Meta-analysis of Value of Statistical Life Estimates

Abstract

Value of Statistical Life (VSL) is one of the most debatable areas in economics. However, VSL is frequently used as a policy instrument for evaluating various safety, health and environmental regulations. Policymakers have to undertake the difficult task of assigning monetary value to the reduction of various health and mortality risks while analyzing safety policies. Compensating wage differential (CWD) for job risks acts as a reference point for valuing mortality risks while VSL serves as a basis to analyze these benefits of risk reduction policies. However, it has been observed in the recent past that VSL estimates vary substantially across various studies. Therefore, it has become necessary for researchers and policymakers to understand the source of this variation in order to aid policymaking. This paper attempts to bring together some of the emerging issues in VSL literature and presents a meta-analysis that is based on 34 observations from 30 hedonic wage-based VSL studies. The results of this meta-analysis show that certain emerging areas in VSL literature such as worker’s compensation benefits, age and long-term health-related job risk require more emphasis and further examination.

Keywords

Valuations of life/injury value of a statistical life (VSL)risk compensating wage differentials (CWD)hedonic wage method

Introduction

Thomas Schelling (1968) was the first person to coin the term ‘Value of Statistical Life’ (VSL) in his essay titled ‘The Life You Save May Be Your Own’. However, according to Banzhaf (2014), the intellectual origins of VSL lie in a political controversy between U.S. Air Force (USAF) and the RAND Corporation. RAND Corporation was responsible for designing the first aerial attack by USAF on the Soviets. They estimated the optimal combination of atomic bombs and bombers to maximize the damage to the Soviets given the constraint on fissile material. However, in this analysis, RAND Corporation failed to weigh the lives of bomber crew, which led to an extensive debate regarding valuing lives (Banzhaf, 2014). Thus, this event underscored the necessity of assessing the lives of individuals. In the present-day scenario, such assessments become necessary when governments of various countries introduce various types of environmental, health and safety regulations which aim at the reduction of fatality risk. Cost and benefit analysis is commonly used by policymakers to judge whether a policy or a project is feasible to implement. To evaluate any policy, policymakers have to weigh the benefits and other factors against the viability, cost of implementation, etc. Valuing life and fatality risk has always been a debatable issue in economics. The aims of this article are to provide a simple introduction to the concept of VSL, discuss about the dimensions across which VSL estimates are observed to vary and present a meta-analysis of VSL estimates to enquire about and understand the sources of this variation. The article starts with a brief introduction to the concept of VSL. It then describes how VSL can be measured by using a standard approach called hedonic wage methodology. The issues with risk and wage data has been briefly discussed after this, which then leads to drawing our attention towards heterogeneity issues of VSL estimates. The next section presents a meta-analysis of VSL estimates using mixed effects regression model and discusses the results of the analysis. The final section of the article highlights the application of VSL in the field of policymaking and touches upon a few criticisms of VSL.

Understanding the Concept of VSL

Value of Statistical Life refers to small risks to life. According to the definition put forward by Borjas (2013, p. 217), VSL is the amount that people are jointly willing to pay for fatality risk reduction in the expectation of saving one life. Economists commonly refer to the monetary value of fatality risk reduction as the ‘value of life’ and therefore VSL is often misunderstood and misinterpreted as an attempt to put a price tag on an individual’s life. Economists do not value life, because life is priceless. What economists do is try to estimate the monetary value that people themselves are willing to pay for fatality risk reduction of a statistical life and not an identified life. The terms statistical life and identified life are not same. The meanings of these two terms can be easily understood with the help of an example. Assume there is a poor sick kid who needs money for an expensive treatment that will prolong his/her life. When people come to know about it, they can easily collect enough money for the treatment. On the other hand, consider an instance where the government is going to implement a public healthcare programme that would reduce the fatality risk for the people of the city and to finance this programme it plans to raise the taxes. In the first example, the life that would be saved is an identified life, whereas in the second example the life that can be saved could be any random individual from the city and therefore an unidentified life, which can be converted into a statistical life (Schelling, 1968).

Economists observed that market decisions of individuals provide information on how they value their health and fatality risks. Such market decisions may involve purchase of hazardous products like cigarettes or getting engaged in risky jobs. However, these activities entail risks to one’s health and life. Clearly, there must be some aspects of such risky activities which make them desirable. These market choices involve implicit trade-offs between risk and money. For economists, these market choices act as a key instrument to develop the estimates of the VSL (Viscusi & Aldy, 2003). Regulators use these VSL estimates to value the benefits associated with risk reduction policies.

There are many methods to estimate the risk–money trade-off an individual is ready to make, which is in turn used to estimate the VSL. Initially policymakers used human capital approach which was based on the foregone or lost income that would have been earned during an individual’s additional productive lifetime. This method was quite popular until 1982, when a controversy arose during the analysis of a proposed hazard communication regulation by the U.S. government. Occupational Health and Safety Administration used human capital approach for the analysis whereby the costs of implementation exceeded the benefits from the regulation and, therefore, it was rejected. This regulation would have made it mandatory to put danger warning labels for hazardous chemicals at workplace. It is evident that such policy would obviously benefit the involved but the human capital approach yielded opposite results. However, when the value of life estimates were used, the regulation was accepted, since the benefits of that regulation exceeded its cost (Viscusi, 2011). This has been a historical event in the context of the application of VSL in policymaking. Later on, governments of other countries also started recognizing and applying VSL estimates for the evaluation of the risk regulations.

Estimation of VSL Using Hedonic Wage Approach

For valuing death risks, the most commonly used method is the compensating wage differential (CWD) method which arises in the labour market. Adam Smith (1776), the father of economics, introduced the concept and theory of CWD. CWD mainly refers to the sum of money that a worker has to be paid to make him accept a small increase in his death risk. In other words, it is the sum of money which a worker requires to pay to realize a small reduction in his death risk. Smith proposed that job characteristics influence the nature of labour market equilibrium. According to this theory, a job is viewed as a package that includes wage and non-wage job characteristics. CWD therefore arises to compensate workers for the non-wage job characteristics. However, it is difficult to empirically identify these wage differentials mainly because the wage of a worker is affected by several factors. Therefore, economists use statistical models which not only take care of a worker’s productivity but also of other non-pecuniary job characteristics. This is done to segregate the wage– risk trade-off from a multitude of factors that may affect the wage of a worker. Hedonic wage approach is used to estimate CWD. This approach studies the equilibrium risk preferences of workers and their corresponding wage levels (Viscusi & Aldy, 2003).

Hedonic Wage Approach

In 1976, Richard Thaler and Sherwin Rosen developed the modern theory of CWD in which they adopted hedonic wage function approach (Thaler & Rosen, 1976). Hedonic wage treats a job as a bundle of attributes such as characteristics of workers, conditions of work and level of the risk of an accidental injury. Other things remaining equal, risky jobs require wage premiums to attract workers. In this approach, wage acts as a key source of information on work-related health and fatality risk. Hence, it reveals the underlying implicit wage–risk trade-off which is used to estimate VSL.

Equilibrium in Hedonic Labour Market

This section briefly explains how the hedonic wage function is derived from the framework of CWD. It is known that the demand for labour decreases with rise in the total cost of hiring workers, because providing safer working conditions to workers is a costly affair for firms. In Figure 1, wage offer curves OC₁ and OC₂ exhibit a positive relationship between risk level and wage. Given any particular level of risk, workers will choose those wage–risk combinations on the market offer curve which offers them the highest wage. The outer envelope of these curves is called the market opportunity locus w(p). Thus, the supply of labour is partially dictated by their choice over wages and risk. By imposing various minor restrictions on the preferences of workers, the labour supply can be well characterized.

Figure 1.

Source: Viscusi and Aldy (2003).

Let us consider Von-Neumann–Morgenstern expected utility approach with two state dependent utility functions. Let good health state represent utility u(w) and ill health (due to injury) represent utility v(w). Good health is preferred to ill health. So,

u (w) > v (w)

(1)

It is assumed that workers prefer higher wage given the risk level; therefore, the marginal utility of income is positive, that is,

u^{'} (w) > 0 a n d v^{'} (w) > 0

(2)

Workers aim is to maximize their expected utility by choosing from possible wage–risk combinations along some market opportunities locus w(p). Therefore, the optimal choice of wage and job risk for a worker is obtained at the point of tangency of the expected utility curve of the worker and the wage offer curve of the firm. Worker 1 maximizes his expected utility at the point of tangency between OC₁ and EU₁ while worker 2 maximizes his expected utility at the point of tangency between OC₂ and EU₂. Thus, all the optimal wage–risk choices associated with a given worker’s expected utility locus must satisfy

Z = (1 - p) \times u (w) + p \times v (w)

(3)

The optimal wage–risk choice is given by

\frac{d w}{d p} = - \frac{Z p}{Z w} = \frac{u (w) - v (w)}{(1 - p) u^{'} (w) + p v^{'} (w)} > 0

(4)

Here, dw/dp is the slope of hedonic wage function which shows that wage–risk trade-off equals the difference in utility levels in the two states divided by the expected marginal utility of income. This estimated slope does not only resemble a worker’s marginal willingness to accept risk but also his marginal willingness to pay for more safety. It also resembles a firm’s marginal cost of more safety as well as reduction in its marginal cost from an incremental increase in risk.

However, it should be kept in mind that the estimated wage–risk trade-off locus w(p) does not indicate how a worker must be compensated for a non-marginal or a large variation in risk. In such a case, worker’s wage–risk trade-off will alter since the appropriate wage–risk trade-off must be made along the worker’s expected utility locus and not the estimated market wage–risk trade-off (Viscusi & Aldy, 2003).

Econometric Model and Formula for Estimation of VSL

The hedonic wage function is obtained from the human capital theory whereby Mincer’s earnings function is modified by incorporating risk into the wage equation. It has been observed that the wage equation is commonly used by researchers to estimate wage–risk relationship in labour market (Viscusi & Aldy, 2003). Equation (5) shows that worker’s wage is a function of fatal risk denoted as p_i, non-fatal risk denoted as q_i, a vector of worker and firm characteristics denoted as X_ki and a random error term ϵ_i.

W_{i} = α + β_{1} p_{i} + β_{2} q_{i} + Σ k γ_{k} X_{k i} + �_{i}

(5)

where β₁, β₂ and γ_k are the parameters that are required to be estimated using regression analysis.

However, the basic hedonic equation suffers from endogeneity and heterogeneity problems, which need to be corrected. Very few studies have corrected these problems. Shanmugam and Madheswaran (2011) has accounted for these problems in their study based on the Indian labour market.

Formula for Estimation of VSL

The wage–risk trade-off is given by the coefficient of the risk variable, that is, ${\hat{β}}_{1}$ , which can be obtained from Equation (5). In the VSL literature, it is common to measure the fatality risk or the average probability of death per 100,000 workers while assuming that, on an average, workers work for 2000 hours in a year. Given this information, the estimated VSL at mean wage level ( $\overset{─}{w}$ ) is estimated using the following simple formula:

V S L = {\hat{β}}_{1} \times (\bar{w}) \times 10, 000 \times 2, 000

(6)

Sources of Data for VSL Estimation

The previous section elaborated the procedure for the estimation of VSL. This section discusses the sources of data on two important variables, that is, risk variable and wage of the workers, which are essential for the estimation of VSL.

Data on Risk Variable

While reviewing VSL studies, it was found that there exists no single source of information from which researchers can obtain the data on risk variable, the details about individual workers and the characteristics of the firm. Hence, information on these components are mostly obtained by matching observations from several sources and by deciding on how the data (which are mostly reported at aggregate levels) can be combined in the best possible way (Dockins, Maguire, Simon, & Sullivan, 2004). Table 1, summarizes a few sources of risk data that are frequently used by researchers in the United States of America (USA) and the Indian VSL studies.

Table 1.

Sources of Risk Data for the USA and India

Country	Source of risk data	Information provided
USA	Society of Actuaries	Average annual fatality risk for 1/1,000 across 37 occupations
	Bureau of Labour Statistics	Aggregate data on fatal and non-fatal injury risk by industries using 2–3 digits ICC
	National Institute of Occupational Safety and Health	All fatal job injuries based on death certificates
	Census of Fata Occupational Injuries	Job fatality by 4 digit ICC from worker’s compensation records, various Federal and state agencies and death certificates
India	Administrative Report of the Chief Inspector of Factories	Number of blue collar male workers, number of deaths and accidental injuries in a year at 2 digit NIC level
	Indian Labour Yearbook	Fatality rates and non-fatal injury rates by industry

Source: Compiled by the authors.

For the job risk variable, two types of risk measures can be used in the estimation of VSL. While one type of risk measure captures the objective risk associated with an occupation or a specific industry, the other pertains to the subjective evaluation of the mortality risk by the workers themselves. It has been a standard approach to use either industry-specific or occupation-specific measures of risk that reflect average fatalities over a period of time (Viscusi & Aldy, 2003). In his grand compilation of studies Viscusi (2003) found that very few studies used the subjective risk preferences of workers. Madheswaran (2004) used the subjective risk preference of workers and found that about 90 per cent of the respondents considered that their job exposed them to hazards.

Organized and Unorganized Sector Risk Data

Another important and closely linked issue emerging in VSL literature concerns the availability of risk data for workers from the unorganized sector. The share of unorganized sector in developing countries is quite huge. Risk data is mostly compiled and reported by the government or government agencies and it pertains only to the objective risk measures for the organized sector workers. Therefore, objective risk data is available for only the organized sector in the developing countries, which leaves out the information on a vast majority of the unorganized sector workers. This is a major shortcoming of the secondary risk data in developing countries. In order to fill this lacuna, what researchers can do is collect data on the subjective risk of workers through primary survey, which can give a rough idea about the wage compensation and risk premiums in the unorganized sectors of the developing countries. However, it has to be kept in mind that a problem of endogeneity may arise between subjective risks and willingness to pay (WTP), which needs to be controlled in order to get an unbiased estimate of the risk coefficient.

Wage Data

Wage is taken as the dependent variable for all labour market VSL studies and, therefore, it is an important variable. It is a common trend among economists to use either weekly or annual hourly wage of workers as the dependent variable in regression analysis. For studies in the USA, researchers had access to the information on the wage of the workers through various surveys like Survey of Working Conditions, Quality of Employment Survey, Current Population Survey by Bureau of Labor Statistics, Panel Study of Income Dynamics and decennial census data. For India, Alberini and colleagues obtained the data on average daily wage of workers from the Occupational Wage Survey (OWS), conducted by the Indian Labour Bureau (Alberini, Cropper, Simon, & Arora, 1999). The OWS reports average daily wage of full-time manual workers, by occupation, for various industries on the basis of 3-digit NIC codes. Madheswaran (2004) obtained wage data from wage bills and records of the firms where the respondents worked. He used after-tax hourly wage of the workers (assuming that a worker worked for 2000 hours in a year) as a dependent variable of the wage regression model.

Heterogeneity Issues of VSL

VSL is observed to vary on various dimensions like individual risk-taking behaviour and individual characteristics such as age, income, gender, race, immigrant status, etc. Thus, there is no uniform VSL. Therefore, VSL estimates have to be adjusted for these dimensions (Viscusi, 2011). In the context of this article, it is very crucial to inform ourselves with these dimensions across which VSL varies before conducting a meta-analysis of VSL estimates. Understanding these sources may in turn aid in understanding the cause of such variation. It is quite similar to what Jiddu Krishnamurti (1973) said, ‘The solution of a problem lies in the understanding of the problem….’

VSL and Individual Risk Taking Behaviour

Many VSL studies have found that people who are risk lovers in their personal life tend to select jobs that are risky. By undertaking such risky activities, this kind of people indicates that they care less about their lives or they implicitly reveal the lesser value they assign to their lives compared to an average individual in a society (Viscusi, 2011). Risk averse people have high wage–risk trade-offs and they are located on the steeper portion of the hedonic wage function. In other words, they have a high VSL. In contrast, risk loving workers have low wage–risk trade-off and they are located on the flatter portion of the hedonic wage function. Therefore, they have low VSL. However, the above discussion is based on a key assumption that the labour market opportunities faced by one worker is independent of the choices of other workers (Viscusi, 2011).

VSL and Segmented Labour Market

In this section, the focus is on the variation in the VSL estimates that arises due to factors such as race, colour, caste, gender of the worker and so on. A study (Viscusi & Hersch, 2008) showed that even the smoking status of the workers does affect the VSL estimates. This article brings together and discusses some of the factors that can cause considerable variation in the estimates of VSL.

The presence of segmented labour market is an emerging issue in the VSL literature. By segmented labour market, what is meant is that within the labour market there are categories that further differentiate and subdivide the workers. Therefore, different workers will face different market opportunity locus and will consequently have different market equilibria. Viscusi (2011) observed this phenomenon and found that it is possible that the CWD of two workers can vary even though they are exposed to the same risk level. It is also possible that one group of workers faces lesser level of risk and earns a higher wage differential, while another group of workers faces higher level of risk and earns a lower wage differential.

Two important factors that give rise to separate hedonic wage functions for workers are the race and the colour of the worker. In many studies undertaken in the USA, another key difference occurs due to the origin of the worker, that is, whether the worker is a US citizen by birth, or is a Mexican, or Hispanic or some other immigrant.

The caste of workers has been observed to considerably affect their CWD and VSL estimates in the context of a developing country like India. Madheswaran (2004) found that the caste of a worker had a positive and significant influence on their wage. This result implied that the workers belonging to backward castes earned a higher wage premium compared to workers from other castes in the blue collar manufacturing industries. Shanmugam and Madheswaran (2011) found similar results which highlights the fact that the workers belonging to lower castes tend to earn more in risky jobs.

Discrimination in job market opportunities on the basis of the gender of workers has been one of the most controversial and debatable issue for a long time. As more and more women are becoming part of the labour force, it is very important to study and enquire about the evidence of a segmented labour market based on the gender of workers. Costa and Kahn (2004) found that in 1940, male workers had an average risk exposure that was nearly 4 times higher than that of female workers and in 1980 it was higher by two times.

Cultural aspects can also play a significant role in this context. Brajer and Rahmatian (2004) found that an ancient cultural tradition from Iran called ‘the Diye’ strongly affected the WTP for reduction in premature death. In this tradition, compensation is provided by the perpetrator to the victim or his/her family in case of unintentional death of an individual.

VSL and Union Status of Workers

Other than negotiating wages, unions of workers can also play an effective role in demanding better working environment, greater job safety, higher compensation in case of injury or death of workers, higher pensions, etc. Unions can also play an important role in informing workers about the level of risk involved in a particular job. While international evidences send a mixed signal, results from the U.S. labour market based VSL studies indicate that a worker’s membership of a union and higher wage–risk trade-off is positively correlated (Viscusi & Aldy, 2003). Madheswaran (2004) observed that unions influenced fatal risk negatively but have a positive influence on non-fatal work-related injury risks. Besides, member workers were found to be better aware of the risks associated with their jobs compared to non-member workers.

VSL and Workers’ Compensation Benefits

Workers’ compensation can play a major role in the estimation of VSL. Excluding workers’ compensation benefit from estimation of wage premium for risky jobs will lead researchers to biased results. There are two forms of compensation given to any worker—one is ex-ante compensation and the other is ex-post compensation. Ex-ante compensation comprises CWD for undertaking job risks while ex-post compensation takes the form of workers’ compensation benefits like medical outlays and certain portion of their lost wages.

VSL and Age

Earlier studies on VSL had very little evidence to support the statement that VSL varies with age. As more studies focused on enquiring about the relationship between age and VSL, new results emerged. This in turn created more controversies on this issue among the researchers. While most recent studies have shown an inverted-U relationship between age and VSL, Alberini, Krupnick, Cropper, Simon and Cook (2001) found little, evidenced on the variation of WTP with the age of the respondent. Blomquist (2004) suggested the use of multi-period model with uncertain lifetime since it is more realistic and helpful in deriving an individual’s WTP for changes in fatality risks at various phases of his life. Aldy and Viscusi (2004) made a significant contribution in the life-cycle VSL literature and have altered the standard life-cycle models.

Older people are expected to pay less for fatality risk reduction compared to younger ones, because they have fewer remaining expected life years. However, the theory may not consider lower VSL for older people (Krupnick, 2007). In this context, many economists proposed that VSL should be rather converted to Value of Statistical Life Year (VSLY). VSLY estimates are based on how a worker values the discounted years of remaining life. If in each year we assume a time-invariant VSLY, then

V S L = \frac{V S L Y}{r} - \frac{1}{{(1 + r)}^{​}} [\frac{V S L Y}{r}]

(7)

Here, r is the rate of interest used to discount worker’s years of life; L is the worker’s remaining life expectancy. From Equation (7), VSLY can be obtained as follows:

V S L Y = \frac{r V S L}{[1 - {(1 + r)}^{​}]}

(8)

Aldy and Viscusi (2007) found that the WTP for fatality risk reduction in younger cohort will depend on the lifetime income. On the other hand, when VSL is adjusted for the cohort effects, older workers were found to have higher VSL (Aldy & Viscusi, 2008). Researchers are trying to explore possibility of multiple equilibria with respect to the age of workers. Incidentally, the position of a worker on wage–risk curve is observed to change as he ages (Aldy & Viscusi, 2007). On the other hand, employers can offer different wage–risk combinations according to the age-specific safety productivity. Thus, not only a worker’s expected utility curve but the firm’s wage offer curve also varies with the worker’s age (Aldy & Viscusi, 2007). Another noteworthy study was undertaken by Aldy and Viscusi in 2008, wherein they attempted to look at the relationship between age and VSL from the point of view of life expectancy.

Long-term Health-related Job Risks and VSL

Another important yet controversial issue related to estimation of VSL deals with long-term health-related job risks. Workers themselves place certain discount rates on inter-temporal death risks. A young worker stands to incur more loss because of his/her death compared to that of an old worker. Therefore, appropriate discount rate is required to estimate long-term health benefits. Health is not directly traded in an inter-temporal market. In the absence of perfect capital markets, risk free rate of time preference of society cannot be used as an appropriate discount rate for all benefits. The question that remains is whether to use the same rate or different rates of time preference for health impacts. This issue is solved by various studies empirically.

Income and Wealth Effects on VSL

Previous studies sought to establish theoretically as well as empirically that there exists a positive relationship between a worker’s income level and his VSL. Since VSL should rise with per capita income of a country, poorer developing countries tend to have lower VSL compared to developed countries (Viscusi, 2011). In any society, the finest jobs pay more to workers. However, one question still lingers: Is it possible for wage to increase as the riskiness of job increases? Hedonic wage theory has long been criticized on the basis of this contradiction. According to the hedonic wage theory, a job is treated as a bundle of attributes and one such attribute is the job risk. Let us consider two jobs: the job of a CEO of a company and that of a machine operator. The CEO is paid far higher than a machine operator working on the factory floor. However, if we look at these two jobs from the perspective of having health or death risk, the machine operator’s job is riskier than the CEO’s job. According to the hedonic wage theory, the machine operator should ideally be paid more than the CEO for undertaking such job risks. However, in reality, finest jobs like that of the CEO are paid more in the society. Therefore, the contradiction that arises here is that the finest jobs are paid more rather than riskier jobs.

It is known that safety is a normal good which implies that as wealth of an individual increases, his demand for safety should increase. Hence, a richer individual will demand for a safer job. This also shows that wealth of a worker affects the job risk choice of an individual (Madheswaran, 2004). In the Indian context, Shanmugam and Madheswaran (2011) found wealth to have a negative and significant influence on fatal as well as non-fatal job risks.

Meta-analysis of VSL Estimates

Meta-analysis involves application of various statistical procedures to a set of studies for integrating and synthesizing them so as to utilize all the information they contain. It is different from a simple literature review since it provides a basis for an in-depth scientific analysis of the results of various studies. As mentioned earlier, VSL estimates are observed to vary from one study to another. In order to investigate and understand the major sources of this variation, a meta-analysis of VSL studies is performed here. Different studies use different samples, therefore in many cases variation in results arises due to variance in sampling or variance in estimation. Although methodological differences are observed to cause certain amount of variations, several unobserved and uncontrollable factors also influence the outcome. In such a scenario, the application of mixed effects regression model is found to be suitable for meta-analysis since it can account for this unobserved heterogeneity. Mixed effects regression model hypothesizes that estimation variance is not the only source of variation (Bellavance, Dionne, & Lebeau, 2009). The mixed effects regression model is given as follows:

V S L_{j} = β_{0} + \sum_{k = 1}^{p} β_{k} ​ X_{j k} + u ​_{j} + e ​_{j}

(9)

VSL_j is the dependent variable which has to be estimated for each of the m studies in the sample. β₀ is the constant, X_jk is the vector of characteristics of study j which estimates VSL_j, β₁, β₂…. β_P are the regression coefficients which capture the fixed effects of study characteristics on VSL. This model has a joint error term containing a random error (u_j) and an estimation error (e_j) which are independently distributed with zero mean and variance, σ²_u and σ²_VSLj respectively.

According to Raudenbush (1994), ordinary least squares (OLS) is unsuitable for estimating Equation (9) since mixed effects regression model is based on the assumption of heteroscedasticity. Thus, weighted least squares (WLS) should be used for estimation of mixed effects regression where optimal weights are given by

w j = \frac{1}{σ_{V S L j *}^{2}} = \frac{1}{(σ_{θ}^{2} + σ_{V S L j}^{2})}

(10)

If σ²_θ is zero, then the fixed-effects model will be adequate and the optimal weight will be, W_j = 1/ σ²_VSLj. Value of σ² _VSLj can be calculated from data contained in the studies itself. However, the value of σ²_θ is not given in these studies and so needs to be estimated.

An alternative to using Equation (10) as weight is to use the inverse of sample size of studies as weight for meta-analysis. In our meta-analysis, inverse of sample size of the VSL studies has been used as weights for estimating Equation (9) using WLS.

Analysis of Sample

In this study, a meta-analysis of 34 observations obtained from 30 hedonic wage studies has been performed using mixed effects regression model, following Bellavance et al. (2009). The sample comprises a few studies from Bellavance et al. (2009) and a few recent studies from developed as well as developing countries. Since estimated VSL values are used as dependent variable, those studies from which VSL could not be calculated (due to absence of information on certain variables) were removed from the meta-analysis. The explanatory variables included in the analysis are average probability of death, average income, white worker’s sample, union sample, worker’s compensation for injury, year of the publication of the study, age of workers and square of the age of workers.

Steps Followed During Meta-analysis

STEP I: Using the following formula, VSL for each study in the sample is re-estimated.

V S L = \frac{\hat{φ} * (S a m p l e a v e r a g e o f a n n u a l i n c o m e)}{U n i t o f p r o b a b i l i t y o f d e a t h}

(11)

Here, $\hat{φ}$ is the coefficient of risk of death.

STEP II: Re-estimated values are then converted into USD 2005 (in the present study, 2005 is considered as the base year for conversion of VSL estimates to US dollar). For non-American studies, the purchasing power parity (from Penn World Table 7.1; Retrieved [April 23, 2016] from University of Pennsylvania: https://pwt.sas.upenn.edu/php_site/pwt71/pwt71_form.php) has been used as the exchange factor to convert the respective country’s currency to USD. For American studies, consumer price index (United States Government Printing Office, 2013) has been used to adjust the VSL and average income values to USD 2005.

STEP III: Finally, the regression is run according to Equation (9).

Results of Meta-analysis and Discussion

The dependent variable used in our analysis is the log of VSL which has been estimated from each study in the sample. As shown in Table 2, the log of average income has a positive and significant relation with the log of VSL, which implies that people with high income has higher WTP.

Table 2.

Results of the Meta-analysis

Variables	Coefficient	t value
Constant	14.14846 *	8.82
Average income (Log)	0.2576024**	2.69
Average probability of death	−0.4437536*	4.86
Compensation to workers	−0.6617426**	1.98
White workers sample	0.8888838***	1.65
Union sample	0.3663709***	1.78
Age	−1.318973**	2.22
Age²	1.511033**	2.18
Year of Publication	0.00713**	2.31
N	34
REML estimate of between study variance ( $\overset{?}{τ}$ ²)	0.89
% residual variation due to heterogeneity (I²_res)	100%
Proportion of between study variance explained (Adjusted R²)	65.72%
Joint test for all covariates	8.91
With Knapp-Hartung modification (Prob > F)	0

Source: Compiled by the authors.

Notes: *, **, *** indicates statistical significance at 1%, 5% and 10% level, respectively.

While the theory tells us that there exists an ambiguous relationship between VSL and average probability of death, our results imply a negative and significant relation between these two variables. Bellavance et al. (2009) also observed a similar relationship in their meta-analysis for some of the specifications.

Compensation to workers has a negative and significant relationship with VSL. This result is in line with the results obtained in meta-analysis by Bellavance et al. (2009). Studies that incorporate measures for worker’s compensation yield, on average, a VSL which is lesser than the studies that do not. Thus, this result confirms that workers who derive benefit from compensation for job injury demand lesser risk premium for a risky job.

A sample that entirely comprises white workers was observed to have higher VSL compared to other samples; however, the estimated parameter is statistically not significant. Therefore, we cannot come to any conclusion regarding the presence of racial discrimination in job market. This result is, however, contrary to the one obtained by Bellavance et al. (2009).

It has been mentioned earlier that international evidence of effect of unionization on workers’ WTP has been quite mixed. In our analysis, union sample was observed to have a positive relationship with VSL. However, the estimated parameter was not significant. Bellavance et al. (2009), on the other hand, found a negative relationship between these two variables though their parameters were not significant as well.

The effect of a worker’s age and age squared on VSL is found to be significant. While age is found to have a negative relationship with VSL, the square of age has a positive relationship with VSL. Our results show a U-shaped relationship between age and VSL which implies that VSL rises with the age of workers. This result is in line with the results of earlier VSL studies.

The year of publication is observed to positively and significantly influence VSL estimates. This implies that VSL estimates tend to increase when the year of the publication of the study is later than earlier.

The aim of conducting meta-analysis of VSL studies is to investigate the major sources of variation observed in VSL estimates from different studies. The unobserved heterogeneity is found to be a major cause of variation in VSL estimates (since I²_res is 100%). The model explains 65.72 per cent of the variance between two different studies; on the other hand, the remaining variance (given by $\overset{?}{τ}$ ²) appears small at 0.89. Thus, it can be concluded that unobserved heterogeneity plays an important role in causing variations in the VSL estimates across different studies. A joint test is also conducted to test the null hypothesis that the coefficients of the covariates are all zero. The result indicates that we can reject the null hypothesis and say that the covariates are all non-zero.

Conclusion

The concept of VSL has long been misunderstood and misinterpreted. Most of the criticisms are based on the misperception that VSL attempts to assign a price tag on human life. Lisa Heinzerling viewed estimation of VSL to be impractical from the technical point of view (Brannon, 2004–05). According to The Sceptical Economist (2014), problem arises in the calculation of VSL, because economists make a bold assumption that individuals are informed about the risks associated with a job. In reality, such information is hardly available. Besides money and probability of death, there are many other factors which affect an individual’s daily life decisions (The Sceptical Economist, 2014). However, leaving all these criticisms aside, regulators and researchers continue to use and apply VSL worldwide for the evaluation of various risk regulations. VSL approach has facilitated meaningful risk regulation policies for a very long time. Therefore, it is considered as a correct economic approach, which places a considerable value on the lives that will be saved unlike other approaches such as the present value of lost earnings approach (Viscusi, 2011).

At present, the application of VSL is not restricted to risk regulation analysis but is also used for various other areas of health and environmental studies. The availability of more and better data on fatal risk and other surveys has further expanded the application horizon of VSL. Policymakers and researches are focusing on heterogeneity of VSL which is an emerging issue in the VSL literature. Several works have already been undertaken in this area leading to more refined VSL estimates that have aided policymaking and policy evaluation. However, many dimensions are yet to be explored. Meta-analyses aid in understanding the variability in the estimates of VSL across studies and our meta-analysis shows that unobserved heterogeneity is the main cause behind this variation. The results from our meta-analysis also show that the log of average income, average probability of death, year of publication, compensation for injury, worker’s age and square of worker’s age significantly influence the VSL estimates. On the other hand, white workers’ sample and union sample did not show significant influence on VSL estimates.

While many studies about VSL exist for the developed countries, the number of VSL studies available for developing countries like India is still less. One area that is yet to be explored is related to the risk preferences of the workers from unorganized sector in developing economies. In the Indian context, very few VSL studies are available. Those studies have highlighted various issues like worker’s wealth effect, effect of worker’s compensation benefits, or effect of worker’s age on VSL and long-term health-related job risk. However, a huge gap still exists in India that needs to be filled by undertaking more VSL studies. There is also the need to update and re-estimate what has already been done in the Indian studies so far. One area that is still underexplored in the Indian context is the effect of age on VSL with respect to various age groups through using fatality risk data by age and industry (which is considered to be a better measure to determine the effect of age on VSL) as is done by Aldy and Viscusi (2008). The presence of segmented labour market in India on the basis of various dimensions too needs to be further examined. Many researchable areas in the field of VSL are yet to be studied further. Such issues will be of great interest to analyze and in turn provide better safety, health and environmental policies.

Footnotes

Acknowledgements

This manuscript has been derived from Working Paper No. 362 by ISEC, Bangalore issued in 2016 and it is titled as ‘Value of Statistical Life: A Meta-Analysis with Mixed Effects Regression Model’, written by the same authors. We are thankful to Prof. K. R. Shanmugam for his valuable comments and suggestions on this paper.

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