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\begin{Large}{\bf

 Convictions versus Conviction Rates: The Prosecutor's Choice }
\\(The American Law and Economics Review,  11(1): 47--78 (Spring 2009))

\end{Large}

\bigskip

 \bigskip


 \bigskip


 \bigskip




May 19, 2009

\bigskip
   Eric Rasmusen,      Manu Raghav, and  Mark Ramseyer

\bigskip


\bigskip


\bigskip


{\it Abstract}

 \end{center}

\begin{quotation}

  It is natural to suppose that a prosecutor's conviction rate-- the
ratio of convictions to cases prosecuted-- is a  sign of his
competence. Prosecutors, however,  choose which    cases to
prosecute.
If they prosecute  only the strongest cases, they will have high
conviction rates. Any system which pays attention to conviction
rates,
as opposed to the number of convictions, is liable to abuse.  As
a
prosecutor's budget increases, he  allocates it between
prosecuting
more cases and  putting more effort into existing cases.  Either
can
be socially desirable, depending on particular circumstances. We
model
the tradeoffs theoretically in two models, one of a benevolent
social
planner  and one of a prosecutor who values not just  the
number of
convictions but the conviction rate and unrelated personal goals.
We
apply
the model to  U.S. data drawn from county-level crime statistics
and a
survey of all state prosecutors by district.  Conviction rates  do
have a small negative correlation with prosecutorial budgets, but
conditioning on other variables in regression analysis, higher
budgets
are associated both with more prosecutions and higher
conviction
rates.

\end{quotation}

 \bigskip

\noindent
 Eric Rasmusen: Indiana University, Kelley School of Business,
BU 438, 1309 E. 10th Street, Bloomington, Indiana  47405-1701.
Office:
(812) 855-9219. Fax: 812-855-3354.
\href{mailto:erasmuse@indiana.edu}{
Erasmuse@indiana.edu},  \hspace{5 pt}
\url{http://www.rasmusen.org}.


\medskip
\noindent
Manu Raghav:  DePauw University, Department of Economics
and
Management, 7 E. Larabee St, Greencastle, Indiana 46135.
\href{mailto:manuraghav@depauw.edu}
{Manuraghav@depauw.edu}.

\medskip

\noindent
Mark Ramseyer: Harvard Law School, Cambridge, Massachusetts
02138.
\href{mailto:ramseyer@law.harvard.edu}{
Ramseyer@law.harvard.edu.  }


\bigskip


 We thank Michael Baye, Matthew  Gens, Louis Kaplow,  Reed
Smith,
and  participants in
talks at the 2005 Midwest Law and Economics Association
Conference,
IUPUI,  Emory University, Harvard Law School,   and   Indiana
University for helpful comments.

\newpage

%%-----------------------------%----------------




\parindent 24pt
\parskip 12pt

\noindent
 \section{Introduction}
 \setcounter{page}{1}
\begin{quotation}
\begin{small}


Attorney Thomas Broderick Jr. will formally announce Tuesday
that he is a candidate for the Democratic nomination for
Madison County prosecutor. He was chief deputy for Prosecutor
William F. Lawler Jr. for more than seven years, claiming a 98
percent personal conviction rate on cases ranging from drugs to
murder.\footnote{Attorney Says He'll Be Candidate for
Prosecutor, {\it Indianapolis Star}, January 17, 1998, at N02. }

\end{small}
\end{quotation}


%\setstretch{2}


	So reads a typical press release in an election for county
prosecutor.  The prosecutor's role  is one of the most important
in  criminal justice, yet   one of the most
neglected by scholarship.  Standing between the criminal and the
police on the one hand
and the courts and prisons on the other, the prosecutor has two
jobs.
One, as  candidate Broderick stresses, is to convince the court to
convict.  The
other is to decide whom to prosecute.

	Other things equal, it is good to have a prosecutor who
wins
cases.  But other things are not equal.  A prosecutor might
choose to
prosecute innocent enemies.  He might  choose to
prosecute very few cases.  Were he to spend an entire year
prosecuting
one case, he could easily obtain not a 98 percent conviction rate,
but
100 percent.   A high win rate might actually  tell us the
prosecutor
is slack, not tough. As a former Securities and Exchange
Commission
examiner said after the SEC
failed to respond to explicit warnings about the Madoff Ponzi
scheme:
\begin{quotation}
\begin{small}



During my time at the SEC, I heard the excuses about why cases
that
we, the examination staff, uncovered failed to warrant actions by
the
enforcement staff. Too small -- best wishes to the small investors
involved. Too complicated -- derivatives would fall under this
category. Too politically connected -- don't want to rock the
boat. In
many ways, the Madoff fraud shares the same excuses for not
being
uncovered as the unmitigated disasters that were Fannie Mae and
Freddie Mac, AIG and the subprime mortgage lenders.

Part of the problem is that enforcement likes to bat 1.000,
because of
the way the system works, meaning they like to bring only cases
they
will win. I recall that anything less than that resulted in warning
letters or no actions whatsoever.\footnote{Eric W. Bright, ``It
Isn't
Surprising That SEC Missed Madoff's Scam,'' letter to the editor,
{\it
The  Wall Street Journal,} 22 December  2008,
\url{http://online.wsj.com/article/SB123000061773329323.html
}.}

\end{small}
\end{quotation}

	In this article, we focus on the  problem of case selection:
whether to allocate resources broadly over many cases  or
intensively to a few cases.  Consider the contrast between the
United States and Japan.  Conviction rates are high in both
countries, but higher in Japan.  American state prosecutors win
85 percent of their felony cases and 90 percent of their
misdemeanors.  Federal prosecutors   win 90 percent.  Yet
Japanese prosecutors win 99.9 percent.\footnote{The conviction
rates of U.S.  state prosecutors are obtained from the Census of
Prosecutors, 2001, and for federal prosecutors   from           \url{
http://www.ojp.usdoj.gov/bjs/pub/press/fcjt03pr.htm}.  For
Japanese sources, see Ramseyer and Rasmusen (2003), chapter 6.
}  This is not due to any biased judicial incentives:  judges do not
suffer a career hit for acquitting defendants (Ramseyer and
Rasmusen, 2001).

	The contrast between U.S. and Japan holds true even
among contested cases.  Although Japanese law does not allow
plea bargains, defendants can confess, and those who do confess
receive
lighter sentences. Among Japanese defendants who   contested
their guilt in 2004, 98.4 percent were convicted.  Among  U.S.
federal defendants who contested their guilt in 2003, only 83
percent were convicted. \footnote{Table 12 of \url{
http://www.ojp.usdoj.gov/bjs/pub/pdf/fcjt03.pdf}} Japanese
courts really do seem tougher.

	Might not case selection explain the difference better than
toughness?  Suppose Japanese prosecutors drop all but their
strongest cases.  After all, their resources are stretched thin.
Japan employs about 1,200 prosecutors.  With twice the
population, the U.S. employs 32,000.  Japan has less crime, but
that hardly explains the difference.  Each year, police in the U.S.
make about 14 million non-traffic arrests--- 438 crimes per
prosecutor.  Japanese police clear about 1.4 million Criminal Code
violations per year---   1,166 crimes per prosecutor.  The number
of crimes is low in Japan, but the number of prosecutors is even
lower.

	Differences in prosecution rates (prosecutions/crimes)
reflect this difference in workload.  In 1994, U.S. state courts
convicted 870,000 people of felonies, and the federal courts
another 44,000. Combining  this with
the FBI's index crime arrests figure  yields  a 42 percent
prosecution/arrest ratio for serious crimes (note that the 14
million arrests figure mentioned earlier includes misdemeanors;
there are many fewer  index crime arrests).  In Japan, by contrast,
prosecutors only
prosecuted only 14 percent of the 1.4 million people arrested for
Criminal Code violations in 2002.  The comparison is crude -- the
crime data is not strictly comparable.  But it does suggest that
Japanese prosecutors allocate their scarce time to the cases with
the strongest evidence.\footnote{For a dissenting view, see
Saxonhouse (2001). For more on the prosecutorial system in
Japan, see the book by Johnson (2001). } The  lower budget  in
Japan leads to a  lower prosecution rate.   Might the lower budget
in Japan be raising the conviction rate
(convictions/prosecutions)?

	In civil cases, Priest and Klein (1984) famously explained
how the selection dynamic affected verdict rates.  We may
observe plaintiffs winning 90\% of the cases, they noted, but
that does not tell us about the strength of the plaintiffs' cases
generally.  Instead, it tells us only about their strength in the
subset of cases in which the plaintiffs could not agree with the
defendant about what would happen at trial. Quite possibly (but
not necessarily), the plaintiffs would have lost most of their claims
-- but because they knew they would lose they settled out-of-
court for small sums.

	The same dynamic applies to criminal cases.  Conviction
rates at trial (convictions/prosecutions) do not tell us about the
strength of the prosecutors' cases generally.  Instead, the rates tell
us only about the strength of those few cases the prosecutor
chooses to pursue and the defendant chooses to contest. In the
U.S., only 4\% of the defendants go to trial.  In the data
discussed above, 61\% of the actual trials were bench trials and
39\% jury trials.  85\% ended in guilty verdicts and 15\% in
acquittal.\footnote{This implies the percentage of hung juries
was close to zero, which is surprising. A different study of state
and federal courts found that 6.2\% of juries were hung.
Nicole L. Mott, ``Part II:  The Role of Juries in State Courts,''  p.
101,  Brian Ostrom, Neal B. Kauder, and Robert C. LaFountain,
``Examining the Work of State Courts, 2001''(Williamsburg, VA:
National Center for State Courts, 2002) \url{
http://www.ncsconline.org/D_Research/csp/2001_Files/2001_
Part_II_Secti on .pdf}.}   A better measure than conviction rates
at trial of a prosecutor's effectivness is his ability to handle plea
bargains (which after all, have a greater bearing on the aggregate
punishments he is able to obtain):  how many defendants plead
guilty, and what sentences do they receive?

	Rather than examine trial conviction rates, in the article
below we look at conviction rates for all cases pursued, whether
through plea bargaining or through trial.  This has more meaning
than trial wins alone, because if a prosecutor does not even
pursue a case he has failed to punish the criminal.  A high
conviction rate might still indicate lenient plea bargains, but he
has at least done better than if he had dropped the case.  We will
also look at a prosecutor's choice of cases to prosecute.  In Japan,
prosecutors apparently prosecute only the easiest cases, and we
ask whether the same  pattern exists in the U.S.

	We begin with two models of prosecutorial behavior:  (a) a
prosecutor who, as social planner, maximizes criminal sentences,
and (b) a prosecutor who balances that goal against a high
conviction rate and other personal objectives.  We then use  data
on the level of counties and state prosecutor districts  to explore
the relationship between prosecution rates, conviction rates, and
budgets.  Ultimately, we show that elected prosecutors may
respond to electoral pressure by using resources to raise their
conviction rates beyond the levels a social planner would choose,
and thereby (by reducing the number of cases pursued) to lower
the total punishment imposed.


\bigskip

\noindent
 \section{The Literature on Prosecutors}

A complaint at least as old as Forst   \&    Brosi (1977) is that
prosecutors have been insufficiently studied, either with theory or
empirics.  That article did build a theoretical model of the
prosecutor as attempting to achieve penalties based on the
seriousness
of the crime and the past record of the criminal given limited
resources and the possibility of plea bargaining. Their article built
on  the simpler model of Landes (1971) and   tested the theory
with regressions on   data at the level of the individual.  Other
studies have also
looked at the variables determining which cases are prosecuted.
Myers
\& Hagan (1979)  and  Albonetti, (1986) run regressions on
individual
case data,  and Rainville  (2001) uses a survey of 77 prosecutors
that
asked them about their policies of when to prosecute and when
to drop.
Boylan (2004) looks at the internal workings of federal prosecutor
offices and finds that more experienced U.S. Attorneys put
priority on
crimes that have longer prison sentences. Glaeser, Kessler \&
Piehl
(2000) look at how cases are
handled depending on whether the defendant is represented by
private
counsel instead of a public defender. They find that in districts
with
high  lawyers'  salaries, defendants represented by private counsel
are more likely to be prosecuted by assistant U.S. attorneys than
by
state and local prosecutors.

The closest  paper to the present one  is Rhodes  (1976). Rhodes
looks at how the prosecutor uses increased budget to bring more
prosecutions and reduce the leniency of his plea bargains. He uses
data from 84 federal district courts to run regressions of case
dispositions on prosecutor resources and variables such as the
fraction of defendants with criminal records in the district. He
acknowledges the endogeneity problem for the budget, but does
not
attempt to instrument for it.

 Boylan (2005) has  examined  the incentives of federal
prosecutors by
seeing what affects the likelihood that  U.S. Attorneys (the chief
attorneys in federal prosecution offices)  later become federal
judges, a common career goal for them. He finds that a
prosecutor's
conviction rate  and number of indictments do not affect
prosecutor's
success, but the number of prison months achieved does matter.
As we
will see, the results of Boylan (2005) tend to support our model
below
of the prosecutor acting purely in the public interest, in his special
context of    U.S. federal prosecutors.  Federal prosecutors are
appointed, not elected,   and the crimes they prosecute are
different
from those in the state courts, so in the empirical work below we
should not be surprised if state prosecutors are differently
rewarded.

Miceli  (1996)  makes the important point that prosecutors (like
judges) might well care about fair sentences but not about
deterrence,
in contrast to legislatures. Thus, prosecutors might frustrate
Becker-style high penalties, infrequently imposed and cheap to
administer, because those penalties seem unfair  for  the
individual
criminals who  do  get caught. Snyder (1990) looks empirically at
one
example where prosecutors and courts might be expected to
differ from
the legislature: their response to an increase by Congress in
penalties for federal antitrust laws.

    Other work exists on optimal prosecution, but that is  a rather
different topic than prosecutors themselves. Miceli (1990), for
example, looks at how prosecutors and judges (together) should
behave
when the guilt of defendants is uncertain.  That is an important
topic, but it is a different one from the decision of how to
allocate
resources or operate under institutional constraints. Much
economic
analysis has also been  done concerning plea bargaining (e.g.
Adelstein  (1978a, 1978b), Reinganum (1988, 1993), and Baker
and
Mezzetti (2001)), but less on the career incentives of prosecutors
as
agents of the public. One exception is Huber and Gordon
(2002), which
formally models the incentives of elected prosecutors. They focus
on
what  performance measure voters should reward in order to
induce the
prosecutor to  investigate the guilt or innocence of
defendants rather than bring  to court cases he knows are
dubious. In
the present model, our focus will be quite different: it will be on
the prosecutor's decision, forced by
limitations of time and resources, to drop some cases, prosecute
others, and to prosecute some  more intensely than others.



\bigskip

\noindent
 {\bf 3.   A Model with the Social Planner as Prosecutor}

 In this section, we construct a model of
prosecutorial decisionmaking,  in which there are no agency
problems:
the prosecutor is the social planner.\footnote{The starting point
for
this model is the graphical argument    in Appendix A, Chapter 6
of
Ramseyer and Rasmusen (2001).  This model formalizes that
argument
and extends it to consideration of agency slack, length of
sentence,
and the conviction rate.} There is a continuum of potential  cases
that could be prosecuted, all involving the same
crime.\footnote{The
assumption of a single type of crime is innocuous. If there were
$n$
types of crime, with different values for conviction, we would
simply
repeat the same objective $n$ times with different production
functions and a  value coefficient   for each, e.g, a  tough
conviction function but a value of 10 for convicting  a murderer,
an
easy conviction function but a value of    2 for convicting a thief.
}
Thus,  each case has  little importance compared to the
aggregate.
Potential cases will be indexed by  ``strength,'' $\theta$, where
$\theta$ varies from 0 to 1.   The number of cases of type
$\theta$ is
represented by a density function $f(\theta)$, where
$\int_0^\infty
f(\theta) d\theta  =1$, which is to say that  we will normalize so
the
universe of possible cases has size 1.    Figure 1 shows one way
the cases might be distributed.    If the prosecutor decides to
prosecute the strongest  1/3 of cases, he takes all the cases  in
the
interval $[\overline{\theta}, 1]$ in the diagram.\footnote{We
will
assume that the interval [0,1] only includes cases the prosecutor
would like to win. Thus, a case having $\theta= .1$ does not
mean the
defendant  was less likely to have committed the crime, only that
the
prosecutor's evidence is weak. As with adding different types of
crimes, adding concerns about whether convictions for some
crimes are
really in the public interest would not  alter the model's
implications.} We will assume that the prosecutor has no
evidence of
innocence that he
can conceal from the court, so a case being ``strong'' also means
that
the defendant is more likely guilty, and that the resources available
to the prosecutor are low enough that he is far from being able
to
prosecute cases too weak to be worth prosecuting from a social
point
of view.  (Put differently, we will assume that social welfare is
maximized by spending the entire prosecutorial budget rather
than
leaving some unspent.)





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     {\bf  Figure 1:  The  Distribution  of Cases}
\end{center}





    Each case that  is    prosecuted incurs a fixed cost of $C$ and
a
variable cost of $e$. Let the probability  of winning a   case be
\begin{equation} \label{e1}
{\rm Probability \; of  \; winning} =  \theta   P(e(\theta))
\end{equation}
 where   $P(0) =0$,  $P'>0$,    $P''<0$,  and  $P(\infty) <1$, so
additional effort always helps but there are decreasing returns to
effort in a particular case and the prosecutor can never be sure of
winning.



\begin{center}
 \includegraphics{pros2.pdf}
     {\bf  Figure 2:  The Conviction Function}
\end{center}

\vspace*{36pt}


   These assumptions say  that    the probability of winning is
scaled
up or down by case strength, $\theta$, as shown in Figure 2.  If
effort
  is the same for two types of case, the type with the bigger
$\theta$ has the greater probability of winning.\footnote{The
specification also implies that      if effort  is the same for two
types of cases, the  marginal product of effort  is bigger for the
case with the bigger $\theta$, as illustrated by the tangents in
Figure 2.   That effect is not important to our
results.}

Let  the social planner's ideal sentence be $\overline{V}$, and
the
expected penalty for a case of strength $\theta$ be
\begin{equation}
\label{e1} {\rm Expected \; Penalty} =  \theta V(e(\theta))
\end{equation} where   $V(0) =0$,  $V'>0$,    $V''<0$,  and
$V(\infty) <\overline{V}$,   so additional effort always helps,
but
there are decreasing returns to effort in a particular case and  no
amount of effort can achieve the optimal penalty.\footnote{We
are
ruling out the possibility that the social planner would exert high
effort so as to fool the court into imposing a sentence higher
than
$\overline{V}$ with some probability, thus putting the expected
sentence at $\overline{V}$. }

  We have  assumed that the social planner's ideal sentence is
$\overline{V}$. Let us denote his     valuation of an expected
sentence of $  \theta S$ by $  \theta U(S)$ with $U'>0$ and
$U''<0$ in
the range $[0, \overline{V}]$, so  the penalty has  diminishing
marginal returns. We can then write  $  \theta V(e(\theta)))=
\theta
U(S(e (\theta)) ) $ for the  expected value to the social planner
of
exerting effort $e$ on case $\theta$, where  $V(0)$, $V'>0$
$V"<0$,
since $V$ is an increasing concave function of another
increasing
concave function.
\begin{center}
 \includegraphics[width=180mm] {pros3.pdf}
{\bf  Figure 3:  The   Prosecutor's    Choices of $e(\theta)$ and
$\overline{\theta}$  }
\end{center}

\bigskip

  We now come to our social planner prosecutor's problem. He
has  two
choices. First, he must decide which cases to prosecute. He will
want
to prosecute the easier cases first,  which  amounts to choosing a
lower cutoff $\overline{\theta}$ for the interval of types
$[\overline{\theta}, 1]$ that he prosecutes.  The spending on
this
weakest case prosecuted is the extensive margin, analogous to
the
least fertile land  under cultivation.   Second,  he needs to pick
the $e(\theta)$ function,   which shows how much he spends on
each
type $\theta$ of case.  this is the intensive margin, analogous to
the
marginal unit of fertilizer used on cultivated land.   Figure 3 shows
these choices graphically. His   decisions lead to the following
four
expressions.
  \begin{equation} \label{e1a}
  {\rm Prosecution \;rate}  =     \int_{ \overline{\theta}
}^1  f(\theta)
 d\theta = 1-     F(\overline{\theta}).
   \end{equation}


  \begin{equation} \label{e1b}
  {\rm  Number   \;of \; convictions }  =
\int_{\overline{\theta}}
^1  f(\theta )    \theta  P(e(\theta)) d\theta.
   \end{equation}


\begin{equation} \label{e1d}
  {\rm   Value  \;of \; convictions }  =
\int_{\overline{\theta}}^1  f(\theta )       \theta  V(e(\theta))
d\theta.
   \end{equation}



    \begin{equation} \label{e1c}
  {\rm Conviction \; Rate}   = \left(\frac{1}{   1-
F(\overline{
\theta}) }
\right)
\int_{\overline{\theta} }^1 f(\theta)   \theta P(e(\theta))
d\theta
   \end{equation}


  The   prosecution rate sums up the amount of case density,
$g$,  above the prosecution threshold, $\overline{\theta}$. In
Figure
1, $(1-   F(\overline{\theta})) = 1/3$.


 The   number  of convictions  sums up  over all the prosecuted
cases
(those on interval $[\overline{\theta}, 1]$) the density  $f$ of
the
number of cases of each type,   times the  probability of
conviction
for that type, $   \theta P$.

 The   value of convictions  sums up  over all the prosecuted
cases
(those on interval $[\overline{\theta}, 1]$) the density
$f(\theta)$
of the number of cases of each type,   times the value of
convictions
for that type, $   \theta V$.

 The  conviction rate   is the  average probability that a
prosecution
results in a conviction (which will be larger than the probability
that a {\it crime} results in a conviction). It divides the
probability of convictions from equation  (\ref{e1b})   by the
number
of cases prosecuted, from equation (\ref{e1a}).

  What is the prosecutor's payoff function?  Let us start with the
social planner as prosecutor. He would  maximize the value  of
the
expected penalties, i.e.,
 \begin{equation} \label{e2}
  \int_{\overline{\theta}}^1  f(\theta)     \theta   V(e(\theta))
d
\theta
\end{equation}
 by choice of $e(\theta)$ and  $ \overline{\theta}$, subject to
the
budget
constraint,
 \begin{equation} \label{e3}
  \int_{\overline{\theta}}^1  f(\theta )   [e(\theta)   + C]
d\theta  \leq B.
\end{equation}

\noindent
The Lagrangian for the maximization problem is
 \begin{equation} \label{e4}
\begin{array}{ll}
  {\displaystyle  \stackrel{  Maximize}{ e(\theta),
\overline{\theta}
}  \;\;
\; \;\;  L} &  {\displaystyle   = \int_{\overline{\theta}}^1
f(\theta
)
\theta  V(e(\theta)) d\theta } \\
    & \\
 & {\displaystyle  +  \lambda \left\{ B  -
\int_{\overline{\theta}}^1
f(\theta )   [e(\theta)   + C]  d\theta  \right\} .}  \\
\end{array}
\end{equation}



\noindent
Note   that
   \begin{equation} \label{e7.5}
 \frac{d L}{d B} =      \lambda ,
\end{equation}
 which is to say that $\lambda$ equals the marginal value of
relaxing
the budget constraint.


There are  two  optimality conditions if we are at an interior
solution
(that
is, if not all or no cases are prosecuted).
First,  there is the choice of
the $e(\theta)$ function. For each value $\theta$ in the interval
of
prosecuted cases, $[\overline{\theta}, 1]$,
    \begin{equation} \label{e4a}
  \frac{d L}{d  e(\theta)}  = f(\theta )   \theta  V'(e(\theta) )
-
\lambda
f(\theta
)   (1)   = 0.
\end{equation}

 Rearranging,  we see  that the marginal payoff of effort has to be
the same for each case prosecuted. For any $\theta$,
 \begin{equation} \label{e5}
   \theta V'(e(\theta)) = \lambda.
\end{equation}




\noindent
 The second  optimality condition  concerns the choice of
$\overline{\theta}$. It says that
  \begin{equation} \label{e6}
\begin{array}{ll}
{\displaystyle \frac{d L}{d  \overline{\theta}} } &
{\displaystyle  =
  -   f(\overline{\theta})  \overline{\theta}
V(e(\overline{\theta})) +
\lambda
f(\overline{\theta}) [e(\overline{\theta}) +C  ]=0}\\
\end{array}
\end{equation}
 This implies that
  \begin{equation} \label{e7}
   \frac{ \overline{\theta} V(e(\overline{\theta}))   }{
e(\overline{
\theta})  +
C}   =  \lambda
\end{equation}

The amount  $ (e(\overline{\theta})+F)$ is the total cost of a
case of
the marginal strength  $\overline{\theta}$, and
$\overline{\theta}
V(e(\overline{\theta}))$ is the total benefit. Thus, we   have that
$\lambda$ equals the average cost of the marginal case.
Moreover,
since we found earlier that the marginal benefits of all cases
prosecuted are equal and equal to $\lambda$, we can conclude
that
the marginal benefit of any case prosecuted equals the average
benefit of the marginal case.

This implies that at the margin, the prosecutor would get the
same
benefit from adding a new case as he would from increasing
spending on
an existing case. It also implies that as his budget increases
(which
reduces $\lambda$) he will prosecute more cases as well as
spending
more on existing cases.  The result will be to increase the
expected
penalty from both   the new cases brought and the increase in
success
of the old cases. Formally:

\noindent
{\bf  Proposition 1:} {\it In the Social Planner Model,   an
increase
in the budget, $B$, increases   effort on existing cases ($e(\theta)
$
for $\theta >\overline{\theta})$, increases    the   prosecution
rate
(reduces $\overline{\theta})$, and  increases     the  size of the
expected penalty.}


\noindent
{\it Proof.} We will use the implicit function theorem. First,
differentiating
first-order condition (\ref{e4a}) gives us:
 \begin{equation} \label{300}
  \frac{d^2 L}{d  e(\theta)^2}  =   f(\theta)  \theta
V''e(\theta)
<0
\end{equation}
and
\begin{equation} \label{301}
  \frac{d^2 L}{d  e(\theta)  d \lambda }  =     -        f(\theta)
<0,
\end{equation}
Since
\begin{equation} \label{302}
  \frac{de(\theta)}{d \lambda  }  =   - \left(\frac{\frac{d^2 L}
{d  e(
\theta)
d \lambda }}{   \frac{d^2 L}{d  e(\theta)^2}} \right) =  -
\left(\frac{
-
f(\theta)}{ f(\theta)  \theta  V''(e(\theta))} \right),
\end{equation}
 we can conclude that $  \frac{de(\theta)}{d \lambda  }<0$.
Equation
(
\ref{e7.5}) and the  concavity of $L$ tells us that $\lambda$ falls
with
$B$.
Thus, $  \frac{de(\theta)}{d B  }>0$.

 Second,  dividing  first-order condition (\ref{e6}) by
$f(\overline{\theta})$,
we know it is true that we can define a variable $Z$ such that
  \begin{equation} \label{e303}
\begin{array}{ll}
Z & {\displaystyle  =
  -     \overline{\theta}V(e(\overline{\theta})) + \lambda
  [e(\overline{\theta}) +F  ]=0}
\end{array}
\end{equation}
Differentiating  expression (\ref{e303}) and substituting from
equation
(\ref{e5}) for $\overline{\theta}V'(e(\overline{\theta})) $
yields
  \begin{equation} \label{e304}
\begin{array}{ll}
\frac{dZ}{d\overline{\theta}}   & {\displaystyle  =
  -     V(e(\overline{\theta}))-
\overline{\theta}V'(e(\overline{\theta}))
e'(\overline{\theta})  + \lambda
   e'(\overline{\theta})  }\\
 & \\
 & {\displaystyle  =
  -     V(e(\overline{\theta}))- \lambda e'(\overline{\theta})  +
\lambda
   e'(\overline{\theta})  }\\
 & \\
 & = -V(e(\overline{\theta}))>0.
\end{array}
\end{equation}
and
\begin{equation} \label{e305}
\begin{array}{ll}
\frac{dZ}{d \lambda }   & {\displaystyle  =
   e(\overline{\theta}) +C  >0  }
\end{array}
\end{equation}

Since
\begin{equation} \label{306}
  \frac{d\overline{\theta}}{d \lambda  }  =   -
\left(\frac{\frac{dZ}{
d \lambda }}{   \frac{d^Z}{d  \overline{\theta} } } \right) =
-
\left(
\frac{ -
V}{   e(\overline{\theta}) +C } \right),
\end{equation}
 we can conclude that $  \frac{d\overline{\theta}}{d \lambda
}>0$.
Equation (
\ref{e7.5}) and the  concavity of $L$ tell  us that $\lambda$ falls
with
$B$.
Thus, $  \frac{d\overline{\theta}}{B  }<0$.



Finally, the amount of expected penalty rises because it is an
increasing
function of   $e(\theta)$ and a decreasing function of
$\overline{\theta})$.
Q.E.D.


   We cannot, however, say how an increase in the budget will
affect
the
conviction rate.


\noindent
{\bf Proposition 2:} {\it  In the Social Planner Model, an
increase
in
the budget, $B$,  might either
increase or
reduce the conviction rate.}


\noindent
{\it Proof.} From equation (\ref{e1c}) the conviction rate is
 \begin{equation} \label{307}
  \left(\frac{1}{   1-     F(\overline{\theta}) }
\right)
\int_{\overline{\theta} }^1 f(\theta)   \theta P(e(\theta))
d\theta
   \end{equation}

  We have seen in Proposition 1 that when the budget increases,
$\overline{\theta}$ falls (reducing the conviction rate) and
$e(\theta)$ rises (increasing the conviction rate). Since these
changes have opposite effects on expression (\ref{e1c}), the
outcome
is ambiguous. From the proof of Proposition 1,  the effect of
$\overline{\theta}$ is proportional to $\frac{  V}{
e(\overline{\theta}) +C}$ for marginal changes.  Thus, if  the
fixed
cost $C$ is large enough,  the change in $\overline{\theta}$ will
have
little impact, and the conviction rate will rise. The proof of
Proposition 1 also tells us that the effect of changes in
$e(\theta)$
is proportional to $\frac{   1}{    \theta  V''(e(\theta))}$. Thus,
if
$V''(e(\theta)) $ is large enough for all $e(\theta)$,  the change in
$e(\theta)$ will have little impact, and the conviction rate will
fall.  Thus, depending on $C$ and $V''$ the conviction rate might
increase  or might fall after a budget increase. Q.E.D.

 Proposition 2 says that the conviction rate can fall  because two
things matter  as   the budget $B$   increases. First,  how big is
the
fixed cost of prosecuting a case, $C$? If it is big, then most of
the
extra budget will go to existing cases, and the  conviction rate will
rise, not fall. Second, how big is $ \overline{\theta}
V''(e(\overline{\theta})$, the change in the marginal product of
spending on existing cases? If it is very  negative, then there are
sharply diminishing returns to increased spending on existing
cases.
As a result, the average  probability of success should fall when
the
budget rises, because the extra money goes mostly to prosecute
new,
weaker, cases.

An increase in the fixed cost $C$ is similarly ambiguous. It will
certainly increase $\overline{\theta}$, reducing the prosecution
rate
(which is easily proved  using the method of Proposition 1 and
the
fact that $\frac{dZ}{dC} = \lambda >0$).  It reduces the
amount of
variable effort $e$ that the prosecutor can expend for a given
number
of prosecutions, so the effect on his objective, total value of
prosecutions, cannot but be negative. The effect of a higher fixed
cost on the conviction rate, however,  is ambiguous. The average
product of the marginal case will fall, and the prosecutor will shift
resources towards the other cases.  He has to spend more on the
fixed
cost of those cases, so we cannot tell in the end, however,
whether
spending on the marginal effort $e$ increases or falls. Normal
intuition tells us that the conviction rate should fall if costs rise,
but if it is a fixed cost, that might not be the case. Instead,  it
could happen that the prosecution rate
 falls so much that enough resource are freed up for the
remaining
cases that the conviction rate rises.

   Particularly surprising is the possibility that  having juries more
hostile to police and prosecutors might raise the conviction rate.
This can be interpreted as an increase in $C$ if it means more
must be
spent to attain even a base-level probability of success. The effect
of a district having more skeptical juries might as a result be to
make prosecutors more selective about which cases they bring, to
such
an extent that they win a higher fraction of them.  This is just
one
way in which skeptical juries could enter the model, and the way
they
affect variable costs, fixed costs, and  the probability of success is
crucial. If the effect is to multiply the expected penalty $\theta
V(\theta)$ by a constant such as 0.8, for example, the conviction
rate
will fall without any effect on the prosecution rate or the
allocation
of effort. But the possible perverse effect of $C$ on shows how
difficult it would be to predict the effect of variables such as
political climate on conviction rates.  We must resort to empirical
estimation unless we are
willing to make assumptions about how the prosecution rate
affects the
conviction rate.


\bigskip
\noindent
{\bf   The
Political
Model: The Prosecutor as Agent of the Voters


      When the prosecutor uses the social welfare function for his
payoff, we can at least predict the effect of a budget change on
the
prosecution rate, if not the conviction rate.  Once agency
problems
are introduced, even this becomes difficult.  Two central
problems  in
any principal-agent relationship are that (a) the agent will want to
use his effort for personal goals unrelated to the principal's goal,
and (b) if the principal rewards the agent based on a rough proxy
for
the principal's true objective, the agent will pursue the proxy and
not the real thing. The more the principal tries to solve  problem
(a)
with a high-powered incentive contract, the more he must worry
about
problem (b). Here, the principal is the public and the agent is the
prosecutor. Problem (a) takes the form of the prosecutor
wanting to
use his office's resources for things such as perks for himself,
higher wages for his subordinates, and prosecutions or other use
of
office personnel that are intended to further his post-
prosecutorial
career in the public or
private sector.  Problem (b) takes the form of the prosecutor
focusing too much on variables the principal can measure well
and too
little on other goals.

 Here, we will focus on the conviction rate as that well-measured
variable, something some voters will reward despite its
imperfections
as a measure of performance. Prosecutor agents  must please
their
principals, however, whether they are appointed or elected, and
they
have  personal objectives too. If the prosecutor   must run for re-
election   his payoff function  may also include the
conviction rate, as suggested in the Introduction.  An elected
prosecutor--- and perhaps appointed an one  too--- must appeal
to
people
who do not know how many potential cases exist, and so have
difficulty
evaluating the number of convictions he achieves, not to
mention the
average sentence for those convictions. The absolute number of
prosecutions is  by itself meaningless to a voter, and  even the
prosecution rate is
not a helpful number given the small
number of arrests and prosecutions that occur anywhere in the
United
States relative to the amount of crime. A conviction rate,
however,
conveys some meaning, and will have even more impact on
voters who have not thought of the effect of careful selection of
which cases to prosecute.  It will not be the only variable that
affects voters-- if the prosecution rate falls enough, the change
will be noticeable, and by continuity even a small change will
have some impact on voters.  If a person had absolutely no
perception of a temperature decline of  one degree, not even
subconsciously, then   he could not perceive a change of two
degrees, three degrees, or  fifty degrees either.  But voters, even
more than administrative superiors, will look to clear measures
such as conviction rates, as Gordon \& Huber (2002) argue. We
might expect that if an elected prosecutor's term of office is short
or the degree of political competition is high, the prosecutor
would put more weight on acquiring a high conviction rate.
Conviction  rates should thus be correlated with short terms or
frequent turnover in the office, something we will look at below.

We will put  these two agency problems together  into what we
will
call the Political Model. If we denote effort spent on personal
goals
as $e_a$ and the utility of personal goals and slack  as $S(e_a)$
with
$S'>0$ and $S''<0$ (so increased effort devoted to slack  has
diminishing marginal utility), we can write the payoff function as
 \begin{equation} \label{308}
\begin{array}{lll}
 \mathrm{Payoff(prosecutor)} & = &U[  \alpha
\mathrm{(conviction \;
rate)}, (1-\alpha)
\beta(
\mathrm{personal \; goals)}, \\


 & &(1-\alpha)(1-\beta) \mathrm{(convictions )}]\\

& &\\
 &  = &U[  \alpha   W[e(\theta), \overline{\theta}], (1-\alpha)
\beta
S(e_a) , (1-\alpha)(1-\beta)  C[e(\theta), \overline{\theta}] ],
\\
\end{array}
\end{equation}
where
 \begin{equation} \label{e309}
W=   \left(\frac{1} { 1-    F( \overline{\theta}) }
\right)
\int_{ \overline{\theta} }^1  f(\theta) \theta  P(e(\theta))
 d\theta       \;\;\;\;\;\;     C=
  \int_{\overline{\theta}}^1  f(\theta )  \theta   V(e(\theta))
d\theta
\end{equation}

  In the Political Model payoff function, $\alpha$ parameterizes
the
importance of a high conviction rate to the prosecutor and
$\beta$
parameterizes the importance of low effort spent on
prosecutions,
where if $\alpha=\beta=0$ we return to the original model of
the
social planner. These parameters have been introduced in a
particular
way. The parameter $\alpha$  represents political pressure by
voters
who vote on the basis of  the conviction rate. To the extent that
the
prosecutor does not care about them, $(1-\alpha)$, his payoff is
split
between slack effort and the number of convictions. Thus, a way
to
represent loose political pressure is a low value of $\alpha$, and a
way to represent a prosecutor who will follow the public interest
if
he is not under political pressure is with a low value of $\beta$.

  The budget constraint changes   also, to reflect the cost to the
budget of effort on personal goals:
 \begin{equation} \label{e310}
 e_a +  \int_{\overline{\theta}}^1  f(\theta )   [e(\theta)   +C]
d\theta  \leq B.
\end{equation}



\noindent
The Lagrangian for the maximization problem is
 \begin{equation} \label{e311}
\begin{array}{ll}
  {\displaystyle  \stackrel{  Maximize}{e_a, e(\theta),
\overline{\theta}   }
\;\;\; \;\;  L} =  &    U[  \alpha   W[e(\theta),
\overline{\theta}],
(1-
\alpha)   \beta  S(e_a) , (1-\alpha)(1-\beta)  C[e(\theta),
\overline{
\theta}] ] \\
    & \\
 & {\displaystyle  +  \lambda \left\{ B -  e_a-
\int_{\overline{\theta}}^1
f(\theta )   [e(\theta)   + C]  d\theta  \right\} .}  \\
\end{array}
\end{equation}


The Political Model has    three  optimality conditions.  First,
there
is the choice
of
agency slack,  $e_a$:
    \begin{equation} \label{e312}
  \frac{d L}{d  e_a}  =  U_2   (1-\alpha)\beta S'    - \lambda
=
0.
\end{equation}
 The first optimality condition says that the prosecutor's gain
from
extra slack must equal its cost in terms of foregone budget that
might
have been spent on a higher conviction rate or more convictions.


 Second, there is the choice of
the $e(\theta)$ function. For each value $\theta$,
    \begin{equation} \label{e313}
  \frac{d L}{d  e(\theta)}  = U_1 \left(\frac{ \alpha}{1-
F(\overline{
\theta})}
\right)     f(\theta )   \theta   P' +  U_3(1- \alpha)   (1-\beta)
f(\theta ) \theta V'    -
\lambda      f(\theta )   (1)   = 0.
\end{equation}

 Rearranging and combining these last two conditions,  we see
that
the
marginal
payoff of effort must   be
the same for each case prosecuted, as in the social planner model,
though now
the payoff is more complicated. For any $\theta$,
 \begin{equation} \label{e314}
  U_2 (1-\alpha)\beta A'    = U_1 \left(\frac{ \alpha}{1-
F(\overline{
\theta})}
\right)
\theta   P' + U_3(1- \alpha)   (1-\beta) \theta V' = \lambda.
\end{equation}

      As $\alpha$ rises, the prosecutor slackens  less and puts
more
effort
into each case he chooses to prosecute, for a higher conviction
rate.
Unlike earlier, however, it is the probability of conviction in  a
case,
$P$, rather than the expected penalty, $V$, which matters, so we
would
expect the number of generous plea bargains to increase. As
$\beta$
rises, more effort is diverted to personal goals, leaving less for
prosecutions. As $\lambda$ rises, indicating a greater marginal
utility
of money and a lower budget, effort per prosecution and slack
both
fall.




\noindent
 The third  optimality condition  concerns the choice of
$\overline{\theta}$. It says that
  \begin{equation} \label{e315}
\begin{array}{ll}
{\displaystyle \frac{d L}{d  \overline{\theta}} } &
{\displaystyle  =
\alpha U_1
\left( \left[ \frac{f(\overline{\theta}) }{(1-
F(\overline{\theta}))^2}
\int_{\overline{\theta}}^1
f(\theta) \theta P d\theta  \right] -  \left[ \frac{1}{(1-
F(\overline{
\theta}))  }
\right] f(\overline{\theta}) \overline{\theta}
P(e(\overline{\theta}))
\right)
} \\
 & \\
 & - U_3(1-\alpha)(1-\beta) f(\overline{\theta})
\overline{\theta}
V(e(\overline{\theta})) + \lambda f(\overline{\theta})
[e(\overline{
\theta}) +C
]\\
 & \\
 & = 0.\\
\end{array}
\end{equation}

    Again the   the  average   payoff  of expenditure  on   the
marginal
case prosecuted equals $\lambda$, the marginal payoff on any
case. At
the margin, the prosecutor should get the same benefit from
adding a
new
case as he would from increasing spending on an existing case.

Introducing the conviction rate into the payoff function and
allowing
the payoff function to have three arguments instead of one
means  the
objective function is no
longer concave in the prosecution  threshold $\overline{\theta}
$.
Indeed, we can say very little now about comparative statics,
which is
the point of Proposition 3.


\noindent
{\bf Proposition 3:} { \it  In the Political Model, the conviction
rate
and
prosecution rate might
either rise or fall with the budget.     }


\noindent
  {\it Proof.} The conviction rate  depends on the number of
cases
prosecuted, $\overline{\theta}$ . Proposition 2 showed that if
more
cases are prosecuted, the conviction rate can fall.  Here, we have
changed the objective function by adding the conviction rate
directly
and by adding slack. If the weight put on these two objectives is
chosen to be small enough, then Proposition 2's conclusion will
still
apply, because the effect of   these two objective variables on the
optimal choice of control variables is continuous. Thus, the
budget
has an ambiguous effect on the conviction rate.

  We similarly know from Proposition 1 that the prosecution rate
might
rise with the budget. To prove that it can fall, consider the
following example. Suppose that  at the level of the choice
variables
optimized for the initial budget the marginal  payoff to the
prosecutor of convictions, $C$ is falling in the size of the
conviction rate, $W$, so $U_{13}<0$ (e.g. because a higher
conviction
rate  reduces the importance of a  high number of convictions to
win
re-election), and that convictions has greater diminishing payoff
than
the conviction rate ($U_{11}$ is small relative to $U_{33}$). An
increase in the budget will then be spent more on increasing the
effort on existing cases, increasing the conviction rate, and
reducing
the value of convictions, so that the optimal $\overline{\theta}$
would fall: fewer prosecutions would result from a higher budget.
Q.E.D.

  Proposition 2 showed how the conviction rate might either rise
or
fall with increased budget because the prosecutor might start
taking
on more difficult cases.  What is different in Proposition 3's
Political Model is that now the prosecutor has two new
objectives
besides getting convictions: a high conviction rate, and low
personal
effort on prosecution. If we reduce the parameters that represent
the
importance of these new goals to the prosecutor, we will be
increasing
the importance he puts on convictions. The Social Planner Model
is the
limit of the Political Model as the importance of the two new
goals
goes to zero, so  we should expect Proposition 2's  reasoning for
an
ambiguous effect of budget on conviction rate to also be possible
in
the Political Model.  We  might  expect somewhat greater
conviction
rates in the Political Model because the prosecutor cares directly
about the conviction rate, but we also might expect both fewer
prosecutions
and a lower conviction  rate because of agency slack.

 Proposition 3 also says that   it is impossible to say generally how
an increase in the budget will affect the prosecution rate. This is
surprising, because Proposition 1 said that in the Social Planner
Model we could confidently predict that prosecutions would rise
with
the budget. The reasoning there was that an increase in the
budget
allowed improvements on two margins, the extensive margin of
prosecuting more cases and the intensive margin of prosecuting
existing cases more heavily, and that with diminishing returns to
each
margin some of the extra budget should be devoted to each. The
analogous reasoning here would be that although we now have
three
goals (convictions, conviction rate, and slack) instead of just one
(convictions), there are diminishing returns to each and the extra
budget should be spent on all three. But that reasoning is
fallacious.

 The fallacy is that the three goals can interact with each other in
complicated ways, unlike the two means (marginal and extensive)
to the
one goal of maximizing convictions.  The prosecutor's choice
among
goals is like that of a consumer among goods. Just as a higher
income
can result in a consumer buying less margarine, an inferior good,
so a
higher budget could result in a prosecutor ``buying'' fewer
convictions. Instead,  he might spend all of the increase in his
budget, and more, on increasing the conviction rate and
advancing his
personal goals of slack and  non-prosecution goals
that  aid his  re-election.

 \bigskip

The theory just laid out  tells us that from the normative point of
view it is difficult to say whether a prosecutor's conviction rate
should be high (from Proposition 2),  and from the  positive
point of
view it is hard to predict whether it will increase with the budget
(Propositions 2 and 3). If the Social Planner model is not just
normative, but also describes how prosecutors behave,    then a
budget
increase will be spent partly on the intensive margin of
prosecuting
the existing  number of cases with extra resources and partly on
the
extensive margin of prosecuting more cases. The number of
prosecutions
and  convictions
will rise, but the conviction rate
might fall. The Political Model is a positive model which adds the
conviction rate and slack  to the prosecutor's objective function.
These new goals both conflict with spending on the extensive
margin of
new prosecutions. As a result, not only does the effect of
increased
budget on the conviction rate remain ambiguous (Propositions 2
and 3),
but we cannot even say that
the number of prosecutions or convictions will rise (Proposition
3).
Thus, when we look at anecdotal or numerical data, if the
personal
goals of prosecutors depend on the conviction rate we should
not be
surprised if high budgets are not always associated with high
conviction rates.

 Proposition  3   does say that in the Political Model the effect of
a
budget increase is ambiguous for both the conviction rate and
the
prosecution rate, unlike the effect for the prosecution rate in the
Social Planner Model stated in Proposition 1.  Thus, we can try to
test between the two models based on whether prosecutions rise
with
budgets.  Whether the conviction rate  rises with the  budget  is
ambiguous in both models, but we would expect the conviction
rate to
rise more (or fall less) in districts where the Political Model is
more likely to apply because prosecutors are elected rather than
appointed or where they are elected with shorter terms.


\bigskip

\noindent
 \section{State Prosecutions in the United States }

  We can do simple tests for the Social Planner Model versus the
Political Model using state-level data from the United States. In
2001, 2,341  prosecutor's offices handled felony cases  in state
courts of general jurisdiction. They  employed some 79,000
people and
had a budget of about 4.7 billion dollars. The size of offices
increased markedly over the previous decade. From 1994 to
2001,
budgets grew 61\% after inflation, and the number of assistant
prosecutors grew 26\%.   Of these offices, 1,809 employed full-
time
chief prosecutors and 532 had part-time chief prosecutors (in 29
states), an increase  in the number of full-time chief prosecutors
to
77\% compared to the 53\% in 1990. Figure 4,  a map of the
districts
 that also shows which had part-time
prosecutors, shows that as one might expect, districts tend to  be
bigger in rural areas, which also often only have part-time
prosecutors .\footnote{Figure 4
and the
numbers in this and the next three  paragraphs are from
DeFrances
(2002).} 3\% of the personnel were chief prosecutors, 30.5\%
were
assistant prosecutors, and the rest  were  people  such as support
staff
  and investigators.

A full 87\% of chief prosecutors were elected or appointed to
four-
year
terms. Chief prosecutors are elected everywhere except   Alaska,
Connecticut, the District of Columbia, and New Jersey.  In
Alaska,
Delaware, and Rhode Island,  criminal prosecution is the primary
responsibility of the state's attorney-general (appointed in Alaska
and elected in Delaware and Rhode Island), and in the District of
Columbia the U.S. Attorney has jurisdiction over felonies and
misdemeanors.

 Half of the offices received at least 85\% of their funding from
the
county government, and a third relied entirely on the county. Half
received some state funding, and 6\% had their entire funding
from the
state. City governments and grant funds also supplied funding.

 By definition, these offices handle felonies. The percentages also
handling misdemeanors, juvenile matters, and traffic violations
were
91\%, 89\% and  84\%. 51\%,   handled felony appeals
(including 77\%
of the 34 with district populations of over one million), and
54\%
represented the government in civil suits.\footnote{The
distribution
of these by size is interesting. The percentage of  full-time
prosecutors representing the government in civil suits was 41\%
in
districts with over a million people, 27\% in districts with
between
250,000 and 1,000,000 people, and 51\% in districts with fewer
than
250,000 people.  The figure for part-time prosecutors, however,
was
75\%. }




\hspace*{-92pt}

\begin{center}
 \includegraphics{pros4.png}
{\bf  Figure 4:  Prosecutorial Districts in the United States}
\end{center}

\section{Data and Endogeneity  }

If we are to study American prosecutors, our  unit of observation
should be  the  state prosecutorial district.   These are often the
same as   counties, but not always. Alaska, for example, has just
one prosecutorial district but many counties; and some counties
are split between prosecutorial districts.

Our task  is to see what explains conviction rates and
prosecution rates in this data.

  Our source for variables involving  prosecutors' offices (e.g.
conviction rates, salaries) is the 2001 National Prosecutor's Survey
by the Bureau of Justice Statistics,   U.S. Department of Justice.
This was a mail survey with telephone follow-up. 96\% of the
2,341 offices responded, and 98 did not. The Survey tried to find
data on the 98 missing offices from secondary sources such as
the county budget office. It also imputed missing values for
various variables using the ``hot deck'' method, which copies
values from observations having similar values for non-missing
variables. These included 288 values for felony cases closed
(DeFrances (2002)). The  National Prosecutor's Survey occurs
every few years, but  in the 1992, 1994, 1996, and 1998 surveys it
only included a   sample of about 300 districts, not the entire
population of districts.

 Our second major  data source is   the  2001 Uniform Crime
Reporting Program Data of the   Federal Bureau of
Investigation, U.S. Dept. of Justice.  This is published annually
and contains detailed data on the amount of   crime  reported to
police.

 The crime data, which is arranged by county,  is   well known
and much used; the prosecutor dataset, arranged by prosecutorial
district, is much less known. Our dataset merges them, using the
district as the unit of observation.

 Another of our variables is  a measure of political conservatism:
the percentage of the November 2000 presidential vote  received
by George Bush.  We also obtained demographic information,
from the Bureau of Census. This includes information about
population, racial composition of counties, per-capita income,
median rent, population density, housing density, educational
attainment of population in different counties, the age
composition of population in different counties, and the total
area of each county. Each of these variables was at the county
level, which we   aggregate to the level of prosecutorial districts
that usually consist of several counties.

To see if the predictions from our theoretical model match  the
real world, we run regressions  to explain the conviction rate
(felony convictions divided by the  number of prosecutions times
100) and the number of felony prosecutions.  We run a tobit
regression  to explain  the conviction rate because it always lies
between 0 and 100 (\%), but least squares for the number of
prosecutions.  We use the number of prosecutions rather than
the rate because    we do not have data for all   reported felony
crimes,  just the number of index crimes, whereas our
prosecutions number is for all felonies.

We dropped districts for which the  Census of Prosecutors data
reports conviction rates  of greater than 100\% or less than 0\%.
We also dropped   the top 5\% of the districts by population.
These districts are drastically different in size from the rest,
ranging in population from 537,000 to 9.51 million compared to a
median of 34,000 for the other districts. We are left with 1,625
observations, of which about 1,400 contain values for all the
variables we use in the regressions.

Table 1 shows summary    statistics for the regression variables.
In addition, note that 338 of the districts are in metropolitan
areas and  14 of them have appointed prosecutors.




\vspace*{40 pt}


\begin{center}
 {\bf Table 1: Summary Statistics for 1,625 Prosecutorial Districts
in 2001 }
\end{center}
{\footnotesize

  \begin{longtable}{lrrrrrr}

\hline
  &           &      &          &       &   &    \\
   &           &      &     \multicolumn{3}{c}{Percentile}       &
\\
  Variable   &Mean &    Min &       25th    &   50th      & 75th
&        Max \\

  &           &      &          &       &   &    \\

\hline

  &           &      &          &       &   &    \\


Conviction Rate (\%) &    83&    0  &      79 &      90  & 96  &
100  \\

Number of Felony Cases Closed &  599&      1  &      75  & 218
&     573  &  15,639  \\

Prosecutorial Budget (\$1,000s) & 777 &   6   & 125   & 298   &
750 & 15,500  \\

Number of Index Crimes    &  457 &     0  &      26  &     131  &
461  &   8,856  \\

Chief Prosecutor's Salary (\$1,000s) &  77 & 11  &  52  &  80 &
99   & 150   \\

Years in Office &  9.14 &      0  &       4  &       7  &           14 &
40  \\

Term of Office &    4.11 &    1  &       4  &       4  &             4 &
10  \\

Population  &     444  &  12,905  &  34,010  &  67,577.46 &
82,341  & 537,484  \\

Presidential Vote for Bush in 2000 (\%) &   59 &    19  &      51
& 58  &           66 &      91 \\

African-American Population &  7,271&     0  &      60  &     517
& 3,842  & 326,097  \\

   &           &      &          &       &   &    \\

  \hline

\end{longtable}
}


The prosecutorial budget and the number of index crimes might
be endogenous, influenced by the conviction and prosecution
rates. We have accordingly  used instrumental variables in our
regressions. As instruments for the prosecutorial budget, we have
used the aggregate pay of the full time and part time employees
and the total personal income of the district's population, both in
logs. These instruments are   correlated with the prosecutorial
budget because the prosecutor gets most of his budget from the
local government (a first-stage regression has an $R^2$ of .22
and an $F$-statistic significant at well beyond the .01 level). They
are beyond his control, unlike the prosecutorial budget, which he
can request to increase   in response to the difficulty of
prosecution or conviction.

As instruments for    crime the percentage of people who are
employed, the percentage of males with no schooling, and
housing
density, all in logs. The first-stage regression has an $R^2$ of .44
and an $F$-statistic significant at well beyond the .01 level.  We
did not use the percentage of African-American population as an
instrument for crime because it is in the main regression.  Since
African-Americans are disproportionately both victims and
perpetrators of crime, we might well expect a direct effect of race
on the prosecutor's conviction rate (as Butler (1995) suggests),
though its direction  would be unclear.

\section{Regression Results}

 Proposition 1 says that in the Social  Planner Model an increase
in the prosecutor's budget increases his prosecution rate   for a
given crime rate, but Proposition 2 says that his conviction rate
might either rise or fall. Proposition  3   says that in the Political
Model the effect of a budget increase is ambiguous for both the
conviction rate and the prosecution rate.  We can   test between
the two models based on whether prosecutions rise with
budgets.  Whether conviction rates rise with budgets is
ambiguous in both models, but we would expect the conviction
rate to rise more (or fall less) in districts where the Political Model
seems more likely to apply because prosecutors are elected rather
than appointed or where they are elected with shorter terms.

 Table 2 looks at how   the conviction rate is related to   the
prosecutor's budget, the number of prosecutions,  and other
variables. Regression (1) uses tobit. Regression (2) uses tobit
with instrumental variables for both budget and crime as
described above, while    Regression (3)  instruments only for the
prosecutorial budget.  Maximization of the likelihood in
regression (2) would not converge, so Stata's two-step estimator
was used instead. The coefficients from two-step regressions are
not comparable to the standard maximimum likelihood
technique, but it is valid for the purpose of testing the
instruments.   A Wald chi-squared test applied to regression (2)
cannot reject the exogeneity of budget and crime. We therefore
prefer regression (3), in which only the budget is endogenous,
and where      a Wald test  does reject exogeneity of the budget
(at the 2.1\% significance level).\footnote{A Wald test on a
regression  which  instrumented only for crime (unreported here)
could not reject its  exogeneity.} We also  tested the residuals
from (3) for correlation with district population and found no
significant correlation.

  In regression (3), the conviction rate does rise  with the
budget, keeping the number of prosecutions (the ``cases closed''
variable) constant. The statistical significance is well above the
5\% level. If one calculates the elasticities of how the conviction
rate changes with respect to the budget,  (using the STATA mfx
command evaluated at the  median),  it is  $ 0.11$: a 10\% rise
in the budget increases the conviction rate by  1.1\%. If the
conviction rate is at the median of  90\%, a rise in the
budget by 10\% will increase it   to  90.90\%. That amount
may not seem large, but given the high rate of diminishing
returns we would expect in the process of prosecution  it is not
surprising. We should note, too, that in regression (1),  which
assumes that the prosecutor's budget does not depend on the
desired conviction rate, the budget is  not statistically significant
as an influence on the conviction rate.

  The conviction rate also falls with the  number of prosecutions
(the ``cases closed'' variable), at a very significant level statistically.
This is not an implication of the theory, but one of its
assumptions. Yet if one calculates the elasticity of how the
conviction rate changes with respect to prosecutions,  it is only
$-0.031$:  a 10\% rise in the number of prosecutions   reduces
the conviction rate by .31\%.   If  the conviction rate is at the
median of  90\%, a rise in the number of prosecutions by 10\%
will reduce it   to about 87.27\%. What our statistical significance
establishes is that the size of the effect  is definitely  small, if
nonzero. This implies that in the average United States district
the prosecutor has a large enough stock of potential cases that
increasing his number of prosecutions would scarcely reduce his
conviction rate. We conclude from the budget and number of
prosecutions coefficients that the  chief limiting factor for  the
number of prosecutions is the fixed cost of an additional
prosecution.

  The chief prosecutor's salary and years in office are insignificant
in Table 2. Whether the chief prosecutor is appointed is highly
significant, both statistically and in magnitude. As the Political
Model would predict, appointed prosecutors, less worried about
poorly informed  public opinion,  have lower conviction rates.
The Political Model predicts that the effect of a longer term of
office would also be negative because it would ease voter
pressure,  but it comes in positive and  marginally significant  at
the 10\% level.  We have no explanation for this.

   The other variables in the regression are control variables which
do not help distinguish between the Social Planner Model and
the Political Model. The amount of crime reduces the conviction
rate in specification (3).  The values of the other control variables
are insignificantly different from zero except for the regional
dummy South, which has a positive effect on the conviction rate
(at the 5.4\% significance level).  In particular, the percentage of
the population that is black does not significantly increase or
reduce the conviction rate.





\newpage

\begin{small}

\begin{center}
 {\bf Table 2:  Explaining the    Conviction Rate across
Prosecutoral Districts}
\end{center}


\begin{longtable}{lrrr}

   &   (1)     &        (2)    &      (3) \\

   &     &           &      \\

\hline

   &     &           &      \\

  Prosecutorial Budget (\$ millions) &     0.13 & -7.07 &   27.32**
\\

  &    (0.21) & (1.18) &     (2.31) \\


   &     &           &      \\

                        \hline

   &     &           &      \\

Number of Felony Cases Closed (1,000s) & -2.72*** &
-0.71 &  -.01*** \\

   &       (3.96) & (0.45) & (2.15) \\

Chief Prosecutor's Salary (\$10,000s) &    -0.56** & 0.13 &    -.01
\\

     &      (2.43) & (0.30) &   (1.27) \\

Chief Prosecutor Appointed  (= 1)&      -4.56 & 21.39 &
-114** \\

    &      (0.71) & (0.90) &    (2.29) \\

        Years in Office &   0.15* & 0.18** &       0.11 \\

     &     (1.96) & (2.09) &    (0.92) \\

         Term in Office &      1.06 & 0.79 &      1.70* \\


 &    (1.59) & (1.09) &       (1.65) \\

    &     &           &      \\

                        \hline

   &     &           &      \\

Number of Index Crimes (1,000s) &  0.13 & 7.39 &      -23.55**
\\

   &                 (0.13) & (1.02) &                 (2.11) \\

 Metropolitan Area  (=1)&    -1.26 &-2.75 &   2.75 \\

   &        (0.87) & (0.87) &       (1.14) \\

 African-American Population (\%) &       -0.07 & -0.21* &
0.35
\\

  &     (1.33) & (1.73) &       (1.52) \\

   Constant &     85.37*** & 82.95*** &                75.59*** \\

   &      (25.81) & (23.79) &    (17.08) \\

   &     &           &      \\

\hline
\end{longtable}





  \noindent
  {\small   {\bf \it Notes:} These are marginal effects. 2001 data;
n=1,625.   In parenthesis are z-statistics.  *, **, and *** indicate
significance at the 10\%, 5\%, and 1\% levels. All regression use
tobit with an upper limit of 100 for the conviction rate and
include 3 regional dummies. In regression  (1)   no explanatory
variables are  endogenous, in regression (2) prosecutorial budget
and index crime are endogenous, and in regression (3) only
budget is endogenous.    Regression (2) uses Stata's twostep
option.    Robust standard errors are used in (1) and (3).    See
the text for discussion of the  instruments and  the two-step
option. }

\end{small}

        \bigskip

 Table 3 looks at how the number of prosecutions changes with
the budget and other variables.  Since this variable  is not
bounded as conviction rate is by the upper limit of 100\%, we
use simple least squares instead of tobit.  As  in Table 2,  no
explanatory variables are endogenous in the first column, both
budget and crime are instrumented in the second, and   budget
alone is instrumented in the third. Since these are least squares
regressions rather than tobit, a Hausman specification test  is
appropriate instead of the Wald test. We find we cannot reject
exogeneity of either or both crime and budget (the significance
levels are .45 for budget, .50 for crime, and .50 for both).  We will
take column (1) as our preferred regression. Columns (1) and (3),
however, have similar patterns of statistical significance, so
whether or not we instrument for the budget makes little
difference.

    In Table 3, the prosecutor's budget does  have a highly
significant effect on the number of prosecutions, with an
elasticity of 25\%.  This regression holds constant the
population, which naturally also has a positive effect. The amount
of crime  has an insignificant effect, conditioning on budget and
population. Districts with more crime (but the same budget) do
not  seem to have more prosecutions.

 Of the political variables, the chief prosecutor's salary and
whether he is appointed have no effect on the number of
prosecutions, but the longer he is in office, the fewer the
prosecutions, an elasticity of 6.7\% that is marginally significant.
Thus, this regression lends mild support to the Political Model.

 Of the control variables, being more Republican or  in a
metropolitan area has no significant effect.  The South and West
prosecute more than the Northeast and Midwest. The
percentage African American has a highly significant positive
effect.



 Conviction rates do rise with prosecution budgets and are higher
where prosecutors are elected rather than appointed and where
terms of
office are shorter. The number of prosecutions also rises with the
budget, but falls with the prosecutor's tenure in office.  Overall,
this lends mild support to the Political Model over the Social
Planner
Model.
  As predicted by  both of the models, the conviction rate falls
with
the number of prosecutions, but it falls very slightly.




\newpage

\begin{small}

          {\bf Table 3: Explaining   the Number of  Prosecutions
across Prosecutorial Districts
 }




\begin{longtable}{lrrr}

         &      (1)    &     (2)   &(3)  \\

 &   & &  \\
\hline

      & & &  \\
   Prosecutorial Budget (millions) &                179.15** &
-82.36 &                 390.33* \\

                        &                 (2.55) &
(0.35) &                  (1.70) \\


                        \hline

 &   & &  \\

Chief Prosecutor's Salary (\$10,000's) &                   -7.86 &
5.70 &                  -10.77 \\

                        &                 (0.76) &
(0.27) &                 (0.96) \\

Chief Prosecutor Appointed (=1) &                 -570.46 &
439.21 &                -1263.96 \\

                        &                 (1.25) &
(0.39) &                 (1.34) \\

        Years in Office &                  -4.29* &
-4.58 &                  -4.88* \\

                        &                 (1.74) &
(1.62) &                 (1.74) \\

         Term of Office &                   27.13 &
3.68 &                   39.61 \\

                        &                 (0.83) &
(0.08) &                 (1.24) \\

                        &   & &  \\
                        \hline
 &   & &  \\
             Population (1000's) &                 5.59*** &
5 &                  3.72** \\

                        &                 ( 4.02) &
(1.33) &                 (2.16) \\

 Number of Index Crimes (1,000's)  &                   69.47 &
548.75 &              -41.63\\

                        &                 (0.46) &
(0.77) &                 (0.23) \\

  African-American Population (\%) &                18.75*** &
14.20** &      20.74*** \\

                        &                 (5.41) &
(2.42) &                 (4.91) \\

  Vote  for Bush in 2000 (\%) &                    2.65 &
0.34 &                    4.18 \\

                        &                 (1.18) &
(0.09) &                 (1.28) \\

      Metropolitan Area (=1) &                  -68.39 &
-78.22 &                  -35.65 \\

                        &                 (0.89) &
(0.92) &                 (0.49) \\

               Constant &                 -192.92 &
10.9 &                 -336.72 \\

                        &                 ( 1.05) &
(0.03) &                 (1.33) \\
                         &   & &  \\

                        \hline

\end{longtable}


 \noindent
 {\small {\it{ \bf  Notes:}  }  2001 data; n=1,625.     In
parenthesis are z-statistics. *, **, and *** indicate significance at
the 10\%, 5\%, and 1\% levels. All regression use least squares
and include 3 regional dummies, not reported.     In regression
(1)   no  explanatory variables are endogenous, in regression (2)
prosecutorial budget and index crime are endogenous, and in
regression (3) only budget is endogenous. Robust standard errors
are used.    See the text for discussion of the instruments.  }



\end{small}

  The theoretical model established that the  conviction rate
might actually fall with the prosecutor's budget because the larger
budget causes him to pursue more difficult cases.  The conviction
rate regressions in Table 2   would not detect this, because they
condition on the number of cases prosecuted. If instead we
simply look at the correlation between the conviction rate and
the prosecutor's budget per capita, it is -.029, which is indeed
negative, but tiny. Another approach is to see how the
conviction rate depends on the budget when we condition on all
of our explanatory variables except for budget. This can be done
by repeating    regression (3) of Table 2 but   omitting cases
prosecuted as an explanatory variable. The result is that the
budget comes in in positive and significant, as before. That
makes sense, since the effect of the number of prosecutions we
found in regression (3)  was   tiny, an elasticity of -3.1\%, so that
if the elasticity of prosecutions with respect to budget is   the
25\% we found in Table 3's regression (3), the negative indirect
effect of more prosecutions   is still less than the positive  direct
effect of the budget on the conviction rate. Thus, the theoretical
possibility of the conviction rate falling with an increase in the
prosecutor's budget is not an empirical actuality in  our U.S. data.

\noindent
{\bf Conclusions}

     A prosecutor's high conviction rate may not be a sign that  he
is tough on crime and doing a good job. Instead, he might just
be taking easy cases and letting too many criminals go without
prosecuting them. We have explored the implications of this idea
for a theory of prosecutors and used it to think about
prosecutions in Japan and the United States. Prosecutors have a
choice between using extra resources to prosecute their current
level of cases harder, the intensive margin, or to increase the
prosecution rate, the extensive margin. A social planner would
want both margins increased if the budget expands. Prosecutors
are agents of the voters, however, which
means
they are subject to the temptation of agency slack and that they
are
monitored using imperfect summary statistics such as the
conviction
rate. As a result, we do not necessarily see increases in the budget
spent on increasing the conviction rate, or even on increasing the
prosecution rate.  This may be particularly true for prosecutors
who
are elected, as in most of the United States, but appointed.
Japanese
prosecutors seem, with their low budget compared to American
prosecutors, to have low prosecution rates and high conviction
rates.
We do   find  that higher budgets are associated with  both
higher
amounts of prosecution and higher conviction rates conditioning
on the
amount of prosecution.  We  also find that appointed
prosecutors in
the United States do have lower conviction rates than elected
ones
and  that those with longer tenure in office prosecute fewer
cases.




\newpage



\textbf{References}

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Framework for Analysis,'' 53  {\it New York University Law
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783-834.

  Albonetti, Celesta.   1986.     ``Criminality, Prosecutorial
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Decision
Making in Felony Case Processings,''  24  {\it  Criminology}
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Baker, Scott and  Claudio Mezzetti.   2001.     ``Prosecutorial
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Plea Bargaining, and the Decision To Go to Trial,'' 17  {\it
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of
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Boylan, Richard.     2004a.     ``Do the Sentencing Guidelines
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Boylan, Richard.   2004b.     ``Salaries, Turnover, and
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Boylan, Richard.   2005.     ``What Do Prosecutors Maximize?
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