Therefore, the VSL is the value calculated in Becker’s Eq. (5) above discounted at an interest rate r as shown in Eq. 6 below:
$${\text{VSL}} = V/r$$
(6)
The VSL developed by Becker; represents the tradeoff between wage and risk is given by the marginal costs for investments in longevity and equals full wealth, including the value of leisure weighted by the parameter that indicates the extent of the concavity of the utility function (i.e., the extent of the diminishing marginal utility of consumption that is equal to γ (Details on pp. 382–385, Becker 2007).
The investments in health are made because the increase in life adds to lifetime wealth; however, there is a diminishing marginal utility of consumption when the homogeneity degree is less than one (γ < 1) and investments in health are greater than investments made to maximize wealth. Investments are made because additional spending on consumption adds marginal utility while spending on health adds years to life and average utility; since average utility is greater than marginal utility, total utility increases. A greater utility function's concavity causes a smaller γ, and thus more benefits to invest in health and survive longer.
Using the above VSL formula, the calculations differ for people with different wealth, age, survival probability, and other factors. These factors are embodied in the utility function and reflected in γ that appears in the formula derived for calculating the VSL. The VSL measured by using the formula above depends on the following parameters:
-
1.
The interest rate r
-
2.
The extent of the diminishing marginal utility of consumption that is equal to γ which is between 0 and 1.
-
3.
The full wealth that depends on the wage rate and the total amount of available time for work and leisure.
The novelty of the VSL formula is that the willingness to pay to reduce the chances of dying considers not just the foregone income, but the lost utility that also includes the value of leisure time, as Beker (2007) indicated and that most of the statistical value of life comes not from foregone earnings, but from the loss of leisure time, and differences between average and marginal utilities.
Moreover, the above formula can be used to represent a single individual VSL with his (her) own time preferences, wage rate, interest rate, the extent of the diminishing marginal utility of consumption that is equal to γ, and most important, leisure time, which all depends on the individual's age, occupation, gender, and many other factors reflected in γ.
The introduced VSL formula can be used with averages for individuals or a specific group of people, gender, ethnicity, or population. Hence, a more accurate and valid judgments can be made by comparing the VSL across different groups or even for a specific individual over time.
The VSL developed by Becker (2007) is consistent with values measured by other methods, is believed to solve most, if not all, the deficiencies in the traditional VSL, as Becker's formula does not deviate from the recommendations of a utilitarian or social welfare function. Economists must consider both functions when calculating VSL regarding COVID-19, as suggested by previous research (Adler et al. 2020). A study of the optimal lockdown policy to control fatalities in a pandemic found that the intensity of the lockdown should depend on the fatality rate and the VSL assumed and measured by the cost–benefit analysis in previous literature (Alvarez et al. 2020).