Marginal deterrence and multiple murders.

Tollison, Robert D.

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One must calculate a penalty in terms not of the crime, but of its possible repetition. One must take into account not the past offence, but the future disorder. Things must be so arranged that the malefactor can have neither any desire to repeat his offence, nor any possibility of having imitators. Punishment, then, will be an art of effects....

Michel Foucault (1977, p. 93)

1. Introduction

Study of the deterrent effect of capital punishment has become a staple of the economic literature emphasizing marginal behavior. The preponderance but not the totality (Fox and Radelet 1989; Fagan 2005) of empirical evidence in this ever-growing literature is that higher arrest, sentencing, and execution probabilities--marginal deterrence--all lower the murder rate (Mocan and Gittings 2003; Zhiqiang 2004; Zimmerman 2004). In a well-executed empirical study using county-level data, Dezhbakhsh, Rubin, and Shepherd (2004) show that between 1977 and 1996, higher arrest, sentencing, and execution probabilities all lower the murder rate (18 fewer murders, with the margin of error at plus or minus 10). (1) Shepherd (2004) shows, moreover, that even "domestic" homicides and other "crimes of passion" may be deterred. Less studied has been the effects of execution methods on murder rates, although Zimmerman (2003) has shown that executions conducted through electrocution have a significant effect on deterrence using state-level data between 1978 and 2000. But one critical issue remains: Does capital punishment deter all kinds of murder? Specifically, given the principles of marginal behavior and deterrence at the margin, does capital punishment deter multiple murders?

The purpose of this paper is empirically to apply the concept of marginal deterrence to the effects of executions on multiple murders using state-level data between 1995 and 1999. We find, using data in part provided by the Federal Bureau of Investigation, that multiple murders are not deterred by execution in any form, quite possibly because the marginal cost of murders after the first is approximately zero. Although our research does not aim to cover old ground, we provide, in the course of our investigation and for purposes of comparison to multiple murders, additional empirical evidence on the effects of execution and method of execution on murder rates. (Our results here generally conform to the conclusions found in the extant and growing literature.)

In an initial section we offer a brief history of the economics of deterrence and capital punishment and establish a hypothesis relating to marginal deterrence and multiple murders. In the following two sections we present empirical tests related to single murder, forms of punishment, and multiple murder. We include a discussion of the use of marginal deterrence in an attempt to ameliorate particular forms of multiple murder. Finally, we speculate on how or whether, in a contemporary social and political environment, extensions of marginal deterrence would be possible for homicides in general and multiple murders specifically that are now punishable by a less costly death penalty.

2. Murder and Marginal Deterrence: A Brief History

The practice if not the theory of deterrence of all kinds of criminal acts is, of course, ancient. All societies have sought to restrict and punish rampant murder with a lowering of benefits and an increase in costs to perpetrators. All manner of "costs" accompanied the crime of murder during ancient and medieval times both under private systems of justice (e.g., the Germanic and early Anglo-Saxon frankpledge system) and under public systems, such as in those found in later Anglo-Saxon jurisprudence and continental systems. This history is bloody--to modern eyes "uncivilized"--and, in the case of the medieval Inquisitions, as we will see, creative in its applications of marginal deterrence.

The modern economic conception of crime and punishment undoubtedly originated in substantive form with the "incentives-based" utilitarian philosophy of Jeremy Bentham (1931) and his brilliant secretary Edwin Chadwick (1800-1890). In two seminal essays Chadwick developed what we now call the economic theory of crime (1829), an incentives-based reform of the criminal justice system (1841). (2) Chadwick focused on economic crime--robberies--and developed an institutional analysis of factors that would restructure marginal incentives of perpetrators (thieves) as summarized in the following general relationships:

[Marginal Cost.sub.Criminal Acts] = [Marginal Benefits.sub.Criminal Acts]

or

[Marginal Benefits.sub.Property Crimes] = Prob.(Apprehension) x (Cost from Apprehension) + Prob.(Conviction [Apprehension) x (Cost from Conviction) + Prob.(Punishment [including severity] [Conviction) x (Cost from Punishment).

The marginal calculation is instantly recognizable as the one underlying the modern (Becker 1968) "economics of crime" discussed in the introduction above. (3) More to the point of the present study, Chadwick appeared to recognize that these principles also applied to murder and capital punishment. Capital punishment in England, immediately before Bentham's and Chadwick's time, had an extremely bad reputation in the populace because death sentences were not geared to marginal deterrence and were imposed for far lesser crimes than murder (Zaller 1987). (4) For example, novelist-criminologist Henry Fielding was an unabashed defender of capital punishment. In 1749 Fielding supported the execution of one Bosavern Penlez, a British sailor, who had caused a riot in a house of prostitution. Despite pleas from the jury that convicted him and from the public at large, he received the ultimate punishment, and Fielding (1749) wrote a spirited defense of the sentence. Capital punishment for "crimes" such as petty theft and "riots" in whorehouses had been eliminated by the time of Chadwick's evaluation of the criminal justice system. Thus, although he did not focus on the link between severity of punishment, that is, death, and deterrence in cases of murder, Chadwick clearly recognized such possibilities and tradeoffs among apprehension, conviction, and punishment. (5)

The "economics" of crime thus established in the nineteenth century was reincarnated in modern economic theory in the second half of the twentieth century. (6) Although the seminal modern contribution was that of Gary Becker (1968), an important elaboration of the idea was made by George Stigler several years later (1970). Specifically, Stigler emphasizes the necessity for "optimal" marginal deterrence. In this situation, ill-established penalties would not have a deterrence effect. As Stigler argues, "... the marginal deterrence of heavy punishments could be very small or even negative ... if [for example] the offender will be executed for a minor assault and for a murder" (1970, p. 527). If an eye is to be plucked out or a foot chopped off for stealing $5 or $5 million, a thief might as well opt for the higher payoff. Thus, the establishment of marginal costs is necessary to marginal deterrence, or, in Stigler's words, "The penalties and chances of detection and punishment must be increasing functions of the enormity of the offense" (1970, p. 530). (7)

That the marginal severity of punishment applied to property and other crimes is a deterrent is empirically verifiable. The existence of two--and three-strike laws of California, where the laws are seriously enforced, are a case in point. In a county-level study of the full deterrence effect of this legislation, Joanna M. Shepherd (2002a) studied the impact of these laws on all offenders (not simply those committing their last strike). Empirically she finds that, because strike laws may deter individuals contemplating committing their first offense, approximately 8 murders, 3,952 aggravated assaults, 10,672 robberies, and 384,488 burglaries were deterred in California over the first two years of the legislation. Set against this benefit was the substitution of larceny and auto theft (nonstrike offenses). (8) Numerous other studies (e.g., Trumbull 1989; Marvell and Moody 1995; Shepherd 2002b) would also appear to firmly establish the effectiveness of marginal deterrence in other forms of legislation and penalty structures as well. (9)

The Nature of Murder and Marginal Deterrence

The issue of the impact of marginal deterrence for murder, for some rather obvious reasons, has not been studied very extensively. There are numerous objections to the argument that murder would respond at all to forms or margins of punishment. For example, many criminologists and sociologists might be willing to grant that property crimes might respond to economic incentives but object stridently to the fact that homicide might react similarly. We argue that although some murders might be categorized as "acts of (irrational) passion," a number, perhaps a large number, of them might be analyzed as calculated and rational. Consider these briefly.

Some murders are clearly calculated. Murders are demanded and supplied in our economy just as are drugs and sex. Like the property criminal, the killer is a middleman who steals the life of the victim and sells it to the murder contractor. Further, such crimes respond to traditional economic theory: Higher costs in the form of higher probabilities of detection, conviction, or execution will reduce "supply," causing a price increase and a reduction in the quantity demanded of murders for hire.

Other murders, those so often cited in the sociological literature, are the so-called "crimes of passion." Without rational calculation, or so the story goes, capital punishment (or other forms of deterrence) could not elicit a rational response. As a matter of analysis and "law," culpability is reduced without advance deliberation and planning. But, economically, the absence of advance de liberation and planning does not mean that price is irrelevant. So-called "crimes of passion" seldom occur in the midst of large crowds. A wife fed up with her husband's cheating may kill her husband, but rarely in the midst of a cocktail party where she observes his dalliance. She will wait until the guests have departed or set up a murder later in hopes of a good alibi. Although the utility gain of a very public murder might outweigh the cost of an increased probability of conviction, stealth and secrecy are often used. This suggests, and modern evidence clearly supports (Shepherd 2004), that even murders regarded as "passionate" may contain a large rational element in their calculation and might be deterred.

Multiple Murders

Many have hypothesized that marginal deterrence works in the case of single murders. The studies of Ehrlich (1977), Layson (1985), Dezhbakhsh, Rubin, and Shepherd (2004), and others, mentioned above, would suggest that the death penalty raises the cost of committing murder because the usual alternative is a prison sentence of some length all the way up to life in jail. An empirical finding that capital punishment reduces murder means that the reduced number of people committing their first murder is greater than the additional murders committed in the presence of capital punishment. And here is the paradox. When murders are rationally calculated--through murder-for-hire, for example-and not a one-time event, the marginal cost of additional murders after the first is zero discounting that the probability of capture might increase with additional crimes. Because the typical murderer commits only one murder, the restriction on the severity of capital punishment has little relevance to the problem of efficient deterrence. When the same punishment is levied for one or ten murders, there is no marginal deterrence, and there is at least anecdotal evidence that multiple murders are on the rise in part because of the rapid rise in largely professionally enforced drug-related homicides (Teasley 1991). (10) This means that capital punishment in its existing forms may be inadequate to deal with certain increasingly important forms of homicide such as multiple murders and serial killings.

Consider, for example, drug-related killings. High returns may accrue to the killer, making the enterprise perfectly rational from his or her perspective. Contract enforcement, predation on other dealers' or sellers' territories, and witness "hits" are only three reasons for a thriving specialized market for professional assassins. Even if, as our data have suggested, single first-degree murders are reduced by the increasing severity of capital punishment, capital punishment may have no effect on multiple murders. Serial killers who receive great utility from their "spree" or "planned" or "sniper" murders are unlikely to be much deterred by the relative cost between life in prison or a sentence of death after two murders. The fact is that existing criminal codes fail to elicit a positive price for multiple murders. Only the first " premeditated murder is subject to a penalty--execution. Killings beyond the first are, in effect, free.

The failure of the present system to create marginal deterrence in cases of multiple murders may be shown (or at least suggested) in Figure 1. Figure 1 contrasts the number of murders with the deterrence effect. Assuming that the death penalty is a perfect deterrence to murder number one, the marginal cost of the second and third through N murders committed is effectively zero. (11) When the probability of arrest rises with additional murders, this effect is mitigated to a certain extent. However, the marginal effect of this cost on a murderer who intends to commit multiple murders ex ante is probably washed out by the asymptotically declining costs of additional murders after the first. Clearly, as the number of murders increases to two (N?), deterrence falls to zero.

[FIGURE 1 OMITTED]

The central, and uncomfortable, issue is whether this effect carries any empirical support and, if so, whether some kind of positive price could be imposed for multiple homicides. Unfortunately, little work has been conducted on multiple murders in an economic context. (12)

3. Setting the Stage: Empirical Tests on Single Murders and the Form of Execution

As a prelude to an empirical analysis of the determinants of multiple murders, we test two propositions using state-level data for 1995-1999. Two questions have become staples of the traditional literature, the first more than the second: (i) whether, with nondeterrence factors considered, and assuming that a single murder is a negative function of cost and a positive function of benefits, capital punishment affects the overall murder rate; and (ii) whether a more "costly" method of punishment has an additional or marginal deterrence effect on the overall single murder rate, other things being equal.

The use of state-level data in our tests is, we believe, an improvement over the use of national data, because of the obvious heterogeneity of deterrence practices across states. (13) Execution in Alabama, for example, may have no deterrent effect outside the state, even though it deters future potential Alabama murders. In addition, we hypothesize that the penalty of execution by electrocution is marginally more costly as punishment than lethal injection, presently the method or choice in virtually all executions in the United States. Certainly this is the way that human rights and other groups, including anti-capital punishment groups, view the matter. We specify the following equation (14):

Murder rate = [alpha] +[[beta].sub.1] (Poverty) [[beta].sub.2] + (Nonwhite) + [[beta].sub.3](Gradrate) + [[beta].sub.4](Unemployment) +[[beta].sub.5](Metro) + [[beta].sub.6](POA) + [[beta].sub.7](Death) + [[beta].sub.8](Executionlag) + [[beta].sub.9](Electrocution) [[beta].sub.10](Year) + [19.summation over (j-11)][[beta].sub.j]([Regional Dummy.sub.j-10]) + [epsilon],

where all variables are defined as in Tables 1 and 2.

The first five explanatory variables are standard in models of capital punishment and proxy opportunity cost of engaging in criminal behavior, and the next three focus on costs imposed by our justice system--the deterrence variables that are the focus of the tests. The first five independent variables as described in Table 1 all carry the expected signs. Poverty, Unemployment, Nonwhite, and Metro are expected to be positive, and Gradrate is expected to be negative.

The deterrence variables also enter into the estimation of the cost of criminal behavior. Because the dependent variable is the rate of murders and nonnegligent manslaughters, we construct a rough proxy of the probability (on average) of perpetrators of these crimes being arrested. POA (probability of arrest) is equal to the number of arrests for murder and nonnegligent manslaughter divided by the total number of those crimes reported in the observed state during the same year. This is not a true probability because the value of POA can exceed unity: it is possible that during a particular year, more arrests occur than criminal acts if some of the arrests pertain to crimes committed in previous years. Nonetheless, this variable does measure the effectiveness of law enforcement, and therefore the coefficient is expected to be negative.

Death is a dichotomous variable equal to one if capital punishment in the observed state is legal; equal to zero otherwise. Any behavior is dependent on costs and benefits. This variable is meant to indicate which states have available the ultimate form of punishment for punishing murderers. A negative and significant coefficient would indicate that the death penalty serves as a deterrent to murder. A shortcoming of this variable lies in the fact that some of the states that allow death sentences have not executed anyone in the last 25 years. Connecticut, Kansas, New Hampshire, New Jersey, New York, and South Dakota are all death-penalty states where no death sentence has been carried out since 1976.

Whether the state allows for executions may not be as important as whether the state actually carries out death sentences. To this end, we include as a regressor Executionlag, which is the number of executions carried out in the observed state in the previous year. Individuals who are contemplating committing murder are more likely to consider the possibility that if caught their punishment could entail the death penalty if they reside in Texas rather than New York. Executionlag contains more information related to the cost of committing a crime than Death. Both of these variables are expected to have a negative impact on the murder rate.

In addition to the potentiality that the possibility of execution deters crime, it is also plausible that the method of execution may factor in the criminal's decision-making process. In other words, the costs to the commission of murder may not be totally uniform. Some forms of capital punishment may carry a higher "cost"--in terms of perceived brutality on the part of the potential murderer--and therefore greater deterrence than others. In 19th and 20th century U.S. practice, firing squad, hanging, the gas chamber, electrocution, and (most recently) lethal injection were the principal methods of punishment. (15) Lethal injection has actually replaced other methods. Other forms are likely deemed, at least marginally, more costly from an offender's perspective. Human rights groups often decry electrocution as being brutal, painful, and inhumane. Nonetheless, we predict that states that use the electric chair as the method of execution will have lower murder rates ceteris paribus. We expect the coefficient of Electrocution to be negative.

In addition, we consider two interaction terms (not specifically posited in Equation 1): Interactl and Interact2. Interactl is the product of Death and Nonwhite. Interact1 is included to see if the death penalty is particularly effective in deterring nonwhites from committing murder. Critics of the death penalty have long maintained that the sentence is handed down in a discriminatory fashion, with white defendants having a greater likelihood of receiving a prison term than nonwhite defendants. If so, the death penalty may be less of deterrence to whites than to others, and the coefficient of Interact1 will be negative. Interact2 is the product of Executionlag and Electrocution. We hypothesize that the combination of many executions with the most painful method of execution still in use will prove to be an effective deterrent to murder. If so, the coefficient of Interact2 will be negative.

Finally, we note that our sample consists of five years of observations (1995-1999) on each of the variables for each of the 50 states plus the District of Columbia. (16) The Year variable is simply a time trend composed of these five years for each state. Further, to try to account for some of the geographic variation in murder rates, we group the states according to the nine regions defined by the U.S. Bureau of the Census. The resulting dummy variables are listed as the last nine entries in Table 1.

Before we turn to a discussion of the empirical findings, there are a number of econometric issues that must be addressed. First, rather than attempting to model the murder rate by state, we model the log odds of being murdered (17) because this latter measure avoids the implicit truncation problems of the former. Thus, our dependent variable in the analysis to follow is

y = ln [P(murder) / 1-P(murder)]

Second, we must consider the panel nature of our data. Rather than estimating a traditional fixed- or random-effects specification, (18) we opt for a multiplicative heteroscedasticity approach as discussed by Greene (2000, pp. 518-20). (19) This model involves jointly estimating a regression function and a variance function. By incorporating the regional dummies into both the regression and variance function specifications, we can incorporate both the differential intercept aspect of a traditional fixed-effects model concurrently with the cross-region variation in the disturbance variance aspect of a traditional random-effects model.

Empirical Results: Murder Probability and the Form of Execution

Table 3 provides maximum-likelihood estimates from the multiplicative heteroscedasticity model (20) of various specifications of the basic regression function posited in Equation 1 and the corresponding variance functions. (21) The numbers in parentheses in Table 3 are t statistics, but because they are only asymptotically valid, they could as well be viewed as standard normal deviates. The numbers at the bottom of the table are summary statistics: the chi-square statistic tests the joint significance of all of the slope coefficients in both the regression and variance functions; Log L is the logarithm of the overall likelihood function; and BP Test is the Breusch-Pagan statistic testing for heteroscedasticity in the initial OLS estimate of the regression function.

In general, the regression function results in Table 3 conform rather closely to our a priori expectations and to results found in other studies. The trend variable Year is statistically insignificant in half of the specifications. In those where it is significant, it is uniformly negative, indicating that the log odds of being murdered was generally falling during the 1995-1999 period. (22) The four variables included in the regression function to measure the opportunity cost of criminal behavior all perform precisely as expected. (23) Metro, Nonwhite, and Unemployment are uniformly positive and statistically significant at the 0.01 level in all of the regression function estimates in which they were included, whereas Gradrate is always negative and statistically significant at the 0.01 level. As anticipated, the more urban the state, the greater its nonwhite population, the larger its unemployment rate, and the lower the proportion of its citizens graduating from high school, the greater are the log odds of being murdered in that state.

The variables of interest that proxy the cost of criminal behavior within our criminal judicial system also generally performed as expected, although the effect of the mere presence of the death penalty was somewhat surprising. The probability of arrest variable is negative as anticipated and statistically significant at least at the 0.10 level in all regression function specifications in which it was included. The more effective law enforcement, the lower the log odds of being murdered, ceteris paribus. However, the negative effect anticipated for the death penalty is not present. The estimated coefficient on Death is positive in all models in which it was included and statistically significant at the 0.01 level in all except Model 5. Apparently, the mere presence of the death penalty provides no deterrence per se, and in death penalty states that continually eschew invoking it, our results suggest that it may even increase the log odds of being murdered. On the other hand, for states that actually do execute people, we see the predicted deterrent effect, as the lagged executions variable is negative and statistically significant at the 0.01 level in all specifications in which it was included. Increases in the lagged number of executions significantly decrease the log odds of being murdered. Furthermore, the deterrent effect appears to increase with the costs (in terms of pain) associated with the particular method of execution. The Electrocution dummy variable has negative and statistically significant (at least at the 0.10 level) coefficient estimates in all regression function models in which it was employed. (24) This result parallels those found by Zimmerman (2003), that is, that the severity of execution form is a marginal deterrent to murder. (25) Finally, the interaction variables showed mixed results. Interaction 1, the product of the death penalty dummy with percentage nonwhite had the posited negative and significant (at the 0.05 level) effect. Interaction 2, the product of the Electrocution dummy with lagged executions, failed to produce any statistically significant results. (26)

The regression function results on the regional dummies suggest that a fixed-effects specification may have some merit. The base region is taken to be the Pacific region, and it, along with possibly the New England and Mid-Atlantic regions, has the lowest log odds of being murdered. The coefficients on the New England and Mid-Atlantic dummies are mostly insignificant, indicating no significant difference in the log odds of being murdered between them and the Pacific region. These results do not hold uniformly, however, as Model 3 suggests the log odds of being murdered are significantly higher in New England, and Models 5 and 6 indicate that it is significantly lower in the Mid-Atlantic than in the Pacific Region. The remainder of the regional dummies are almost all statistically significant (at the 0.10 level or better) and positive, indicating a higher log odds of being murdered for these regions than for the Pacific region. The exceptions are the South Atlantic region in Model 5, the West North Central region in Models 4 and 5, and the Mountain region in Model 5.

Now let us briefly examine the variance function results. The intercept of the variance function is labeled SIGMA in Table 3. If none of the explanatory variables in the variance function turns out to be statistically significant, the antilog of this estimate is the estimated homoscedastic variance of the regression function. However, the results on the regional dummies in the variance function suggest the appropriateness of a random-effects specification. Generally speaking, all regions included in the variance function specifications demonstrated a smaller variance than the Pacific region, with coefficient estimates that are almost all negative and statistically significant, at least at the 0.10 level. (27) Overall, all eight models appear to fit the data very well. The chi-square statistics for all eight models indicate that the null hypothesis of null-slope coefficient vectors for the regression and variance functions can be rejected at any reasonable level. In addition, the Breusch-Pagan statistics clearly indicate the presence of heteroscedasticity in the initial OLS estimates of the regression functions.

4. Marginal Deterrence and Murder: Empirical Tests

In order to develop a test of whether capital punishment or execution affects multiple murder rates, data on multiple murders in all states had to be assembled. The Federal Bureau of Investigation collects data on multiple murders. We developed a dependent variable from the FBI's Supplementary Homicide Report for our test period 1995-1999 that includes all multiple victims from single, multiple, or unknown offenders. Our test equation includes MULTIPLE, which equals multiple murders as the dependant variable. We begin by assuming that multiple murders are determined by the same factors that determine the probability of being murdered. Thus, our tests are all variations on the following equation:

MULTIPLE = [alpha] + [[beta].sub.1] (Poverty) + [[beta].sub.2](Nonwhite) + [[beta].sub.3](Gradrate) + [[beta].sub.4](Unemployment) + [[beta].sub.5](Metro) + [[beta].sub.6](POA) + [[beta].sub.7](Death) + [[beta].sub.8](Executionlag) + [[beta].sub.9](Electrocution) + [[beta].sub.10](Guns) + [[beta].sub.11](Year) + [20.summation over (j=12)] [[beta].sub.j]([Regional Dummy.sub.j-11]) + [epsilon]. (3)

Because MULTIPLE is the number of incidents of multiple murders occurring in a state in a given year, we estimate Equation 3 using maximum-likelihood methods assuming that we are sampling from a Poisson distribution. (We have no data on number of victims.) This assumption is made because we are dealing with count data and because a counting process can be shown to follow a Poisson distribution under some fairly general conditions. (28) We further implicitly assume that the logarithm of the expected value of multiple murders can be expressed as a linear function of the explanatory variables in Equation 3. Thus, the coefficient estimates of Equation 3 can be interpreted as the percentage change in expected multiple murders caused by ceteris paribus unit changes in the various explanatory variables, or in the case of dummy variables, caused by the presence of the relevant characteristic. Finally, note that because, under these assumptions, both the mean and variance of multiple murders depend on the explanatory variables in Equation 3, including the regional dummies allows us to account for the potential for both fixed effects and random effects arising from the panel nature of our sample.

The first six columns of Table 4 present Poisson parameter estimates from various forms of Equation 3; asymptotic t statistics (standard normal deviates) are in parentheses. The time trend variable Year is always negative and statistically significant at least at the 0.05 level in four of the models, indicating that, ceteris paribus, there was a downward trend in multiple murders during the later 1990s.

The variables that proxy opportunity cost usually behave as expected. Metro and Poverty are positive and significant in all models. The graduation rate is negative, as expected, but statistically insignificant in all models. Nonwhite is also negative but statistically significant at the 0.01 level in all models. Although this result is in contrast to our findings in Table 3, it may make sense for multiple murders: It is often alleged that most serial killers are white and male. Finally, Unemployment is always negative and is statistically significant in four models. This result was not anticipated and was not what was found for the probability of being murdered in Table 3.

The estimated regional effects are quite uniform. All regional dummies in all models are negative and statistically significant at least at the 0.05, and usually at the 0.01, level. This indicates that, after the effects of the other explanatory variables have been taken into account, the Pacific region has significantly more multiple murders than any other region.

Finally, the results for the variables that proxy the cost of crime as imposed by our criminal justice system are, generally speaking, problematic. Only the results for Guns show the anticipated deterrent effect; all of those estimates are negative and statistically significant at the 0.01 level. (29) The apparent effectiveness of "right-to-carry" legislation as a deterrent to multiple murders may reflect the fact that in such states the first potential murder victim, as well as subsequent potential victims, enjoys the right to carry a firearm. Thus, the perpetrator faces a potential cost from these would-be victims that does not decline at the margin.

One possible explanation for these general results is simultaneity bias, that is, joint determination of the number of multiple murders with the probability of arrest, the (lagged) number of executions, the presence of the death penalty, and death by electrocution. There is a clear case for suspecting a simultaneous relationship between multiple murders and the probability of arrest: The more efficient law enforcement, the more likely is a criminal to be arrested before he commits additional murders, i.e., MULTIPLE =-f[P(arrest)],f' < 0. But equally, because of the increased attention that multiple murders receive from media and law enforcement, the probability of being arrested is more likely if a murderer claims more than one victim, i.e., P(arrest) = g(MULTIPLE), g' > 0. Similar arguments can be made for executions, the death penalty, and electrocutions. In order to reduce the deleterious effects of simultaneity bias, we created instrumental variables for the probability of arrest, the death penalty, and lagged executions. (30) We created these instruments as follows: first, we (arbitrarily) deleted Poverty from the MULTIPLE model in order to help identify the instruments. Next we posited models in which the probability of arrest, the death penalty, and lagged executions were each determined by Year, Nonwhite, Unemployment, Poverty, and Population. Then we estimated equations for the probability of arrest and lagged executions by OLS and for the death penalty using probit. The predicted values from these equations became our instruments. Column 7 of Table 4 presents Poisson estimates of the multiple murders model using these instruments in lieu of their corresponding natural measures.

This instrumental variables approach results in some improvement in the intuitive appeal of the model. The P(Arrest) effect is now at least negative even though it is statistically insignificant at the 0.10 level. The Death penalty and Electrocution are now statistically insignificant, although they remain positively signed. These results ameliorate to some degree the seemingly perverse results in the first six columns, to the extent that now, the supposed deterrent variables simply do not affect the number of multiple murders. These results, however, are still unsatisfactory; Executions Lagged is still positive and highly significant, confronting us with the improbable implication that the more executions we had last year, the more multiple murders we can expect this year. We acknowledge that the instrumental variables we created were crude, but we think that they can be taken as indicative of what a more sophisticated analysis of the simultaneity issue would reveal. It is fair to suggest that correcting for simultaneity may well improve the P(Arrest) results, but it is unlikely to reveal any deterrent effects of executions, the death penalty, or electrocutions.

What, then, can account for the perverse results found in the first seven columns of Table 4? Our prior analysis of marginal deterrence provides an answer. Rote application of a model geared to explaining the probability of being murdered to the case of multiple murders may involve a misspecification problem, at least from a deterrent perspective. It makes perfect sense to suggest that increases in multiple murders should increase the number of executions, the probability of adopting the death penalty, and even the probability of execution by electrocution. However, if the marginal cost in terms of punishment of any murders beyond the first is indeed zero, then there is no motivation for including these "deterrent" variables in a model explaining multiple murders. There is indeed a positive relationship between multiple murders and each of these "deterrent" variables, but the statistical results we find in Table 4 are the result of an incorrectly reversed causal specification, not an actual behavioral relationship. Put more simply, no one would be surprised to find a positive and significant coefficient on multiple murders in an equation explaining executions, or one explaining the death penalty, or one explaining electrocution. (31) What we observe in Table 4 is this positive relationship showing up as significant in our statistical analysis, but the estimated models posit the causal flow backward. This argument explains why instrumental variables estimation will not eliminate the anomalies in Table 4. The causal flow between multiple murders and the three legal "deterrents" above is not bidirectional. Causality flows from multiple murders to each of the deterrents, and not conversely. The models estimated in the first seven columns of Table 4 assume the reverse of this causal flow.

The results presented in column 8 of Table 4 provide Poisson estimates of a model employing the instrumental variable for P(Arrest) and deleting Executions Lagged, Death, and Electrocution as variables explaining multiple murders. First note that now the P(Arrest) variable has the appropriate negative sign and is statistically significant at the 0.01 level. The criminal judicial system does provide some deterrence; more efficient police protection and "right-to-carry" gun laws reduce multiple murders. It is also worth noting that the anomalous results for unemployment are now reversed, the estimated coefficient being positive and statistically significant at the 0.01 level. Furthermore, the coefficient estimate for the graduation rate is negative as expected but now is statistically significant at the 0.10 level. All other behavioral variables are statistically significant and have their expected signs, and the regional effects (except West South Central) are again negative and statistically significant, at least at the 0.10 level. These results, taken together, make considerably more sense than those in columns 1-7.

Overall, our analysis of multiple murders produces an important result with respect to marginal deterrence: None of the models suggests even remotely that either the death penalty or executions have a deterrent effect on multiple murders. This results leads to conjecture concerning how deterrence might be accomplished.

Marginal Deterrence and Murder

Our empirical results pose some rather disconcerting questions. Until relatively recently, gas, electrocution, and lethal injection were among the methods of execution used in U.S. capital murder punishments. (32) All except lethal injection have gone into disuse. Our data and tests suggest that, marginally at least, execution by electric chair was more costly to those sentenced to die and was a more significant deterrent to single murders. (This opinion has been that of human rights, anti--death penalty, and other groups as well as those who have had a choice between lethal injection and electrocution as the method of punishment.) But, as our empirical analysis of multiple murder rates shows, a "terrible paradox" exists in that multiple murders are not significantly deterred by any form of capital punishment. How, within existing societal institutions, could marginal deterrence for these most horrendous of crimes be established? As usual, history provides some instruction.

The concept of marginal deterrence is ancient. Anglo-Saxon and other systems of jurisprudence used it, but, as suggested in an earlier section, the apotheosis of its use was during the medieval and, most especially, the Roman and Spanish Inquisitions that followed the Protestant Reformation. In the medieval past, for example, gradations of punishment were clear attempts at "efficient deterrence." The physical torture meted out by the Spanish Inquisition has been well documented and, in addition to torture and death, included confiscation, imprisonment, exile "from locality," scourging, galleys, and reprimand. Describing a case of relatively small marginal consolation, Burman (1984, p. 153) notes that "The ultimate penalty, again as in the medieval Inquisition, was the stake, reserved for unrepentant or relapsed heretics. The inquisitors attempted to the last moment to convince even relapsed heretics to confess and save their lives. If this last-minute confession took place during the auto de fe, they were given the benefit of strangulation before burning." Scott (1949, pp. 71-2), describing an auto de fe of 1690 in Madrid, is even more explicit:</p> <pre> In the great square was raised a high scaffold; and thither, from seven in the morning until the evening, were brought criminals of both sexes; all the Inquisitions in the kingdom sending their prisoners to Madrid. Twenty men and women out of these prisoners, with one renegade Mahometan, were ordered to be burned; fifty Jews and Jewesses, having never before been imprisoned, were sentenced to a long confinement, and to wear a yellow cap; and ten others, indicted for bigamy, witchcraft and other crimes, were sentenced to be whipped and then sent to the galleys; these last wore large pasteboard caps, with inscriptions on them, having a halter about their necks, and torches in their hands. </pre> <p>But the Inquisition employed mental torture as well, and was global in its reach. Speaking of French practice, Foucault (1977, p. 40) notes that the torture imposed by the Inquisition was "a regulated practice, obeying a well-defined procedure.... The first degree of torture was the sight of the instruments." The ensuing degrees of torture assumed an array of methods: the ordeal of water, the ordeal of fire, the strappado, the wheel, the rack, and the stivaletto--all used to break alleged heretics. Torture, or even the prospect of torture, under a legal system that bestowed vast power on the Roman Catholic Church, was a particularly vivid means of raising the cost of membership in a rival sect. To be sure, such marginal deterrence was not the exclusive province of the Roman Catholic Church. It was commonly practiced all over the European continent, in England, and in the Americas, including colonial New England. (33)

Although no one wants a return to the inquisitions, the fact is that existing criminal codes fail to exact positive marginal prices for multiple murders. Some "painless" execution might be sufficient to deter "normal" murders, but only the first premeditated murder of a series of multiple murders is subjected to a penalty of execution. However, pages might be taken from past systems of punishment in this regard. Combinations of marginal punishments might be devised for multiple murderers, including confiscation of property for wealthy murderers--a principle that appears to motivate deterrence of certain drug-related offenses. (34) Because some (even many) murderers might not hold significant estates, other means of increasing costs before execution could be devised.

As Foucault suggested with respect to Inquisitorial practice, much of torture is "mental" in nature. Historically, execution methods were selected for the pain produced as a means of creating marginality. Hanging would be relatively mild compared to being boiled in oil. Torture was also used. Although major risks in the use of torture were permanent physical injury to the subject or the production of "premature" death, modern scientific methods avoid this problem. An economic word for torture is of course "disutility produced." The return of hard labor under gradations of duress in combination with ultimate capital punishment might be considered. Parts of earlier methods of punishment, generally eschewed in the modern world, are "public humiliation." Punishment, as well as trials, may be made "public." Further, years of positive punishment (n years) could be attached to (ultimate) execution in levies for n murders.

Such forms of punishment might be regarded by many as "barbaric" or "uncivilized." Others argue, on the other hand, that policies that prevent the torture and murder of innocent victims are the essence of judicial "civility." Increasing the costs for multiple capital offenses might deter determined lethal snipers, rampant serial killers, and fanatic terrorists. The vivid emergence of these types of crimes, at the very least, demands a reexamination of the marginal deterrent effects of existing penalty structures.

5. Conclusion

Does capital punishment deter multiple murders? This paper opens that debate by examining the principle of marginal deterrence and its effects when the opportunity cost of an additional murder is approximately zero. However strongly execution variables deter first and only murders, the marginal cost of additional murders is, in effect, zero. Empirically, we find that execution and the death penalty have no significant effect on multiple murders. We do so using state-level data for the years 19951999, applying an econometric technique that combines elements of fixed- and random-effect models. While not attempting to cover old ground, moreover, our study also shows that, for the period we study and given the technique we employ, single murders are deterred by execution variables. Further, we show, adding evidence to the point, that the form of execution--electrocution being considered marginally more "painful" than lethal injection--is an added deterrent to single murders.

Without marginal deterrence, however, multiple murders do not appear to be preventable by execution in any form. Historical examples of marginal deterrence do provide clues to its effectiveness in preventing certain crimes. Following historical illustrations, we explore some of the possibilities of establishing marginal deterrence in the application of capital punishment. Naturally, it is unnecessary to point out that both Type 1 and Type 2 errors must be assiduously avoided and that all punishment must be imposed on the basis of fair and equal justice for all the accused. Although the Eighth Amendment of the U.S. Constitution prohibits "cruel and unusual punishment," that concept is, historically speaking, malleable and could possibly be amended to help provide marginal deterrence of crimes that heinously violate human and judicial "civility."

We are grateful to Professors Richard Ault, Jim Buchanan, Barry Hirsch, Bill Shughart, Mark Thornton, Keith Watson, and Paul Zimmerman for valuable comments on early drafts of this paper. We are also grateful to the editor and referees of this journal for useful comments. Naturally, we stand liable for the product. Research on this paper was conducted, in part, while Ekelund was Vernon F. Taylor Visiting Professor at Trinity University in San Antonio in Spring 2003. He gratefully acknowledges this support.

Received February 2005; accepted July 2005.

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(1) The economic literature in modern times began with Becker (1968) and Stigler (1970), who argued theoretically that murderers and potential murderers make marginal cost-benefit calculations just as are made with property crimes. Naturally, these are empirical questions. The pioneering work of Isaac Ehrlich (1975, 1977) showed that homicide rates varied inversely with the cost of committing murder. Examining the effects of executions on national homicide rates between 1933 and 1969, Ehrlich (1975) found that, other things being equal, one execution prevented or deterred up to eight homicides. In further evidence, based on a cross section of states for 1940 and 1950, Ehrlich (1977) estimated that each execution deterred up to 24 murders. With a similar methodology, Layson (1985) updated Ehrlich's initial study to 1977, reporting that each execution deterred approximately 18.5 murders.

(2) These essays and their development are discussed at length in Hebert (1977) and in Ekelund and Dorton (2003).

(3) Becker develops an expected utility approach from an offense, writing it as EU = pU(Y - f) + (1 - p)U(Y), with EU as expected utility, Y = money value of gain, p the probability of detection and conviction, and f the fine.

(4) Clearly the imposition of capital punishment in England, pre-late 18th century, was not a device of marginal deterrence. Death for stealing a loaf of bread and for murder meant that there was, as Stigler later argued, not deterrence for murder.

(5) Later in his career, Chadwick analyzed murder: see Chadwick (1863). In this essay, Chadwick linked properly crimes to murder and argued that hoarding or the business practice of keeping large sums on the premises was an incentive to murder. He advocates (1863, pp. 402-3) methods of self-protection through the use of banks. Although Chadwick did not believe that capital punishment was much of a deterrent to property crime, he was open to its use in cases of murder. With respect to the former, he believed, after consultations with convicted felons and empirical research, that certainty of punishment was a stronger deterrent.

(6) It should be noted that Bentham was clearly influenced in his views on the "economics" of crime and criminology by the 18th century Italian writer Cesare Beccaria (1712-1769): see Beccaria (1767 [1751]).

(7) As in so many other areas of behavioral analysis, some of the foundation for "marginalism" in this area was in analogy to biology. According to Foucault (1977, p. 99), discussing immediate pre-Revolutionary French thought, "'one sought to constitute a Linnaeus of crimes and punishments, so that each particular offence and each punishable individual might come, without the slightest risk of any arbitrary action, within the provisions of a general law." Citing a late 18th century French source, Foucault continues, noting that tables of genera and species of crimes should be drawn up where crimes are separated according to their objects. "Lastly, this table must be such that it may be compared with another table that will be drawn up for penalties, in such a way that they may correspond exactly to one another" (P. L. de Lacretelle quoted in Foucault [1977, p. 100]).

(8) Shepherd (2002b) also concludes that violent crimes are marginally deterred by the imposition of truth-in-sentencing legislation increasing the minimum sentence length for violent offenders.

(9) An interesting exception is Altrogge and Shughart (1987), who find that the civil penalties levied by the FTC are regressive with larger fines levied on smaller firms.

(l0) Although the evidence is spotty, a number of writers argue that the number of murders committed with unknown motives have risen with an increased incidence of serial murder accounting for most of the rise (Lindsay 1984; Ressler, Burgess, and Douglas 1988). We note that the probability of capture might also decrease with additional crimes if witnesses are killed.

(11) Note that the same result obtains whether one regards multiple murder as "rational choice" or, as established in common law, cases of "blood simple," which assigns irrationality to multiple acts.

(12) Important exceptions are recent and well-executed studies of the impact of "concealed weapon" laws on murder rates and on "public [multiple] shootings" (Lott 2000; Lott and Landes 2000). These laws are yet an additional cost to prospective murderers and would be expected to reduce murder rates. Lott and Landes (2000) show, for example, that arrest and conviction rates and the death penalty reduce "normal" murder rates but that the only policy factor to have consistent and significant influence on "public shootings" is the passage of concealed handgun laws.

(13) Ignoring this heterogeneity can lead to questionable findings. In a test using national-level data, for example, Peterson and Bailey (1991) regress the execution rate on the homicide rate for the period 1976 to 1987 and find no consistent evidence of deterrence.

(14) Our model is unique in its choice of independent variables, although it conforms closely to those used in previous studies.

(15) The electric chair was introduced in New York in 1889 with the first person executed in this manner in 1890. The gas chamber was established in Nevada in 1924. The combination of sulfuric acid and cyanide was used, and death was not speedy. The infamous Texas electric chair "Old Sparky" executed 361 killers between 1924 and 1982.

(16) The sample size is n 5 x 51 = 255. We confine our attention to the five-year span 1995-1999 for two reasons. First, we hope to pick a period sufficiently short so as to retain an intrastate homogeneity of preference for deterrence, while at the same time allowing enough time for the observable intrastate determinants of the murder rate to vary. Five years seemed an appropriate span to accomplish these dual objectives, and the 1995-1999 period is the most recent five-year period for which complete data are available. Second, a subsequent analysis involving multiple murders encounters data availability problems if we extend the period of analysis much past this five-year window.

(17) Data on the murder rate are typically expressed as the number of murders per 1000 (or 10,000 or 100,000) people in the state. Although this view is convenient for obtaining an intuitive feel for how the propensity to murder varies across states, it inherently incorporates econometric difficulties. That is, it ignores left-truncation at zero, and the inflation resulting by multiplying murders per capita by [l0.sup.i] (i = 3, 4, or 5) obfuscates a corresponding right-truncation problem. The murder rate is fundamentally some multiple of murders per capita, or more precisely, some multiple of the probability of being murdered. Because this fundamental measure is a probability, it is bounded by the unit interval--a problem that must be dealt with econometrically. A common method of handling this problem is to subject the probability to a logistic transformation. Converting the probability of being murdered to the odds of being murdered simply involves dividing through by one minus the probability, i.e., odds= ([P.sub.i]/(1 - [P.sub.i]))- This transformation allows the dependent variable to vary between zero and infinity. Taking the logarithm of this ratio allows it to range from positive to negative infinity.

(18) A fixed-effects model assumes that all cross-state heterogeneity can be summarized by differences in the model's intercept, via a set of state dummy variables. A random-effects model, sometimes called an error components model, assumes a common intercept and summarizes all cross-state heterogeneity in a state-specific component of the model's stochastic disturbance, leading to a heteroscedastic disturbance covariance specification and a generalized least-squares (GLS) remedial approach. For our inquiry, the fixed-effects specification, in its natural form, is problematic. Estimating parameters for 50 dummy variables along with 10-plus other explanatory variables using a sample of only 255 observations simply does not allow for any confidence in the robustness of the estimates. An alternative to the traditional fixed-effects model is a first-differences approach, which obviates the need to analyze variables for a given state that do not change over time. Unfortunately, given our current sample, this procedure is not helpful because we lose as many degrees of freedom from the requisite deletion of observations overlapping two states as we do from the traditional fixed-effect specification. Thus, we group the states according to the nine regions defined by the U.S. Bureau of the Census. This aggregation across states reduces the number of required parameter estimates for the fixed-effects specification, but it also spawns potential unobserved heterogeneity within each region and hence blurs the distinction between the fixed- and random-effects specifications. In response to this conundrum, we opt out of the typical either-or approach to the fixed versus random effects question, choosing instead to estimate a model that allows for both types of effects. There is also a one-way versus two-way question dealing with whether to also model heterogeneity over time along with heterogeneity across states. We have chosen to concentrate on heterogeneity across states, assuming a one-way approach, by modeling time effects explicitly in the structural model as a time trend variable (see the Year variable in Equation 1). This approach is typical; see Greene (2000, p. 576).

(19) This procedure involves estimating a regression function, such as Equation 1, and a variance function in which the logarithm of the variance is assumed to be a function of an alternative set of explanatory variables, some of which may also appear in the regression function. The estimation procedure can be viewed as iterative: begin by using ordinary least-squares (OLS) to estimate the regression function, and obtain the residuals from the estimated model. The log of the square of these residuals becomes the dependent variable for the variance function, which is estimated by OLS. Predicted values from the variance function estimate are then used as weights in a GLS estimation of the regression function, the squared and logged residuals of which form the new dependent variable for a new estimate of the variance function, the predicted values of which provide the weights for a second GLS estimate of the regression function. Iteration between estimates of the regression function and the variance function continues until the coefficient estimates of both models stabilize. After convergence, the resulting parameter estimates are maximum-likelihood estimates. One advantage of this approach is that the fixed effects of cross-regional variation in the probability of being murdered can be incorporated into the regression function by simply including a set of regional dummies, and the spirit of the random effects specification can also be incorporated by including regional dummies in the variance function (allowing the disturbance variance to differ across regions). In addition, this approach is more general than either the fixed- or random-effects models in that it also allows heteroscedasticity to arise from more traditional sources, i.e., variables affecting the probability of being murdered.

(20) We employed the program LIMDEP, specifically the HREG option, to estimate these models. This option allows the user to control the maximum number of iterations (we set it at 1000, and all models converged) but not the convergence criteria. The model is judged to have converged when the estimated coefficients change by no more than 10-9 from one iteration to the next. Clearly, model specification plays a role in convergence. Ceteris paribus, the more parameters to be estimated, the more difficult convergence becomes. For instance, using all of the census regions in the variance function sometimes caused convergence problems simply due to the increased number of parameters to be estimated. Similarly, putting Poverty in both the regression and variance equations resulted in both coefficient estimates being insignificant, whereas deleting it from the variance function sometimes resulted in lack of convergence.

(21) Because the regression functions differ, the implied disturbance variances and the corresponding variance functions should also be expected to have differing specifications as well.

(22) Concerning the possibility of intertemporal drift in the variance function, a number of preliminary model estimates revealed no statistical significance of Year in the variance function.

(23) Initially a fifth variable, Poverty, was also posited to affect the murder rate. It turns out that Poverty has a much more pronounced effect on the variance than directly on the log odds of being murdered. See footnote 20.

(24) There were four states that used the electric chair exclusively during the time period studied. It is interesting to note that three of those states--Alabama, Georgia, and Florida--have abandoned using electrocutions exclusively and now offer a choice of lethal injection. In November 2002 the Nebraska Legislature's Judiciary Committee rejected a bill that would have altered the state's method of execution from the electric chair to lethal injection. Nebraska thus remains the only state with the chair as the only method of execution. In a well-designed state-level test of the "form" of execution on deterrence between 1978 and 2000, Zimmerman (2003) found that the deterrent effect of capital punishment is determined primarily by executions conducted by electrocution. None of the other methods in use over this time had statistically significant effects on the percapita incidence of murder. Our model, using regional dummy variables, adds support to this conclusion.

(25) We added an explanatory variable to our regressions that captures the impact of concealed handgun laws in states where applicable over our test period (1995-1999). Our initial results, using the simple murder rate as the dependent variable, showed that concealed gun laws did not significantly reduce the murder rate. Our tests are conducted, however, at the state level and not at the county level as in Lott and Landes (2000), a fact that might help explain their insignificance level. Dezhbakhsh, Rubin, and Shepherd (2004) present an interesting result in this regard, showing that NRA membership is positively related to the murder rate. In the studies below, we focus on the impact of the death penalty and execution on multiple murders, but we fully appreciate the potential impact of extending concealed weapon laws on certain types of multiple murders and, indeed, show that they are important determinants at the state level.

(26) This insignificance may be a result of the interaction variable picking up other types of execution than electrocution. Many states (e.g., South Carolina) that allow electrocution also allow other types of capital punishment and further allow the criminal to "choose his poison." The interaction variable measures only the (lagged) number of executions in states that allow electrocution--not deaths by electrocution. So there is no way to ensure that the method used in these states was the "costliest." Indeed, for states that allow the criminal to choose, the opposite is likely to occur.

(27) Table 3 also suggests that heteroscedasticity in the regression function estimates arises from more traditional sources. In most of the variance function specifications in which they are included, increases in Executions Lagged, P(Arrest). and Poverty statistically significantly decrease the variance of the regression function at the 0.01 level. The same can be said for the presence of the death penalty and the electrocution dummy. Increases in the percentage nonwhite significantly (at the 0.01 level) increase the variance of the regression function.

(28) See, for example, Cameron and Trivedi (1998, pp. 5-6).

(29) There currently are 30 states that have "shall issue" or "right to carry" legislation. These laws allow for qualified individuals to carry a concealed firearm. Our results parallel the results found by Lott and Landes (2000), who, at a county level, found these laws to be a significant cost to "public" multiple murders. The remaining variables in this category all show up as positive and statistically significant at the 0.01 level. Thus, we are in the uncomfortable position of trying to explain how increases in the probability of arrest and the (lagged) number of executions, and how the presence of the death penalty and death by electrocution, can increase the number of multiple murders.

(30) We also tried to create an instrument for Electrocution, but given the limited number of variables to choose from, we were unable to identify any factors that would allow us to predict, even erroneously, a positive probability for having electrocutions. That is, all reduced-form probits we estimated predicted zero values for Electrocution for all states for all years.

(31) Of course there may well be other determinants relevant to the level of punishment. Some of these include the mindset of the offender, the heinousness of the crime, past criminal involvement, characteristics of the offender (for example, white vs. minority, wealthy vs. indigent, and so on), characteristics of the victims (were they all children, elderly, or criminals themselves?), and so on. We are grateful to a referee for pointing out these possibilities.

(32) 0f 432 executions that took place between 1977 and 1997, 284 were by lethal injection, 134 by electrocution, 9 by lethal gas, 2 by firing squad, and 3 by hanging.

(33) Marginal punishments for crimes ranging from "drinking on Sunday" or theft were punished by time in the locks with public humiliation or having a hand cut off to more serious gradations. Medieval punishment for murder and serious crime depended on the nature of the crime and often involved marginally severe punishment before death, e.g., use of the "wheel" before the coup de grace. In some monarchical jurisdictions, plots to overthrow government involved hanging combined with being drawn and quartered. The hung victim was cut down while alive with organs then drawn out, including, for treason or sedition, "heart held high."

(34) Other crimes carry penalties approaching "public humiliation." Publication of names and locations of sex offenders and the wearing of "orange" uniforms in cleanup brigades are forms of such punishment.

Robert B. Ekelund, Jr., Department of Economics, 215 Lowder Business Building, Auburn University, Auburn, AL 36849, USA; E-mail bobekelund@prodigy.net.

John D. Jackson, Department of Economics, 212 Lowder Business Building, Auburn University, Auburn, AL 36849, USA; E-mail jjackson@business.auburn.edu; corresponding author.

Rand W. Ressler, Department of Economics and Finance, University of Louisiana at Lafayette, Lafayette, LA 70501, USA; E-mail rwr5011@louisiana.edu.

Robert D. Tollison, ([sections]) Department of Economics, Clemson University, 201G Sirrine Hall, Clemson, SC, 29630 USA; E-mail rtollis@clemson.edu.

Table 1. Variable Definitions

Murderrate Rate of murder and nonnegligent manslaughter per
 100,000 state inhabitants. Source: UCR, various
 years.
Year The year of the observation.
Metro Percentage of the state's inhabitants residing in
 metropolitan areas. Source: Statistical Abstract of
 the United States, metropolitan areas.
Poverty Percentage of the state's inhabitants who are below
 the poverty level. Source: Statistical Abstract of
 the United States, various years.
Nonwhite Percentage of the state's population who are not
 white. Source: Statistical Abstract of the United
 States, various years.
Gradrate The number of public high school graduates divided by
 resident population for the observed state. Source:
 Statistical Abstract of the United States, various
 years.
Unemployment The unemployment rate in the observed state.
 Source: Statistical Abstract of the United States,
 various years.
POA Probability of arrest: the number of arrests for
 murder and nonnegligent manslaughter in the
 observed state divided by the number of murders and
 nonnegligent manslaughters in that state. Source:
 UCR, various years.
Death A dichotomous variable equal to 1 if the state uses
 capital punishment, equal to 0 otherwise. Source:
 Death Penalty Information Center
 (www.deathpenaltyinfo.org).
Executionlag The number of executions during the previous year in
 the observed state. Source: Death Penalty
 Information Center (www.deathpenaltyinfo.org).
Electrocution A dichotomous variable equal to 1 if the state's
 primary or sole method of execution is the electric
 chair, equal to 0 otherwise. Source: Death Penalty
 Information Center (www.deathpenalty.org).
Interact1 An interaction term equal to the product of Nonwhite
 and Death. Source: See above.
Interact2 An interaction term equal to the product of
 Executionlag and Electrocution. Source: See above.
Multiple The number of incidents of multiple murders in the
 observed state. Source: Federal Bureau of
 Investigation, Supplementary Homicide Reports,
 1995-1999.
Guns A dichotomous variable equal to 1 if the observed
 state has adopted "right-to-carry" or "shall issue"
 legislation legalizing carrying concealed firearms,
 equal to 0 otherwise. Source: CCW Database
 (www.packing.org).
New England A dichotomous variable equal to 1 if the data refer
 to Maine, New Hampshire, Vermont, Massachusetts,
 Rhode Island, or Connecticut; equal to 0,
 various years.
Mid-Atlantic A dichotomous variable equal to 1 if the data refer
 to New York, New Jersey, or Pennsylvania; equal to
 0, otherwise.
South Atlantic A dichotomous variable equal to 1 if the data refer
 to Delaware, Maryland, Washington, D.C., Virginia,
 West Virginia, North Carolina, South Carolina,
 Georgia, or Florida; equal to 0, otherwise.
East South A dichotomous variable equal to 1 if the data refer
 Central to Alabama, Mississippi, Tennessee, or Kentucky;
 equal to 0, otherwise.
West South A dichotomous variable equal to 1 if the data refer
 Central to Louisiana, Arkansas, Oklahoma, or Texas; equal
 to 0, otherwise.
East North A dichotomous variable equal to 1 if the data refer
 Central to Ohio, Indiana, Illinois, Michigan, or Wisconsin;
 equal to 0, otherwise.
West North A dichotomous variable equal to 1 if the data refer
 Central to Missouri, Iowa, Minnesota, North Dakota, South
 Dakota, Nebraska, or Kansas; equal to 0, otherwise.
Mountain A dichotomous variable equal to 1 if the data refer
 to New Mexico, Arizona, Colorado, Utah, Nevada,
 Wyoming, Idaho, or Montana; equal to 0, otherwise.
Pacific A dichotomous variable equal to 1 if the data refer
 to California, Oregon, Washington, Alaska, or
 Hawaii; equal to 0, otherwise.

Table 2. Descriptive Statistics of Variables

Variable Mean Median Maximum Minimum Standard
 Deviation

Murderrate 6.89 5.70 73.1 0.90 8.05
Metro 68.13 70.00 100.00 27.45 20.96
Poverty 12.55 11.80 25.50 5.30 3.78
Nonwhite 16.29 12.57 67.51 1.53 13.88
Gradrate 0.0095 0.0091 0.0311 0.0050 0.0023
Unemployment 4.76 4.70 8.90 2.50 1.26
POA 0.78 0.71 5.11 0.00 0.53
Death 0.75 1.00 1.00 0.00 0.44
Executionlag 1.07 0.00 37.00 0.00 3.34
Electrocution 0.08 0.00 1.00 0.00 0.27
Interact1 11.93 10.82 63.48 0.00 10.91
Interact2 0.1059 0.00 4.00 0.00 0.4701
Multiple 12.516 8.00 124.00 0.00 18.9
Guns 0.59 1.00 1.00 0.00 0.49

Data are aggregated to the state level (plus Washington, DC) for the
years 1995-1999; the number of observations is 255 for each variable.

Table 3. Determinants of the Murder Rate: Maximum-Likelihood
Estimates (a)

 Dependent Variable, ln(P[murder]/
 {1-P[murder]})

 1 2 3 4

Regression Function
 Constant 23.16 17.24 -6.43 150.9
 (0.86) (0.66) (-0.23) (6.67)
 Year -0.015 -0.012 -0.000 -0.078
 (-1.10) (-0.91) (-0.02) (-6.90)
 Metro 0.006 0.007 0.007 0.004
 (4.88) (5.41) (5.41) (3.21)
 Nonwhite 0.018 0.019 0.017 0.022
 (10.45) (11.08) (9.39) (11.93)
 Graduation Rate -90.44 -92.79 -82.46 -81.89
 (-5.21) (-5.29) (-4.72) (-4.56)
 Unemployment 0.173 0.182 0.219
 (7.92) (8.92) (9.22)
 P (Arrest) -0.051
 (-1.65)
 Death 0.273 0.225
 (3.82) (2.74)
 Executions Lagged -0.008 -0.007 -0.007 -0.007
 (-3.19) (-3.03) (-2.98) (-2.71)
 Electrocution -0.095 -0.129 -0.12 -0.106
 (-1.82) (-2.61) (-2.36) (-2.07)
 Interaction 1

 Interaction 2

 New England 0.209 0.116 0.309 0.032
 (1.47) (1.18) (2.45) (0.21)
 Mid-Atlantic 0.143 0.131 0.013 -0.184
 (0.90) (0.82) (0.11) (-1.19)
 South Atlantic 0.702 0.709 0.648 0.228
 (6.29) (6.47) (6.60) (1.84)
 East South Central 0.938 0.972 0.853 0.564
 (8.33) (8.86) (9.11) (5.09)
 West South Central 1.012 1.014 0.877 0.673
 (8.82) (8.79) (8.75) (5.98)
 East North Central 0.885 0.876 0.866 0.469
 (7.72) (7.68) (8.18) (4.12)
 West North Central 0.723 0.766 0.816 0.212
 (4.88) (5.26) (6.66) (1.51)
 Mountain 0.740 0.790 0.626 0.316
 (6.41) (6.98) (6.52) (2.57)
Variance Function
 Sigma 1.297 1.44 1.41 0.418
 (5.13) (5.13) (4.68) (5.42)
 South Atlantic -0.697 -0.745 -0.019 -0.061
 (-0.29) (-2.80) (-0.07) (-0.23)
 East South Central -2.738 -2.677 -1.876 -3.396
 (-7.37) (-7.21) (-4.86) (-8.81)
 West South Central -0.882 -0.668 0.117 -2.212
 (-2.09) (-1.59) (-0.27) (-5.15)
 East North Central -1.538 -1.577 -0.658 -1.103
 (-4.85) (-4.98) (-2.08) (-3.49)
 Mountain -0.765 -0.703 -0.019 -0.372
 (-2.83) (-2.60) (-0.07) (-1.29)
 Executions Lagged -0.091 -0.096 -0.103 -0.054
 (-3.03) (-3.18) (-3.41) (-1.80)
 P (Arrest) -0.653 -0.577 -0.597
 (-3.73) (-3.30) (-3.39)
 Poverty -0.110 -0.132 -0.121 0.083
 (-3.91) (-4.69) (-4.30) (2.98)
 Death -0.844
 (-3.71)
 Nonwhite

 Electrocution

Summary Statistics
 [chi square] 367.00 364.81 380.43 331.55
 Log L -88.12 -89.21 -81.40 -105.8
 BP test 28.80 27.80 29.46 58.92

 Dependent Variable, ln(P[murder]/
 {1-P[murder]})

 5 6 7 8

Regression Function
 Constant 138.9 43.6 47.55 47.50
 (6.60) (1.64) (2.28) (2.27)
 Year -0.072 -0.025 -0.027 -0.027
 (-6.86) (-1.89) (-2.61) (-2.60)
 Metro 0.004 0.005 0.003 0.003
 (3.48) (4.43) (2.50) (2.50)
 Nonwhite 0.026 0.023 0.033 0.033
 (12.36) (10.50) (6.95) (6.94)
 Graduation Rate -52.74 -36.72 -30.50 -31.21
 (-3.50) (-2.60) (-2.32) (-2.31)
 Unemployment 0.148 0.138 0.137
 (6.30) (6.88) (6.47)
 P (Arrest) -0.085 -0.042 -0.055 -0.054
 (-3.64) (-2.13) (-2.64) (-2.61)
 Death 0.103 0.232 0.349 0.347
 (1.56) (3.58) (4.26) (4.23)
 Executions Lagged -0.006
 (-2.83)
 Electrocution -0.087
 (-1.67)
 Interaction 1 -0.010 -0.010
 (-1.97) (-1.96)
 Interaction 2 -0.031 -0.003
 (-1.11) (-0.19)
 New England -0.153 0.015 0.143 0.140
 (-1.09) (0.83) (1.04) (1.01)
 Mid-Atlantic -0.428 -0.229 -0.203 -0.207
 (-2.86) (-1.78) (-1.46) (-1.48)
 South Atlantic 0.204 0.370 0.353 0.354
 (1.63) (3.34) (3.10) (3.11)
 East South Central 0.459 0.616 0.545 0.547
 (3.87) (5.71) (4.85) (4.84)
 West South Central 0.545 0.661 0.646 0.646
 (4.48) (5.93) (5.60) (5.60)
 East North Central 0.372 0.707 0.698 0.695
 (2.99) (5.71) (5.55) (5.50)
 West North Central -0.041 0.342 0.339 0.339
 (-0.29) (2.46) (2.60) (2.60)
 Mountain 0.199 0.294 0.262 0.263
 (1.55) (2.51) (2.14) (2.15)
Variance Function
 Sigma 0.467 0.470 0.476 0.463
 (5.12) (9.06) (9.02) (9.02)
 South Atlantic -0.892 -1.197 -1.315 -1.319
 (-3.12) (-4.31) (-4.59) (-4.60)
 East South Central -3.846 -3.135 -2.449 -2.458
 (-10.3) (-8.82) (-6.73) (-6.76)
 West South Central -2.233 -1.778 -2.171 -2.169
 (-5.29) (-5.08) (-6.19) (-6.19)
 East North Central -1.178 -0.857 -1.071 -1.079
 (-3.71) (-2.71) (-3.38) (-3.41)
 Mountain -0.651 -0.430 -0.501 -0.508
 (-2.38) (-1.62) (-1.89) (-1.90)
 Executions Lagged -0.087
 (-2.90)
 P (Arrest) -0.914 -0.729 -0.736
 (-5.29) (-4.22) (-4.26)
 Poverty 0.023
 (0.79)
 Death

 Nonwhite 0.036 0.043 0.044 0.044
 (4.71) (5.81) (5.97) (5.97)
 Electrocution -2.746 -2.735
 (-7.82) (-7.78)
Summary Statistics
 [chi square] 373.71 394.65 431.08 431.13
 Log L -84.76 -71.29 -56.06 -56.05
 BP test 132.66 54.87 59.76 59.06

(a) Asymptotic t statistics, standard normal deviates, are in
parentheses. Critical values for the standard normal distribution
are 1.645 for [alpha] = 0.10, 1.96 for [alpha] = 0.05, and 2.575
for [alpha] = 0.01, assuming two-tailed tests.

Table 4. Determinants of Multiple Murders: Poisson
Maximum-Likelihood Estimates

 Dependent Variable: Number of
 Multiple Murders

 1 2 3 4

Constant 68.15 100.37 68.15 104.40
 (2.21) (3.25) (2.21) (3.37)
Year -0.034 -0.049 -0.034 -0.052
 (-2.17) (-3.19) (-2.17) (-3.35)
Metro 0.020 0.019 0.020 0.025
 (12.5) (11.7) (12.5) (17.3)
Poverty 0.122 0.130 0.122 0.140
 (15.1) (16.3) (15.1) (17.4)
Nonwhite -0.031 -0.031 -0.031 -0.025
 (-11.9) (-13.2) (-11.9) (-11.0)
Graduation Rate -11.90 -27.39 -11.90 -12.45
 (-0.79) (-1.76) (-0.79) (-0.89)
Unemployment -0.073 -0.109 -0.073 -0.066
 (-2.36) (-3.59) (-2.36) (-2.19)
P (Arrest)

Death 0.505 0.505
 (7.58) (7.58)
Executions Lagged 0.028 0.025
 (7.56) (6.91)
Electrocution 0.434 0.561 0.434 0.480
 (5.29) (6.88) (5.29) (5.82)
Guns -0.586 -0.562 -0.586
 (-11.8) (-11.2) (-11.8)
New England -1.501 -1.843 -1.501 -1.687
 (-14.4) (-19.6) (-14.4) (-18.3)
Mid-Atlantic -1.134 -1.127 -1.133 -1.364
 (-13.1) (-13.1) (-13.1) (-12.4)
South Atlantic -1.315 -1.516 -1.315 -1.582
 (-14.9) (-17.7) (-14.9) (-18.1)
East South Central -1.299 -1.408 -1.299 -1.493
 (-11.6) (-12.4) (-11.6) (-13.0)
West South Central -0.201 -0.494 -0.201 -0.605
 (-2.60) (-5.49) (-2.60) (-6.74)
East North Central -0.563 -0.733 -0.563 -0.397
 (-7.02) (-9.19) (-7.02) (-5.39)
West North Central -1.092 -1.371 -1.092 -0.858
 (-9.00) (-11.4) (-9.00) (-7.74)
Mountain -1.437 -1.460 -1.437 -1.338
 (-18.7) (-18.7) (-18.7) (-17.2)
[chi square] 2406.7 2398.9 2406.7 2271.2
Log L -1457. -1461. -1457. -1525.

 Dependent Variable: Number of
 Multiple Murders

 5 6 7 8

Constant 18.17 41.46 169.82 193.39
 (0.58) (1.31) (3.36) (4.66)
Year -0.009 -0.021 -0.084 -0.095
 (-0.58) (-1.30) (-3.60) (-4.16)
Metro 0.021 0.017 -0.002 0.029
 (13.4) (10.5) (-0.81) (18.5)
Poverty 0.132 0.133
 (16.4) (16.5)
Nonwhite -0.023 -0.025 -0.032 -0.049
 (-11.4) (-9.79) (-11.8) (-22.1)
Graduation Rate 4.14 3.169 -88.37 -25.47
 (0.28) (0.21) (-4.13) (-1.91)
Unemployment -0.037 -0.049 0.223 0.163
 (-1.17) (-1.56) (7.91) (6.63)
P (Arrest) 0.464 0.463 -0.228 -2.156
 (11.2) (11.4) (-0.83) (-9.44)
Death 0.716 0.728 0.215
 (10.4) (10.5) (1.35)
Executions Lagged 0.031 0.448
 (8.06) (25.2)
Electrocution 0.340 0.370 0.120
 (4.13) (4.50) (1.52)
Guns -0.567 -0.592 -0.264 -0.580
 (-11.3) (-11.7) (-4.99) (-12.1)
New England -1.279 -1.246 -0.330 -1.564
 (-12.3) (-12.0) (-3.15) (-17.5)
Mid-Atlantic -1.084 -1.095 -0.367 -0.589
 (-12.4) (-12.5) (-4.37) (-7.29)
South Atlantic -1.177 -1.296
 (-13.1) (-14.4)
East South Central -1.119 -1.262 -0.232 -0.234
 (-9.69) (-10.8) (-2.32) (-2.41)
West South Central -0.175 -0.521 0.666 0.692
 (-2.26) (-5.79) (9.76) (11.1)
East North Central -0.443 -0.484 0.350 -0.120
 (-5.36) (-5.86) (4.41) (-1.69)
West North Central -0.828 -0.975 0.126 -0.272
 (-6.69) (-7.82) (1.12) (-2.67)
Mountain -1.301 -1.351 0.119 -0.829
 (-16.9) (-17.3) (1.38) (-12.0)
[chi square] 2513.2 2571.3 2631.3 1908.9
Log L -1404. -1375. -1345. -1706.