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\begin{document}

\title{Crime and rationality:\ An empirical study of murdered police officers}
\author{Leon Taylor \and Instructor \\
%EndAName
Marshall University\\
Lewis Business College\\
Department of Finance and Economics\\
Huntington, West Virginia USA}
\maketitle

\begin{abstract}
The rate at which police officers were murdered rose over the 1960s,
although the expected penalties were also growing more severe. During that
period, a subjective index indicates, arrest suspects may have been growing
more prone to commit crimes for impulsive or political reasons rather than
out of rational behavior. [\textit{JEL\ K42}]
\end{abstract}

\section{Introduction}

Murders in which no rational gain is immediately evident pose an important
issue for policymakers. The killing of police officers may discourage police
presence in areas perceived as dangerous, many of them poor neighborhoods;
killings in schools may hinder learning. Such murders also pose an
intriguing puzzle for the economic theory of crime. Murders committed in
public --- or against public symbols such as police officers and politicians
--- carry a high risk of conviction and of long imprisonment or of
execution. Why, then, do they occur?

This paper examines the question through an analysis of a subset of murders
for which data are relatively abundant --- the killing of police officers.

Killers of officers are much more likely to be arrested and sentenced to
life imprisonment or execution than are other murderers. From 1986 through
1995, of 967 suspects in the deaths of law enforcement officers in line of
duty in the United States, 80 percent were arrested and charged; 12 percent
were ``justifiably killed,'' reported the FBI; and 6 percent killed
themselves. There were 11 fugitives and 18 pending cases. Of those charged
for whom the FBI knew the final outcome, 70 percent were convicted of
murder; 9 percent, of a lesser charge related to murder; 6 percent, of a
charge not related to murder; and 1 percent died in custody. Nine percent
were acquitted, or their charges were dismissed. Of 539 convicts, the courts
sentenced 21 percent to death; 46 percent, to life imprisonment; and 32
percent, to prison terms of 2 to 396 years.\cite{FBI}. In addition, police
officers pose less lucrative targets for robbery than do most victims of
murder-for-profit.

The distinctive feature of the police officer as an object of murder is his
representation of law and order. The killers of police officers may gain
direct utility from striking out against this symbol.

The point of this study is to test alternative models of police killings.
Perhaps the killer acts impulsively; perhaps he takes into account the
likelihood of life imprisonment or execution but nonetheless favors the
utility from the murder; or perhaps the murders occur not because of
criminal motives but because guns are widely available or because officers
are poorly trained to manage volatile encounters. The study seeks to
determine which of the three models might best account for police killings
in the United States in the latter half of the 20th century.

\section{Model}

The model treats the killing as an economic decision under uncertainty, and
it follows a standard analysis of uncertainty \cite{dreze}. The criminal may
be in one of three behavioral states $i$: (1) premeditation, in which he has
weighed the benefits and costs of murdering an officer; (2) impulsiveness,
in which he has not weighed the benefits and costs; and (3) externally
directed, in which he has no opportunity to kill an officer, either because
he has no lethal weapon or because the officer has averted any deadly
encounter. The probability of state $i$ is $P_{i}$, where $%
P_{1}+P_{2}+P_{3}=1.$

In states 1 and 2, the criminal may resort to one of two actions, denoted as 
$a$:\ Kill the officer ($a=1$) or don't kill him ($a=0$). In state 3, he can
choose only not to kill the officer. When choosing his action, the criminal
knows which state he is in.

In states 1 or 2, the criminal chooses a probability that he will kill the
officer, $\beta $, to maximize his expectation of utility $U$. This is a
positive monotonic function of his rebelliousness $R$ against law and order
as well as of his expected longevity of life outside incarceration, $F$:\ $%
U=U(R,F)$. In the event of the criminal's death, $U=0$.

The utility function estimated takes a Klein-Rubin form \cite{henderson}:

\[
U=\gamma _{1}(a)\ln (R-R_{0})+\gamma _{2}(a)\ln (F-F_{0}) 
\]
where $R_{0}>0$ and $F_{0}>0$ are the minimal levels of $R$ and $F$
necessary to the survival of the full-fledged criminal personality. The
criminal does not choose levels of $R$ and $F$; these are given by
institutions. Rather, in state 1, he chooses whether to let himself enjoy
utility from $R$ and $F$. The parameters $\gamma _{j}(a)$ are thus functions
of his chosen action:

\begin{eqnarray*}
\gamma _{1}(1) &>&0,\gamma _{1}(0)=0; \\
\gamma _{2}(1) &=&0,\gamma _{2}(0)>0; \\
\gamma _{1}(a) &=&\gamma _{2}(a)=0\ \text{if criminal dies.}
\end{eqnarray*}

Through his premeditated choice between killing or not killing the officer,
his utility function in state 1 will change: $\gamma _{1}$ and $\gamma _{2}$
cannot both remain positive. The analysis thus treats his decision as one
that will reshape his personality.

To satisfy data constraints, the analysis assumes that the criminal in state
1 believes that, if he kills an officer, then he will be caught and
sentenced either to life imprisonment or to execution. In either case, $%
F=F_{0}$. The criminal also believes that, if he does not try to kill the
officer, then he will not be sentenced either to life imprisonment or to
execution.

In premeditation, the criminal chooses $\beta $ to maximize expected utility:

\[
\beta \gamma _{1}(1)\ln (R-R_{0})+(1-\beta )\gamma _{2}(0)\ln (F-F_{0}). 
\]
where $\gamma _{1}+\gamma _{2}=1$. Thus

\begin{eqnarray}
\gamma _{1}(1)\ln (R-R_{0}) &>&\gamma _{2}(2)\ln (F-F_{0})\Rightarrow \beta
=1,  \label{Picking beta} \\
\gamma _{1}(1)\ln (R-R_{0}) &<&\gamma _{2}(0)\ln (F-F_{0})\Rightarrow \beta
=0,  \nonumber \\
\gamma _{1}(1)\ln (R-R_{0}) &=&\gamma _{2}(0)\ln (F-F_{0})\Rightarrow 0\leq
\beta \leq 1.  \nonumber
\end{eqnarray}

In state 2, the impulsive criminal expects that killing the officer will
yield utility $U(R,F)$ such that $\gamma _{j}>0,\ j=1,2$. He takes no
account of consequential execution or life imprisonment.

In state 3, the criminal must accept utility $U=\gamma _{2}(0)\ln (F-F_{0}).$

\section{Estimating the models}

\subsection{Estimating arguments in U(R,F)}

$R$ may be a function of several interacting factors. The criminal may view
the killing of a police officer as either a symbolic act of rejection or as
retaliation; and his predisposition to make such emotional decisions may
depend in part on his age.

Rejection is distinct from retaliation. An act of rejection is a
demonstration that the criminal does not belong to the group that the
opponent represents to him. It implies no subsequent acts. An act of
retaliation, on the other hand, demonstrates the consequences to an opponent
if he persists in some policy.

The murder of an officer may symbolize, for the criminal, rejection of
either the rules of society or of those who visibly enforce them. The
criminal believes himself to be acting on behalf of some group to which he
claims membership. His killing of an officer may be an exaggerated
consequence of group beliefs that he shares. Rejection of enforcers may be
evident in a rise in the crime rate in the group, $c$. Rejection of the
rules themselves may be evident in a low rate of voter participation, $1-v$.
Especially before 1964, $v$ also reflects exclusion of African-Americans
through Southern literacy tests, so that the impact of $v$ on the
probability of police murder may partly reflect the identification of the
criminal with African-Americans excluded from voting.

Rather than as rejection, the criminal might instead view the murder of an
officer as retaliation, either for harsh treatment of a group to which he
claims membership or for harsh treatment of suspects in general (such as
shootings by police officers or avertable deaths in detention). Retaliation
is indexed with $s$.\footnote{%
In principle, one may distinguish between group retaliation --- a sense that
officers are most harsh toward suspects who belong to a particular group ---
and general retaliation. In practice, the data available for this study do
not permit the distinction.}

Finally, the share of youths in the population arrested, $y$, reflects the
predisposition of the criminal to make emotional decisions. From 1988
through 1997, the average age of a suspect arrested for the murder of an
officer was 28.

In sum,

\[
R=(1-v)^{\alpha _{1}}c^{\alpha _{2}}y^{\alpha _{3}}s^{\alpha _{4}}. 
\]

$F$ may be computed as the life expectancy of the criminal at his age of
arrest, minus the expected length of incarceration.

\subsection{Estimating the probability of State 3}

$P_{3}$ depends negatively upon the availability of guns in the population, $%
G$. From 1988 through 1997, of 688 officers killed in line of duty, 92
percent were killed with firearms. Indeed, 72 percent were killed with
handguns. The .38 caliber and the 9 millimeter alone accounted for 43
percent of the handgun deaths \cite{FBI}.

$P_{3}$ also depends positively upon government investment in the police, $N$%
. It seems likely that the preparation of the officers for potentially
deadly encounters influenced their chances for survival. From 1988 through
1997, 62 officers killed in line of duty (9 percent) were killed with their
own weapons. More than half of the officers killed by gunshot wounds were
within 5 feet of the attacker. Almost half of the fatal wounds were to the
head, more than 100 occurring despite the use of body armor \cite{FBI}.

A hill-shaped probability distribution for $P_{3}$ seems appropriate. An
increase in the number of guns per capita seems likely to decrease $P_{3}$
by the most when weapons reach younger residents and less likely to decrease 
$P_{3}$ by much when virtually the entire population already owns guns. An
increase in police training and support seems likely to increase $P_{3}$ by
the most when it reaches officers who only occasionally have encounters with
the potential for violence; that is, they would be completely unprepared for
the encounters had they received no training. The study will estimate $P_{3}$
as normally distributed:

\begin{eqnarray*}
P_{3} &=&P\{Y_{1}\geq Y\}=\int_{-\infty }^{Y_{1}}\frac{1}{\sqrt{2\pi }}\exp
\left\{ \frac{-h^{2}}{2}\right\} \ dh, \\
Y_{1} &=&c_{1}-c_{2}G+c_{3}N,
\end{eqnarray*}
where $h$ is a transformation of $Y$ with a normal distribution, a mean of
zero, and a variance of one.

The study presumes that when $Y_{1}>Y$, then the officer will put himself
out of range of death at the hands of the subject whom he is arresting. $%
P_{3}$ is thus the probability that the subject arrested will have no chance
to kill the officer.

The study estimates a separable function for $Y_{1}$ since the marginal
impact of guns on the probability of deadly encounters would not seem to
depend greatly on police preparation. The converse would also seem to hold.

\subsection{Estimating the general model}

$P_{k}$ is the probability that the criminal kills the officer in an arrest; 
$P_{a=1}$ is the probability that the criminal decides to kill the officer.
Then, bearing in mind that the criminal will certainly decide to kill the
officer in state 2,

\begin{equation}
P_{k}=P_{1}P_{a=1}+(1-P_{1}-P_{3}).  \label{estimating pk}
\end{equation}

Equation (\ref{Picking beta}) gives the condition for the criminal to choose 
$a=1$. Values of $F$ and $R$ vary from year to year $t$, so that the value
of the criminal's utility may be treated as a random variable. The
probability that the utility value in a given year exceeds the threshold
value required for a decision to kill may be assumed to follow a normal
distribution. Extremely small or large values of $F_{t}$, which reflects the
median age of criminals in year $t$, are about equally less likely than
medium values, suggesting a hill-shaped distribution. This reasoning leads
to the cumulative function

\begin{eqnarray}
P_{a=1} &=&P\{Y_{2t}\leq Y_{0}\}=\int_{-\infty }^{Y_{0}}\frac{1}{\sqrt{2\pi }%
\sigma }\exp \left\{ \frac{-(Y_{2t}-\overline{Y_{2t}})^{2}}{2\sigma ^{2}}%
\right\} \ dY_{2t},  \label{probit} \\
Y_{2t} &=&(1-\gamma _{1})\ln (F_{t}-F_{0})-\gamma _{1}\ln (R_{t}-R_{0}), 
\nonumber
\end{eqnarray}
where $\overline{Y_{2t}}$ is the mean of $Y_{2t}$ and $\sigma $ its standard
deviation (\cite{ross}). A probit model of \ref{probit} will yield estimates
of $\gamma _{1}$ and of $P\{Y_{2t}\leq 0\}=F_{Y}(0)$. Using these estimates
in a rewriting of (\ref{estimating pk}),

\begin{equation}
P_{k}+P_{3}-1=P_{1}(F_{Y}(0)-1),  \label{really estimating pk}
\end{equation}
enables estimation of $P_{1}$ as the coefficient in a regression.

\section{Data}

All data derive from time series in \cite{commer} and are annual series for
1945 through 1970. This period is an historical evolution from a time of
relative national unity (the post-WWII era) to one of divisiveness (the
1960s), so it seems appropriate to a study of violent actions in which
social cohesion may play an important role. Table 1 describes the series,
and Appendix A has details.

\begin{tabular}{|l|l|}
\hline
\textbf{Variable} & \textbf{Description} \\ \hline
$P_{k}$ & (number of officers killed by felons) / (number of arrests) \\ 
\hline
$N$ & (spending by all governments on police protection) / (number of
arrests) \\ \hline
$G$ & (number of homicides by firearms) / (total number of homicides) \\ 
\hline
$c$ & (number of arrests) / (resident population) \\ \hline
$P_{f}$ & (number of executions for murder) / (number of officers killed) \\ 
\hline
$v$ & percentage of eligible adults voting in most recent Presidential
election \\ \hline
$y$ & (number of persons arrested under age of 24) / (total number of
arrests) \\ \hline
$t$ & (number of homicides through intervention of police) / (all homicides)
\\ \hline
$s$ & (number of nonwhites arrested) / (total number of arrests) \\ \hline
\end{tabular}

\section{Estimations}

\subsection{Estimating P$_{k}$}

The risk of death to a police officer diminished historically but remained
high relative to the risk in other occupations. The risk of murder to the
police officer in the course of arrest fell sharply from 12.7 in 100,000
arrests in 1946 to 1.37 in 100,000 arrests in 1960, then rose slightly but
steadily to 2.33 in 100,000 arrests in 1970.\footnote{%
The low probability in 1960 may owe partly to the inclusion of data for
Alaska and Hawaii for the first time that year.}

Total police expenditures per arrest followed a similar pattern. Real
spending per arrest fell steeply from \$1610 in 1945 to \$530 in 1965 and
then rose to \$620 in 1969. These figures do not express the direct costs of
arrest; they include spending on other police services and on the
maintenance of police buildings. They indicate the value of general
investment in capital that the officer may deploy in an arrest.

Although both the level of investment and the risk of death for an officer
were much lower in 1970 than in the late 1940s, the risk of death by remote
weapons rose. The share of all police murders that occurred through firearms
or explosives rose steadily from .54 in 1945 to .67 in 1970.

One may loosely infer that, although police work remained dangerous in 1970,
it was much less violent than in the late 1940s.

\subsection{Estimating R and F}

The tables below give the results for a log-linear estimation of $R$. The
rate of arrest is elastic with respect to the share of the arrested who are
youths (\textit{YouthsArrested}) or who are nonwhite (\textit{NonWhiteArrests%
}). Both coefficients are positive as expected. The share of all homicides
that occur through the intervention of police (\textit{MurdersWithCops})
relates negatively to the rate of arrest. The interventions may signal
danger to potential criminals and thus induce them to try harder to avoid
either committing crimes or getting arrested.

The rate of arrest also relates negatively to the share of adults who don't
vote in Presidential elections (\textit{NotVoting}). The relationship
remains negative after inclusion, among independent variables, of the rate
of unemployment, which correlates negatively with the rate of nonvoting and
positively with the rate of arrest. One can speculate that the negative link
between the arrest rate and the nonvoting rate reflects the impact of apathy
on voting and on crime.

\begin{tabular}{|l|l|}
\hline
\textbf{Regression Statistics} &  \\ \hline
Dependent variable & \textit{Arrest rate} \\ \hline
Adjusted R Square & 0.949925 \\ \hline
Standard Error & 0.151121 \\ \hline
F & 119.562 \\ \hline
Observations & 26 \\ \hline
\end{tabular}

\begin{tabular}{|l|l|l|l|l|}
\hline
\textbf{Variables} & \textbf{Coefficients} & \textbf{\ t Stat} & \textbf{%
Lower 95\%} & \textbf{Upper 95\%} \\ \hline
\textit{Intercept} & -7.80301 & -5.15274 & -10.95225 & -4.65376 \\ \hline
\textit{NotVoting } & -2.88439 & -6.21009 & -3.85031 & -1.91847 \\ \hline
\textit{YouthsArrested } & 1.05250 & 6.95303 & 0.73771 & 1.36730 \\ \hline
\textit{MurdersWithCops} & -1.07245 & -4.15875 & -1.60874 & -0.53616 \\ 
\hline
\textit{NonWhiteArrests} & 1.57864 & 2.68389 & 0.35543 & 2.80185 \\ \hline
\end{tabular}

In general, the index of rebelliousness fell slightly over late 1940s and
then rose steadily to 1970.

The implied penalty of killing an officer followed a similar pattern. It
fell through the late 1940s and early 1950s, then rose. The penalty depends
partly on how much longer the killer might have expected to live, had he not
risked life imprisonment or execution by killing an officer. The expectation
of additional years of life for an arrest suspect, minus the expected length
of imprisonment, fell from 36.5 years in 33.4 years in 1952, then rose to 42
years in 1970.

These trends raise the possibility that, since the likelihood of a police
death per arrest fell over the 1950s and rose over the 1960s, the impulse to
kill due to rebelliousness may have grown more important, relative to the
penalty of lost freedom, during the Sixties. A more precise conclusion may
hinge upon the estimation of $P_{3}.$

\section{Appendix A:\ Data}

$P_{k}$. Data sources:\ Series H987-989 in \cite{commer}. The number of
officers killed by felons is available in \cite{commer} only for 1961
through 1970. For 1945 through 1960, the study estimates the number of
officers killed by felons. It multiplies the total number of law enforcement
officers killed in line of duty by a factor based on a linear regression.
This regresses, on the year, the share of officers killed by felons, for
years 1961 through 1970, excluding 1968. The equation is \textit{FelonShare
= } - 23.2 + 0.0121 \textit{Year, }$R^{2}$ (adjusted) = .441. T-statistics
are (-2.63, 2.70) on the intercept and coefficient, respectively. The F
statistic (7.31) suggests that the regression provides a more valuable
estimate of the share of officers killed by felons than the mean of shares
for 1961-1970.

The regression excludes 1968 because the share of officers killed by felons
that year was onetime and unusually low. The share in 1968 was .52, compared
to a mean of .61 for 1961-70. The drop was evidently due as much to an
unusually large number of officers killed in accidents as to an unusually
small number of officers killed by felons. From 1967 to 1968, the number of
officers killed in accidents rose, from 47 to 59; the number of officers
killed by felons dropped, from 76 to 64.

Most murders of officers do not occur during arrests. Of 61 officers killed
in situations involving felons in 1998, 22 (36 percent) died during arrests
or while serving arrest warrants \cite{FBI2}. Focusing a study of police
murders on arrests, however, allows clearly for two possibilities:\ The
killer may have acted on impulse; or he may have acted after pre-meditation.
The second-largest source of police murders, investigation of disturbance
calls, seems more likely to involve impulsive killings. To study the link
between criminal behavior and rationality, a focus on arrests seems
appropriate.

$v.$ Data source:\ Series Y27 in \cite{commer}.

$s.$ The share of all homicides that occurred through the intervention of
police. It is computed from Series H977 and H971 in \cite{commer}. Estimates
for 1945 through 1948 are computed from a linear regression of $s$ on the
year for 1949-1970, with an adjustment in the intercept such that the
percentage change from 1948 to 1949 equals the mean annual percentage change
from 1949 through 1970.

The original regression was $s=1.0933-.0054\ Year$, t-statistics (7.8, -7.6)
on the intercept and coefficient, respectively; adjusted $R^{2}=.73$. The
mean annual percentage change from 1949 through 1970 was -1.9138 percent. To
constrain the change from 1948 to 1949 to equal this change, I subtracted
.006224 from the intercept.

\textit{F.} This variable is the suspect's expectation of years of freedom
remaining in his life if he is now arrested. It measures the penalty to the
suspect of killing the officer and consequently suffering life imprisonment
or execution.

Here's how I computed the variable. Let the suspect's age be $s$. Let his
life expectancy at that age be $L(s)$. Let the expected length of his prison
term, in the event of an arrest at time $t$, be $P(t)$. Let the share, in
all arrests, of those of suspects of age $s$ be $A(s)$. Then

\[
F(t)=\sum_{s=10}^{70}A(s,t)[L(s,t)-P(t)]. 
\]

Details follow.

\textit{s, A(s,t) }and \textit{L(s,t).} Historical data are available for
persons arrested in age brackets. For a given age bracket, I estimated the
mean age for those arrested; I assumed that those arrested varied between
ages 10 and 70. I then estimated the life expectancy for a male of that age
of a given racial group by interpolating linearly between the reported life
expectancies for the age just above $s$ and the age just below $s$.

Data are available for life expectancy at birth as well as at ages 20, 40,
60 and 70 in Series B116 through B125 in \cite{commer}.

The life expectancy at age $s$ in time $t$ is defined here as the average
number of years that those of that age would continue to live if they were
subject to the same death rates, for particular age brackets, as prevailed
at time $t$\textit{.} Current death rates --- rather than of either actual
future death rates or projected future death rates --- seem most appropriate
in computing the arrest suspect's expectation of the length of his remaining
life. I have drawn upon current life tables computed for a hypothetical
cohort of a stationary population that is supported by 100,000 annual
births. The computations assume no migration \cite{vital}. The current life
tables compute the ``average number of years'' in the life expectancy at age 
$s$ in this way:\ Estimate the total number of years remaining to those in
the hypothetical cohort who would survive to the $s$th birthday; and divide
this number of years by the number of survivors.

Annual life expectancy data are available, for given ages, back to 1955.
Estimates are also available for the periods 1949-51 and 1939-41. To
estimate annual data for 1945 through 1954, I regressed the natural log of
life expectancy on the natural log of the year. I then calibrated the
regression estimates to those available for 1949-51 and 1939-41 by resetting
the intercept.

I computed two series of life expectancy:\ White and nonwhite. ``White''
includes those reported as Mexican, Cuban and Puerto Rican. \textit{L(s)} is
the weighted average of these two series for a given mean age \textit{s},
where the weights are the share of that racial group in all arrests.

I distinguished between racial groups because both life expectancy and
shares of arrest differ significantly between the groups. I used life
expectancy figures for males since these differ significantly from those for
women, who are rarely suspects in murders of officers. From 1988 through
1997, of 950 suspects in murders of officers, 886 (93 percent) were male 
\cite{FBI}.

\textit{P(t): Expected duration of imprisonment.} This annual series is
calculated as the annual probability of imprisonment times the long-run
length of the prison term actually served.

\textit{Length of prison term}. This was estimated by regressing the number
of prisoners released in year $t$ upon the current value and lags of the
number of prisoners received into prison from the courts. I used the lags as
estimates of effective prison terms. The average prison term is a weighted
average of the lags, where the coefficient values provide the weights. To
determine the number of lags to use, I added longer lags to the regression
until adjusted R-squared exceeded .9. The data come from Series H1135, 1138
and 1144 of \cite{commer}.

I estimated the length of the prison term for two time periods. For
1945-1957, prison terms appear relatively short for inmates of state and
federal penitentiaries:

\begin{tabular}{|l|l|}
\hline
\textbf{Regression statistics} &  \\ \hline
Dependent variable: & \textit{Prisoners released} \\ \hline
Adjusted R Square & 0.92 \\ \hline
Standard Error & 2640.79 \\ \hline
Period & 1945-1957 \\ \hline
Observations & 13 \\ \hline
\end{tabular}

\begin{tabular}{|l|l|l|}
\hline
\textbf{Model} &  &  \\ \hline
\textit{Variable} & \textit{Coefficients} & \textit{\ t Stat} \\ \hline
Intercept & 3061.88 & 0.46 \\ \hline
Received in current year & 0.26 & 1.01 \\ \hline
Received in prior year & 0.74 & 3.21 \\ \hline
\end{tabular}

The estimated average length of the prison term for this period is $%
.26*1+.74*2=1.74$.

For 1958-1970, prison terms appear significantly longer:

\begin{tabular}{|l|l|}
\hline
\textbf{Regression statistics} &  \\ \hline
Dependent variable: & \textit{Prisoners released} \\ \hline
Adjusted R Square & 0.91 \\ \hline
Standard Error & 2200.26 \\ \hline
Period & 1958-1970 \\ \hline
Observations & 13 \\ \hline
\end{tabular}

\begin{tabular}{|l|l|l|}
\hline
\textbf{Model} &  &  \\ \hline
\textit{Variable} & \textit{Coefficients} & \textit{\ t Stat} \\ \hline
Intercept & -100185.81 & -4.13 \\ \hline
Received in current year & 0.83 & 3.68 \\ \hline
Received in prior year & 0.38 & 1.98 \\ \hline
Received two years before & 0.09 & 0.44 \\ \hline
Received three years before & -0.01 & -0.03 \\ \hline
Received four years before & 0.22 & 1.11 \\ \hline
Received five years before & 0.83 & 3.93 \\ \hline
\end{tabular}

The estimated length of the prison sentence is 3.42. I set to zero the
relative weight on prisoners received three years before; its impact on
current prisoners released is relatively insignificant. Thus the estimated
length is $(.83*1+.38*2+.09*4+.22*5+.83*6)/(.83+.38+.09+.22+.83)=\allowbreak
3.\,42.$

\textit{Probability of imprisonment.} This is calculated as the share, of
all criminal cases that disposed of the defendant, that ended in a
conviction of imprisonment. There is no need to calculate a probability of
arrest, since the data concern only suspects on the point of arrest. Data:\
Series H1100 and 1108 of \cite{commer}.

\textit{G.} Homicides by firearms and explosives as a share of all
homicides: Index of gun supply among criminals. Data: Series H975 and H971
of \cite{commer}.

\textit{N.} Real spending by all governments on police protection, per
arrest, deflated by the GNP implicit price deflator. Data for 1951, 1947 and
1945 are linearly interpolated. Data:\ Series H1013, E1 and H999 of \cite
{commer}.

\begin{thebibliography}{9}
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in economics:\ Readings and exercises}. Revised edition. San\ Diego:\
Academic Press, 1989.

\bibitem{henderson}  Henderson, J.M. \textit{Microeconomic theory:\ A
mathematical approach}. 1980. Third edition. New York: McGraw-Hill.

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\end{document}
