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{\large {\bf How Optimal Penalties Change with the Amount of Harm }\\[0pt]
}

\bigskip  Eric Rasmusen \\[0pt]

Published: {\it International Review of Law and Economics} (1995) 15:
101-108.

{\it Abstract}
\end{center}

Intuition tells us that the optimal penalty and court care to avoid error
should rise smoothly with the harm to the victim. This is not always
correct: sometimes the optimal penalty and level of court care increase
discontinuously with harm, even when penalties deter harm and court care
reduces error continuously. This is shown in a model in which the social
cost of crime consists of its direct harm, the cost of court care, and the
cost of false convictions.

{\small Draft: 4.1 (Draft 1.1, July 1991).  \vspace{ 10pt} }

{\small I would like to thank John Lott, Steven Shavell and two anonymous
referees for helpful comments. Much of this work was completed while the
author was Olin Faculty Fellow at Yale Law School and on the faculty of
UCLA's Anderson Graduate School of Management. }

{\small \ \noindent \hspace*{20pt} 2000: Eric Rasmusen, Professor of
Business Economics and Public Policy and Sanjay Subhedar Faculty Fellow,
Indiana University, Kelley School of Business, BU 456,  1309 E 10th Street, 
Bloomington, Indiana, 47405-1701.  Office: (812) 855-9219. Fax:
812-855-3354. Erasmuse@indiana.edu. Php.indiana.edu/$\sim$erasmuse. }

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

\newpage \noindent {\bf 1. Introduction}

Should a crime's penalty rise smoothly in proportion to the crime's
harmfulness? This seems obvious, and has a sound economic intuition behind
it. The optimal penalty is the result of a tradeoff between the penalty's
costs and benefits, and when tradeoffs are made, they usually change
smoothly. If the harm increases slightly, then so, it seems, should the
penalty. If the penalty is imprisonment, then too short a prison term
results in too much crime and too long a prison term results in excessive
spending on prisons. A similar argument can be made to show that the care
the court takes with a particular case should be smoothly increasing in the
harmfulness of the crime involved.

This intuition is deceiving: penalties and court care should not always rise
smoothly with harm. The model below will formalize the idea that the optimal
penalty and court care involve tradeoffs between the penalty's cost and
benefit. The penalty and court care will not actually decline as harm
increases, but they may jump sharply, even when the harm increases smoothly.

The reason for the discontinuous jumps is that increasing the penalty has
the good effect of reducing crime but the bad effect of increasing the
penalty costs on those criminals it still fails to deter. This is the idea
that Louis Kaplow (1990a, 1990b) has used to show why optimal costly
penalties will sometimes be either zero or maximal.  Using a model based on
Polinsky \& Shavell (1984), he showed that the social optimization problem
is not convex and so may have corner solutions. The penalty might, for
example, have very weak deterrence value, so increasing it beyond zero would
increase social expenditure without much reducing crime. Or perhaps as the
penalty becomes more severe, it needs to be carried out so much less often
that the total cost falls and the optimal penalty is maximal. Because of
these corner solutions, there will be some harm level at which the corner
solution jumps from zero to maximal, but for the most part the crime's
penalty should be unrelated to its harmfulness.

The model in the present article will be somewhat different. Standard
assumptions will be made to rule out the corner solutions of zero or maximal
optimal penalties, the social cost of penalties will arise from false
convictions, which can be reduced by greater court care, and the focus will
be on how penalties and court care change with harm.

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

\bigskip \noindent {\bf 2. The Model}

Let a certain type of crime cause harm $h$.\footnote{%
The harm $h$ could be the direct harm to the victim, or it might add the
precautions of potential victims and the effort of the criminal and subtract
the crime's benefit to the criminal. See Ehrlich (1982) for a discussion of
theories of the objectives of criminal punishment, and Friedman (1981) and
Shavell (1985, p. 1244) for discussions of whether penalties should increase
with the benefit to the criminal or the harm to the victim.} The amount $n(p)
$ of this crime depends on the expected penalty, $p$. The probability of
conviction will be assumed to be exogenous, so $p$ represents the penalty
itself.\footnote{%
If $p$ were assumed to have a direct social cost, it could represent the
enforcement level as well, and the model would be little changed.}  Let the
deterrence function $n(p)$ satisfy $n_p<0$, where $Lim (n_p)= - \infty$ as $%
p \rightarrow 0$, and $n> \varepsilon>0$ for some constant $\varepsilon$. 
Under these assumptions, additional punishment always deters more crime but
never drives it below $\varepsilon$, and the marginal deterrence is
infinitely high starting from zero punishment.\footnote{%
It is assumed that $n > \varepsilon$ because otherwise the optimum might
specify infinite penalties that eliminate crime completely--- in which case
there would be no false convictions either. The assumptions that $n_p$ and $%
f_c $ become infinite when $p$ and $c$ are zero exclude the opposite
extreme: that punishment and court care are so ineffective that they should
be abandoned altogether.}

For each harm level $h$, society chooses the punishment, $p$, and the care
that courts take to avoid false convictions, $c$. Depending on the
punishment and court care, society incurs a false-conviction cost $f(p,c)$
per crime, where $f_p >\delta>0$ for some constant $\delta$, $f_c <0$, $Lim
(f_c)= -\infty$ as $c \rightarrow 0$, and $f_{pc}<0$ for all $p>0$. Under
these assumptions, greater punishment increases the cost of false conviction
by at least $\delta$, greater court care reduces false conviction and has
infinite marginal benefit starting from a level of zero care, and the
marginal benefit from care is greater if the penalty is greater.\footnote{%
For discussions of court error, see Png (1986), Posner (1973), Shavell
(1987), and Rasmusen (1994); for more specific discussion of error avoidance
as a goal of justice, see Ehrlich (1982), Rubinfeld and Sappington (1987),
and Kaplow (1994). Note that in the present paper, unlike some of these
studies, ``court care'' is the result of the court's decision, not of the
litigants' decisions on how much to spend.} The cost of false conviction, $%
f(p,c)$, includes such things as the disutility of those falsely convicted,
the risk borne by those who fear they might be falsely convicted, the
deterrence to efficient behavior created by fear of false punishment, and
the public's discomfort in knowing that some punishment is mistaken.%
\footnote{%
To focus on the cost of false punishment, the model assumes that the public
expense of correct punishment is zero--- the punishment takes the form of
Beckerian fines with zero transactions cost. This assumption is easily
relaxed by adding another function increasing in $n(p)$ and $p$, but this
would make little difference to the results.}

Society's problem is to choose $p$ and $c$ to minimize $S$, the sum of the
social costs from $h$, the direct harm from the crime; $f$, the cost of
false conviction; and $c$ the cost court care to prevent false conviction.
Summing these yields equation (\ref{e1}). 
\begin{equation}  \label{e1}
S = n(p)(h+ f(p,c) + c).
\end{equation}

Deterrence has two costs here: mistaken punishment and court care. Because
of diminishing returns to court care, not enough will be spent to completely
eliminate false convictions. Given this, the deterrence benefit from heavier
penalties must be weighed against the false-conviction cost. The deterrence
benefit is greater if the crime causes more harm, so for more harmful crimes
the tradeoff will lead to greater penalties. This is a simultaneous system,
so the greater penalties lead in turn to more spending to avoid false
convictions when the crime's harm is greater.  It can be shown, though I
will not do so here, that not only does the most severe crime not receive an
infinite punishment, but there is no ``bunching'' of penalties at that most
serious crime: penalties and court care rise with harm.\footnote{%
The proof that penalties and court care rise with harm rather than staying
constant is available from the author. The analysis has implicitly assumed
that each crime is independent, avoiding the issue of marginal deterrence: a
higher penalty for crime $h$ causing criminals to substitute to lesser
offences, as described in Stigler (1970). Since marginal deterrence provides
an incentive to steepen the punishment schedule rather than smooth it,
discontinuities would very likely continue to exist in the punishment
schedule. The jury response to higher penalties is still another
consideration (see Andreoni [1991]). A somewhat different issue is how the
severity of harm should be related to the amount of enforcement effort; for
analysis of this, see Shavell (1991) and Mookherjee and Png (1992).}

Mathematically, the difficulty with this optimization problem is that the
minimand is not convex. The government is trying to minimize the cost of
crime, $S(p,c;h)$. If $S(p,c;h)$ were convex, optimization theory tells us
that $p^*(h)$ and $c^*(h)$ would be continuous, but one condition for the
convexity of $S(p,c;h)$ is that $S_{pp}$ be positive, i.e., that 
\begin{equation}  \label{e47}
n_{pp}[h+ f(p)] + n f_{pp} + 2n_p f_p >0
\end{equation}

If inequality (\ref{e47}) is false, then the second-order-condition for the
problem is not satisfied and the implicit function theorem cannot be used to
show that the derivative $p_h$ exists and is positive.\footnote{%
Thus, the approach used in Becker (1968) for comparative statics, which
relied on special assumptions, would fail here.}  Inequality (\ref{e47}) can
easily be false under the model's  assumptions, which said nothing about the
sign of $n_{pp}$ or $f_{pp}$. There is no good reason for supposing that
either $n$ or $f$ is concave or convex; they are more like demand functions,
which might have any curvature, than like cost functions, which are usually
convex. Adding arbitrary convexity assumptions to $n$ and $f$ would require
adding the assumptions that $n_{pp} >0$, $f_{pp}>0$, $f_{cc}>0$, and $f_{pp}
f_{cc} - f_{pc}^2 >0$.  These assumptions imply that there are diminishing
returns to deterrence and care, that false-conviction costs rise more than
linearly with the penalty, and own-effects are stronger than cross-effects.
But even these extra assumptions would not guarantee the validity of
inequality (\ref{e47}), because though the first two terms would then be
positive, the last term would still be negative. The first two terms would
be positive because the marginal deterrence effect would weaken as the
penalty increased, and the marginal cost of each false conviction would
become greater. The last term would be negative because when crime fell,
there would be less false conviction, which might outweigh the fact that
with a higher penalty each instance would be more costly. As a result, the
cost function would still fail to be convex, even though its component
functions would be well-behaved under the additional assumptions. To
guarantee a continuous optimum it would be necessary to assume directly that
(\ref{e47}) is true, which has no justification.

The next section of the paper will construct a simple example to show by
construction that the optimal penalty and court care can be discontinuous in
harm and to develop the intuition behind the outcome.

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

\bigskip \noindent {\bf 3. An Example with Discontinuous Penalty and Court
Care}

The following example will show why the optimal penalty and court care might
be discontinuous in the crime's harm. Let the social cost of false
conviction be 
\begin{equation}  \label{e33}
f(p,c) = p^2 + \frac{ p^2}{c},
\end{equation}
which satisfies all the assumptions made earlier: the cost of false
conviction is increasing at an increasing rate in the penalty, it falls in
court care, and it cannot be completely eliminated, no matter how much court
care is used.

The total social cost of crime, using equations (\ref{e1}) and (\ref{e33}),
is 
\begin{equation}  \label{e35}
S= n(p)\left(h + p^2 + \frac{ p^2}{c} + c \right).
\end{equation}
The full optimization problem is to minimize $S$ with respect to $p$ and $c$%
, penalty and court care. It will be convenient to solve this in stages.
Minimizing (\ref{e35}) with respect to $c$ yields the first order condition 
\begin{equation}  \label{e36}
\frac{\partial S}{\partial c}= n(p)\left(-\frac{ p^2}{c^2} + 1\right)= 0,
\end{equation}
which implies that 
\begin{equation}  \label{e38}
c^*= p^*.
\end{equation}

The optimal penalty depends on the specific functional form for the
deterrence function $n(p)$. Let it be shaped as in Figure 1, which is drawn
using the following function: 
\begin{equation}
\begin{array}{lll}
n(p) & =10+180/p & if\;\;p\leq 6 \\ 
& =10+180/p+.1(p-6)^{2} & if\;\;6<p\leq 12 \\ 
& =10+180/p+.1(p-6)^{2}-.5(p-12)^{2} & if\;\;12<p\leq 20 \\ 
& =3.1+2.1/(p-19.4) & if\;\;20<p
\end{array}
\label{e45}
\end{equation}
This deterrence function says that some crime is deterred by a small
penalty, very little more is deterred by a moderate penalty, and almost all
crime disappears when the penalty is high.

\begin{center}
FIGURE 1 GOES\ HERE

\bigskip 
\end{center}

Equations (\ref{e36}) and (\ref{e38}), still apply, so $c^{\ast }=p^{\ast }$%
. Substituting this into the objective function (\ref{e35}) gives an
objective function already optimized for $c$ and now stated only in terms of 
$p$:\footnote{%
A substitution of this kind would mathematically incorrect if the analysis
then proceeded to take a derivative with respect to $p$, but here the minima
will be found by numerical methods.} 
\begin{equation}
S(p)=n(p)(h+p^{2}+2p)  \label{e46}
\end{equation}
Figure 2 shows the shape of the objective function given the deterrence
function from (\ref{e45}).

\begin{center}
FIGURE\ 2 GOES\ HERE
\end{center}

The $S(p)$ function has two local minima. For very low harm ($h=2$), the
global minimum is clearly at a low penalty, and for very high harm ($h=50$)
it is clearly at a high penalty. For $h=15$ and $h=16$, the two minima are
very close in their social cost. If $h=15$, the two minima are at (3.20,
2,096) and (21.90, 2,121) for {\it (penalty, social cost)}, so 3.20 is the
optimal penalty. If $h=16$, the two minima are at (3.29, 2,162) and (21.91,
2,125), so 21.91 is the optimal penalty. A small change in the harm induces
a jump in the optimal penalty from 3.20 to 21.91, and the care to avoid
false conviction also jumps, since $c^* = p^*$ by (\ref{e38}). Despite these
jumps, the social cost of a crime is a continuous function of its harm. The
social cost increases only from 2,096 to 2,125 going from $h=15$ to $h=16$.
The more harmful crime has much higher false-conviction and court costs, but
is offset almost exactly by a  decline in the amount of crime.

A plausible general phenomenon lies behind the example. For low levels of
harm, what matters most is to have a small penalty that deters some crime
but keeps the cost of false convictions low. Increasing the penalty beyond
that low level has little additional deterrent effect. If the crime's harm
becomes great enough, however, the penalty is drastically increased, because
high enough penalties result in another large drop in the amount of crime.
Optimal court care is discontinuous because it is based on the penalty: if
the penalty is low, care has little benefit and should also be low, but once
the penalty jumps, the benefit from increased care also jumps. The system is
simultaneous but recursive: increased harm does not increase court care
directly, but it increases the optimal penalty, which increases optimal
court care and feeds back to the optimal penalty.

This story can be based on either of two interpetations of the deterrence
function in Figure 1.

In the first interpretation, there always exist marginal offenders,
indifferent about committing the crime, but most potential offenders fall
into one of two criminal groups: casual criminals who are deterred by a
relatively low penalty, and serious criminals who are deterred only when the
penalty becomes very high. When the harm is small, the optimal penalty and
care are small but positive, deterring a large number of offenses without
incurring much cost from each false conviction. The number of offenses and
false convictions is high, but since the harm per crime is small, these are
tolerated. As the harm increases, the penalty and care increase slightly,
but there is still no attempt to deter the serious offender. At some point,
however, the crime has become harmful enough that it worth making the jump
to increasing penalties so drastically as to deter the serious offender
also. Since the penalty increases drastically, it is also worth increasing
court care drastically.

In the second interpretation, a single individual decides how many crimes to
commit. He will always commit more crimes if the penalty declines, but his
crimes generally fall into two categories: crimes of opportunity, which
require little thought, and planned crimes, which require considerable
effort. A small penalty will deter the planned crime, but not the crime of
opportunity. If the crime inflicts little harm, then the optimal penalty is
set low and crimes of opportunity are tolerated, but beyond some threshold,
the jump is made towards deterring both types of crime.

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

\bigskip \noindent {\bf 4. Concluding Remarks}

Proportionality and optimal deterrence are incompatible aims of punishment.
Optimal deterrence may require that a slightly more serious crime receives a
much more severe sentence, if a sizeable increase in the penalty would have
little additional deterrent effect, but would still increase the costs from
false conviction. As the crime becomes more and more harmful, it is
eventually worth increasing the penalty substantially for the sake of the
additional deterrence, however, and there must exist some pair of almost
identical crimes that spans the break-point.

Tort law is also concerned with choosing penalties to deter harmful
behavior. As Png (1986) and Shavell (1987) note, excessive imposition of
liability not only penalizes the innocent, but deters them from useful
behavior that courts might confuse with wrongdoing. Tort law balances the
false-liability costs of penalties against the benefit from deterring
tortious behavior. This balancing can result in optimal judgements either
bigger or smaller than the harm inflicted. Polinsky and Che (1991), for
example, suggest that the penalty should be greater, but so should be the
burden of proof, to reduce the number of cases brought and thus the
transactions cost. The question remains of whether the judgement should
increase smoothly with the harm.

The present model applies to civil liability insofar as deterrence, not
compensation, is its aim. Tort law uses cash transfers as penalties to deter
harmful behavior, but high penalties are costly because when wrongly
inflicted they deter innocent behavior. The model shows that the optimal
schedule for liability need not be continuous. Figure 1 can be relabelled as
the relationship between liability and amount of harmful behavior, where a
small amount of attention will eliminate a large number of accidents, but
further large reductions require substantial changes in behavior. A small
penalty will prevent many accidents, but the penalty must become much
greater if the number of accidents is again to be substantially reduced.
High liability for lower levels of harm would needlessly deter innocent
behavior, but at some point the harm becomes great enough for deterrence of
harmful behavior to take precedence, and a jump occurs in liability.

Whether the offense be criminal or civil, a discontinuous jump in the
optimal punishment is not, of course, a necessary conclusion, only a
possible one, which depends on the deterrent relationship between crime and
penalty. The example showed that such a jump would occur if there is a large
group of potential offenders who are deterred by a small penalty and others
who will not be deterred except by substantially higher penalties. Whether
this is true depends on the particular time and place, but judges should
pause before striking down statutes simply because similar offenses have
very different penalties; a jump in penalties at some harm level may be
appropriate, even though the particular threshold level may seem arbitrary.

%---------------------------------------------------------------
\newpage  \noindent {REFERENCES}

Andreoni, James, 1991, ``Reasonable Doubt and the Optimal Magnitude of
Fines: Should the Penalty Fit the Crime?'' {\it RAND Journal of Economics},
22: 385-395.

Becker, Gary (1968) ``Crime and Punishment: An Economic Approach,'' {\it %
Journal of Political Economy}, 76, March/April 1968, 169-217.

Ehrlich, Isaac (1982) ``The Optimum Enforcement of Laws and the Concept of
Justice: A Positive Analysis,'' {\it International Review of Law and
Economics}, 2: 3-27.

Friedman, David (1981) ``Reflections on Optimal Punishment, or: Should the
Rich Pay Higher Fines?'' {\it Research in Law and Economics}, ed. Richard
Zerbe, 3: 185-205.

Kaplow, Louis (1990a) ``A Note on the Use of Nonmonetary Sanctions,'' {\it %
Journal of Public Economics}, 42: 245-247.

Kaplow, Louis (1990b) ``Optimal Deterrence, Uninformed Individuals, and
Acquiring Information about Whether Acts are Subject to Sanctions,'' {\it %
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Kaplow, Louis (1994) ``The Value of Accuracy in Adjudication: An Economic
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Mookherjee, Dilip and Ivan Png (1992) ``Monitoring vis-a-vis Investigation
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Png, Ivan (1986) ``Optimal Subsidies and Damages in the Presence of Judicial
Error,'' {\it International Review of Law and Economics}, June 1986,
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Polinsky, A. Mitchell and Y. Che (1991), ``Decoupling Liability: Optimal
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Polinsky, A. Mitchell and Steven Shavell (1984), ``The Optimal Use of Fines
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Posner, Richard (1973) ``An Economic Approach to Legal Procedure and
Judicial Administration,'' {\it Journal of Legal Studies}, 2: 410-415.

Rasmusen, Eric (1994) ``Damage Awards When Court Error is Predictable,''
working paper, Indiana University School of Business, June 1994.

Rubinfeld, Daniel and David Sappington (1987) ``Efficient Awards and
Standards of Proof in Judicial Proceedings,'' {\it Rand Journal of Economics}%
, Summer 1987, 18: 308-315.

Shavell, Steven (1985) ``Criminal Law and the Optimal Use of Nonmonetary
Sanctions as a Deterrent,'' {\it Columbia Law Review}, October 1985, Vol.
85, 1232-1263.

Shavell, Steven (1987) ``The Optimal Use of Nonmonetary Sanctions as a
Deterrent,'' {\it American Economic Review}, September 1987, 77: 584-592.

Shavell, Steven (1991) ``Specific Versus General Enforcement of the Law,'' 
{\it Journal of Political Economy}, October 1991, 99: 1088-1108.

Stigler, George (1970) ``The Optimum Enforcement of Laws,'' {\it Journal of
Political Economy}, May/June 1970, 78: 526-36.

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

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