The first two sections of this appendix provide additional details about the contingent-valuation analysis presented in Chapter 8. Section 1 discusses the econometric model used to conduct the parametric maximum-likelihood analysis, while Section 2 discusses the basis for our estimates of the effects of household characteristics on the probability of supporting the hypothetical violence-reduction program. The third section presents alternative estimates for what the American public would pay to eliminate gun violence based on values taken from studies of wage premiums for risky work and from jury awards in personal injury cases.
Contingent-Valuation Estimates: Econometric Approach
Our empirical strategy is based on the framework outlined by economists Trudy Cameron and Michelle James.1 Let Yi equal the (unobserved) WTP value that respondent (i) has in mind when answering the first and second referendum questions in the NGPS. The respondent will answer in the affirmative to the first referendum question (I1i = 1) if the “price”’ of the program in the form of higher taxes (t1i) is not greater than the respondent’s WTP (Yi ≥ t1i). Similarly the respondent will support the program in the follow-up CV question (I2i = 1) if the new price t2i is less than WTP (Yi ≥ t2i), where t2i is equal to double t1i if I1i (p.182) = 1 and half of t1i if I1i = 0. We initially assume that Yi is log-normally distributed (equation E1), which constrains WTP to be positive.
From this setup we can estimate household WTP using the “interval-data” or “double-bounded” model developed by economists Michael Hanemann, John Loomis, and Barbara Kan-ninen.2 The probabilities for the four possible joint outcomes for the first (I 1i) and second (I2i) referendum questions are given in equations (E2) through (E5) where F represents some cumulative distribute function. (Recall that with the NGPS data, t2i = 2t1i if I1i = 1, and t2i = 0.5t1i if I1i = 0).
We obtain estimates for the parameters of this model by applying maximum-likelihood estimation (MLE) to the log-likelihood function in equation (E6). (p.183)
The coefficient estimate for the variables log t1i and log t2i is an estimate for 1/σ, which in turn allows us to identify an estimate b for the parameter β. Calculating the standard errors for mean and median household WTP is complicated by the fact that our estimate for b is really the ratio of two estimates—the estimated value for β/σ divided by an estimate for 1/σ. The usual standard error formula for a linear predictor evaluated at some value of the regressors x0 is given by equation (E7).
In the model without covariates, estimation of the formula in equation (E7) is simplified somewhat because b is a scalar rather than a vector, so V is also a scalar equal to the variance of b, x0 = 1, and (E7) simplifies to (E8).
The complication in our case comes from the fact that b is actually the ratio of two estimates b′/s′, where b′ is an estimate for (β/σ) and s′ is an estimate for (1/σ). In this case the variance for b = b′/s′ can be approximated by the formula given in equation (E9).3
The final complication is that (E9) gives us the variance for the estimated mean of the natural log of Y (WTP), while ultimately we are interested in the variance of the predicted mean of the untransformed WTP. With E[ln Y] = b and Var( E[ln Y]) = V then the variance of E[Y] is given by equation (E10).4 (p.184)
With a point estimate and standard error for b in hand, we can calculate societal WTP. If w1 represents the NGPS sampling weight for household (i), which equals the number of households in the population that each sampled household represents, then estimated societal WTP is given by equation (E11). While b provides an unbiased estimate for the expected value of log WTP, for a log-normal variable the mean of WTP itself will be given by exp(b) × exp(0.5σ2).5
Detailed Results of the Parametric Analysis
The main findings from the parametric maximum-likelihood analysis are presented in Chapter 8 and suggest that the average household in America is willing to pay on the order of $200 per year to reduce the use of guns in assault by 30%.
In addition to these main results, the effects of the different covariates on household WTP are also of some interest, since these coefficient estimates provide information relevant to both the credibility of the CV responses and the distributional effects of programs to reduce or eliminate gun violence. The effects of the household-level covariates included in our maximum-likelihood model are shown in Table D1. The numbers reported in the table represent the difference in the probability of a supportive vote for the group indicated in the left column versus the comparison group. The presence of children within the home increase the WTP of respondents to reduce gun violence; either respondents are willing to pay more to enhance their personal safety when they have children, or they are reporting on household rather than individual WTP, or both. Finally, respondents in households with guns are less likely to support programs to reduce gun violence. (p.185)
Table D1 Coefficient Estimates from MLE Estimates, from NGPS Contingent Valuation Referendum Data
Effect of household background variable on probability of voting “yes” to support program to reduce gun violence by 30%
(White is comparison group)
(South is comparison group)
number of children under age six
number of children between ages 6 and 17
number of adults
(less than $20,000 is comparison group)
Gun in home
Notes: Figures are in 1998 dollars. Author calculations from applying maximum likelihood estimation to equation (3) for the 1997 gun survey data and equation (11) for the 1998 gun survey data, under the assumption that WTP is normally distributed.
(**) Statistically significant at the 5% level.
(*) Statistically significant at the 10% level.
(p.186) Sensitivity Analysis for the Parametric Analysis
In our baseline parametric analysis, we use household-level covariates because we interpret the CV responses as reflections of household (rather than individual) WTP. If different individuals within the home would report different WTP values, then our estimates should still be unbiased (since adults are randomly selected from households) but may be inefficient. To explore this possibility, we reestimated our preferred MLE model after restricting the sample to married respondents and including an indicator for the respondent’s gender. While the coefficient estimate for an indicator variable for husbands is negative and statistically significant, inclusion of this variable serves to reduce estimated mean WTP by less than 7%.
We also find that our estimates are fairly robust to assumptions about the distribution of WTP. Reestimating equation (E6) above (with covariates) under the assumption that WTP has a log-logistic (rather than log-normal) distribution produces an estimated mean WTP of $206. Using a normal distribution, which allows WTP to be negative, produces an estimate of $213.
One concern with these CV data is the possibility that responses to the follow-up CV question are influenced by the initial question. As Cameron and Quiggin note, respondents may become more certain about their response to the second rather than first question because they have had more time to reflect on the public good in question.6 Alternatively, respondents may believe that the first question provides information about the actual average cost of the public good and may then react negatively to the second question that asks the respondent to pay “more than it costs.” The descriptive statistics presented in Table 8.1 provides some evidence to support this second effect. For example, Table 8.1 shows that 69% of respondents who are asked about a $100 tax increase in the first question will pay this much to support the program, though only 51% of those who are asked about a $50 increase in the first question will support a $100 tax increase (76% × 67%).
To address the possibility that the respondent is sensitized by the first CV question, and thus that the first and second (p.187) questions produce observations from slightly different WTP distributions, we follow Cameron and Quiggin and reestimate WTP using a bivariate probit model. The bivariate probit model allows for different means for the first and second WTP values (β1 ≠ β2), as well as separate error processes that have different variances (σ1 2 ≠ σ2 2) and are only imperfectly correlated (Corr[u1i, u2i] = ρ 〈 1).7 While the bivariate probit model affords greater flexibility than the MLE model given by equation (E6), this strategy comes at the cost of less precise estimates8 and makes interpretation of the results somewhat complicated. Our bivariate probit estimates suggest a mean WTP of $309 for the first referendum question and $209 for the second. If responses to the first CV question are more accurate, then the estimates presented in Chapter 8 may somewhat understate societal WTP.
Another concern that commonly arises with CV studies is that of “protest zeroes,” defined as cases in which the respondent rejects the hypothetical market scenario even though her true WTP exceeds the stated “price” of the referendum.9 The proper definition of protest zeroes is complicated in our application. Fairly uncontroversial is the case of tax protestors— those respondents who object to financing the program out of tax revenues, but who would be willing to pay the stated amount to achieve a 30% reduction in gun violence if the program were financed by some other means. One possibility is to identify as tax protesters the 24% of respondents who “strongly agree” with the survey question that “taxes are too high.” When we reestimate our model without these respondents in the sample—which is the preferred method for dealing with protesters10—our estimate is only 13% higher than the preferred estimate of $239 reported in Chapter 8.
More complicated are cases where the respondent objects to the mechanism for reducing gun violence, rather than the mechanism for financing the program. The NGPS asks about programs that target the illegal use or transmission of firearms, which in turn should reduce gun violence holding the overall crime rate constant. Respondents who object to these interventions should only be counted as protest zeroes if alternative interventions exist that could plausibly reduce gun crime without (p.188) reducing the overall crime rate, which is a debatable proposition. In any case, we classify as intervention-protesters those who “strongly disagree” that “the government should do everything it can to keep handguns out of the hands of criminals, even if it means that it will be harder for law-abiding citizens to purchase handguns.” Excluding these respondents produces only a 7% increase in WTP compared with the $239 figure from Chapter 8.
The Quality-of-Life Approach
Previous studies attempt to approximate what the public would pay to reduce gun violence by importing estimates for the value per statistical life or injury from generic studies of marketplace behavior, or from jury awards.11 This approach has become increasingly common in the health-economics literature. We use this approach to value unintentional shootings and gun suicides, since our contingent-valuation survey provides no information on gun injuries that do not result from an assault. We also estimate the value of reducing gun assault as a basis for comparison with our contingent-valuation-survey results.
We begin by discussing some of the complications that arise from trying to infer what people would pay to reduce gun violence from other studies of the value of life saving and workplace safety.
Nonrepresentativeness of Gunshot Victims
Studies of marketplace behaviors rely on samples of people who may have different attitudes toward risk than those who are at high risk for gunshot injury. For example, estimates for wage-risk tradeoffs are typically derived using samples of workers in their late 30s or early 40s. In contrast, most victims of gun homicide are quite young, while victims of gun suicide are frequently quite old (Chapter 2). If WTP to reduce the risk of death is inversely related to one’s age, as many economists assume, then wage-risk studies may overstate the value of one statistical (p.189) life to those at highest risk for gun suicide and understate the value for those at greatest risk of gun homicide.
One solution to this problem is to convert estimates for the value of one statistical life into estimates for the value of one statistical life-year.12 The best available evidence suggests an estimate of $70,000 to $175,000 per year of life,13 with $100,000 taken to be a reasonable mid-point.14 What remains somewhat controversial is whether the life years that are saved by public policy interventions should be “discounted” to account for when the life saving occurs. Almost all economists agree that a program that costs $10,000 today and produces $10,000 in cash benefits ten years from now is not a good investment, since the $10,000 could instead be placed in an interest-bearing account that would yield some greater amount in the future. The same logic leads most economists to discount the dollar value of any future health benefits that are achieved by government interventions. Yet not everyone agrees with this logic. Some observers are (rightly) concerned that discounting the value of lives saved off into the future favors those who are currently alive at the expense of future generations—benefits to people’s health or the environment that occur 100 or 1,000 years from now will receive very little weight in an evaluation that discounts future benefits at some nonzero rate.15 A compromise solution is to discount health benefits experienced in future years by people who are currently alive and avoid discounting when the health benefits of programs accrue to future generations.16
Yet even if we adjust for age, victims of gun violence are still not representative of the population of workers who are used to estimate these value-of-life figures. An alternative to using the average value for average workers is to use values estimated from more comparable populations. The adjustments could be of two forms. First, the observation that some gun-assault victims have chosen very risky occupations (for example, drug dealing) suggests that they place a relatively low value on their lives. Several studies have found that workers in high-risk occupations are a self-selected group who require relatively small compensation for those risks. A review by Harvard economist Kip Viscusi shows that while the willingness-to-pay to avoid (p.190) one statistical death among average workers ranges from $3.75 to $8.75 million, estimates derived from workers in high-risk jobs suggest a value-of-life on the order of $1 to $1.5 million.17 Such people are also more likely than others to smoke or drive unbuckled, further evidence that they place a relatively low value on their lives.18 If those who are at high risk for gunshot injury are more like workers in risky occupations than average workers, then by their own preferences a lower-than-average valuation is appropriate. Thus, one adjustment that we can make is to multiply estimates for the value of one statistical life by the ratio of the value to workers in risky jobs to the value for the average worker (1.5 ÷ 3.75 = 0.4).
Whether this first adjustment is appropriate for the case of suicides is not clear. Determining what people who are at high risk for suicide—that is, at high risk of intentionally shooting themselves—would pay to reduce the risk that they are shot raises complicated questions about what we mean by people’s preferences, and which preferences should be taken seriously. Someone who shoots himself essentially reveals that at the moment that the trigger is pulled, he is willing to pay zero (or even some negative amount) to avoid being shot. Yet many survivors do not attempt suicide again,19 suggesting that there may be a certain element of impulsiveness and regret associated with at least some suicide attempts. People who suffer from some mental or physical illness that puts them at high risk for gun suicide may be willing to pay some amount to reduce the chance that they will be able to act on such an impulse in the future. It is in this sense that it is meaningful to talk about people’s willingness-to-pay to reduce their risk of gun suicide and to compare this payment amount with the compensation required by workers in jobs with higher risks.20
Another possibility is to also adjust for income. By one estimate, the value that individuals place on their own lives increases by about 10% for every 10% increase in their income.21 Given the income differences in average income between gunshot victims and other people documented in Chapter 2, the value of risk-reductions to those at high risk of gunshot injury (p.191) may be as much as 37% lower than for the general population. Thus we might further adjust our estimates by multiplying by (1 − 0.37 = 0.64). Because workers in risky jobs may also have below-average incomes, adjusting for both preferences and income may lead us to understate the value of health and safety to those at high risk of gunshot injury.
Some analysts have argued that the government should use a uniform standard across population groups in regulating risk, without regard to income or preferences of the individuals involved. Risk regulation can then serve as a form of implicit income redistribution, giving the poor more safety than they would be willing to pay for (if given the choice of more safety versus more of other things). Our own preference is to value the benefits of public programs as the beneficiaries actually value them and handle income redistribution separately, rather than hiding implicit choices about redistribution within the details of cost-benefit analysis.
Studies of marketplace behavior estimate the wage that people would require to take a different job with a slightly higher risk of injury or illness but all else held constant. Because these studies assume that all else is equal between the two jobs, the issue of offsetting increases in risks from other workplace hazards is not an issue. Yet a central theme from Chapter 3 is that any consideration of the costs of gun violence must take the possibility of weapon substitution seriously. If some criminal assailants or suicidal individuals substitute other weapons for guns, then any reductions in gun injuries that are achieved may be offset in part or whole by some increase in the number of nongun injuries. Because of the possibility of weapon substitution and the replacement of gun injuries with nongun injuries, what workers will pay to reduce the risk of a fatal workplace accident by 1/10,000 may be different from what people will pay to fund a program that reduces their risk of gunshot injury by 1/10,000. To account for the possibility of weapon (p.192) substitution, our quality-of-life estimates include some adjustment for the offsetting disutility that people receive from an increase in nongun injuries.
In sum, extrapolating the value of reducing or eliminating the value of gun violence from data on workplace behaviors or jury awards is subject to a number of errors and requires a number of assumptions that may or may not hold. Yet in the absence of contingent-valuation survey data on what the public would pay to reduce rates of unintentional shootings and gun suicide, the quality-of-life approach is our only hope for placing a value on changes in such injuries. Developing estimates along these lines is the subject of the next section.
Quality of Life Estimates
Extrapolating from previous studies of the value of life suggests that the value of eliminating gun violence in America in 1997 is in the range of $20 to $50 billion, equal to between $170,000 and $400,000 per gunshot injury. Different assumptions yield higher and lower estimates, as shown in Table D2. In our judgment, the estimates presented in the third and fourth columns are the most appropriate though, as noted above, even these estimates are likely to be incomplete given that wage-risk studies are likely to ignore many of the benefits to society from reducing or eliminating gun violence.
The first column of Table D2 contains estimates that are similar in spirit to those presented by economists Ted Miller and Mark Cohen, who have developed the only previous WTP estimates for the value of eliminating gun violence.22 As with the Miller and Cohen calculations, those presented in column one of our table ignore the possibility of weapon substitution. Unlike the Cohen and Miller study, these initial estimates do not discount the value of health benefits that occur in the future (in order to highlight the effects of discounting for those who disagree with this adjustment).
The estimate for fatal gunshot injuries comes from multiplying an estimate for how long the average victim of a gun homicide, suicide, and unintentional shooting would have lived had (p.193)
Table D2 Estimates for What the American Public Would Pay to Eliminate Gun Violence in 1997, Calculated Using the Quality-of-Life Approach
Baseline, ignore weapon substitution (1)
Baseline, account for weapon substitution (2)
Adjust (2) by discounting future health outcomes using 3% Rate (3)
Adjust (3) for risk aversion of gunshot victims (4)
Adjust (4) for income of victims (5)
(N = 21,578)
Notes: Figures in 1998 dollars.
In principle, one could adjust the assumed life expectancies underlying these estimates to control for the fact that gunshot victims have other sociodemographic characteristics that contribute to shorter life spans than those of other people of the same sex and age. But this raises a number of difficult issues: If what people are willing to pay for a reduction in the risk of death is related to life expectancy, how are expectations of remaining life expectancy formed? Do individuals form these expectations by examining mortality patterns among people of the same sex and age, or is the relevant comparison group defined more narrowly with respect to other sociodemographic characteristics? Are these expectations rational (that is, equal to actual life expectancies on average)? Almost nothing is known on these points, and as a result our default is to use the more simple life expectancy estimates. Using this approach, our calculations suggest that the value of eliminating gun violence in 1997 is on the order of $115 billion, or around $1 million per gunshot injury.
Accounting for the possibility of weapon substitution reduces the estimate to between $41 and $88 billion, with a mid-range estimate of nearly $70 billion (column 2 of Table D2). The additional uncertainty with these modified estimates stems from uncertainty about what would happen to the number of suicides in an environment in which guns are not readily available for misuse.
When we make the additional adjustment of discounting the value of future health benefits using a 3% discount rate (column 3), our estimates are on the order of around $50 billion. (These estimates will be subject to a slight error because for computational simplicity we discount the lifetime gains in health for the average gunshot victim, rather than calculate the average of the discounted gains in health for all gunshot victims). Adjusting (p.195) for potential differences in risk aversion between gunshot victims and the populations used to estimate the value of a statistical life in previous studies shifts these estimates downward to around $20 billion.24 We prefer these estimates, which suggest a value per gunshot injury of between $170,000 and $430,000, because they account for the opportunity cost of investing funds in public programs that produce benefits off into the future. Whether the higher or lower of these numbers should be preferred is not clear.
While these estimates are likely to be incomplete (Chapter 4) and rest on a number of assumptions and extrapolations (above), they nevertheless suggest that the benefits of reducing gunshot injuries are far greater than simply the direct, tangible costs from medical spending and lost productivity considered in Chapters 5 and 6.
(6.) Cameron and Quiggin (1994).
(20.) Thanks to Will Manning for this point.
(24.) As noted above, we adjust column (3) for differences in attitudes toward risk among the population at highest risk of gunshot injury by multiplying by 0.4, which is the ratio of the estimated value of a statistical life for workers in high-risk jobs, divided by the value obtained from more general samples of workers.