% JUne 15, 1994 MORE INTUITION. % April 20, 1993, June 10, 2001. \documentstyle[12pt,epsf]{article} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %TCIDATA{OutputFilter=LATEX.DLL} %TCIDATA{LastRevised=Sat Jun 09 21:51:19 2001} %TCIDATA{} %TCIDATA{CSTFile=article.cst} \input{tcilatex} \begin{document} \baselineskip 16pt \parindent 24pt \parskip 10pt \titlepage \vspace*{12pt} \begin{center} {\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