\documentclass[12pt, fleqn, usenames,dvipsnames]{article} \usepackage{amsmath} \usepackage{eurosym} % \usepackage[fleqn]{nccmath} % \usepackage{amsthm} %\usepackage{amssymb} \usepackage{graphicx} \usepackage{hyperref} \usepackage{verbatim} \hypersetup{breaklinks=true, pagecolor=white, colorlinks=true, linkcolor= blue, hyperfootnotes= true, urlcolor=blue } \urlstyle{rm} \usepackage{fancyhdr} \pagestyle{fancy} \fancyhead{}\fancyfoot{} \rhead{\thepage} \newcommand{\margincomment}[1] {\mbox{}\marginpar{\tiny\hspace{0pt}#1}} \newcommand{\comments}[1]{} \renewcommand{\baselinestretch}{1.2} % \oddsidemargin -.1in %\evensidemargin -.1in %\textwidth 7in \begin{document} % \begin{raggedright} \parindent 18pt \parskip 10pt \titlepage %\vspace*{12pt} \begin{center} \begin{Large} {\sc **On the Efficiency and Desirability of Disclosure of Whether Someone Has a Clean Criminal Record \\ \footnote{(Steve--- here's a new possible title--- we should stress the positive aspect of being able to prove a clean record. Also we should include not just employers but volunteering and marriage, if we can fit them in.}\\ %\bigskip Eric Rasmusen and Steven Shavell May 1, 2016 }%end of small capitals \end{Large} {\it Abstract} \end{center} \begin{footnotesize} werwerwewerr This is not ready for circulation. \noindent Eric Rasmusen: Dan R. and Catherine M. Dalton Professor, Department of Business Economics and Public Policy, Kelley School of Business, Indiana University, Bloomington Indiana. \href{mailto:erasmuse@indiana.edu}{ Erasmuse@indiana.edu}. Phone: (812) 345-8573. Messaging: 8123458573@vtext.com. \noindent Steven Shavell: This paper: \url{http://www.rasmusen.org/papers/crime-records-rasmusen-shavell.pdf}. (not posted yet) { \noindent Keywords:sdfsdf.} \noindent { We would like to thank sdfsdfsd for their comments. } \end{footnotesize} \newpage \noindent {\sc 1. Introduction} Not written yet. \noindent \underline{ 2. Basic assumptions: crime-averse and crime-prone people. } The total population is of size 1 and is made up of two mutually exclusive groups of individuals, crime prone and crime averse. Crime-prone people are a fraction $\beta$ of the population, and crime-averse make up the remainder, $\gamma$. People live for two periods, youth and maturity, earning a wage of $w$ in each period. We assume zero discounting. In the youth period, a crime-prone person has the opportunity to commit a harmful act, causing harm $h>0$, where $h$ does not vary across people. If a crime-prone person commits the harmful act, he enjoys a benefit $b$, where $b$ is distributed uniformly on $[0,h]$.\footnote{It might well be that we can relax the uniform density assumption and just use a general f(b). THat is all that is needed for the comparison between regimes. THe only possible use of the uniform density is on the comparative statics on detection rpobability $p$ for the full-information case. Eric put in the assumption because he thought it would allow him to solve for $p^*$, but it turns out solving is still to hard.} If there is a type of crime-prone person who is indifferent between committing a crime and not, we will denote his type by $b^*$. It will be seen when we define social welfare that the commission of the harmful act lower social welfare by $h- b$, which is positive, so it makes sense to call the act a crime. Crime-averse people have $b \leq 0$ and hence never commit crimes. We can think of them as people who don't enjoy any benefits $b$ from committing the act, or who would experience disutility from doing so (like from beating someone up, or from stealing...due to guilt...whatever). If someone commits a crime in youth, he is instantly detected with probability $p$ and suffer a jail sentence $s \leq 1$. There are no legal errors once someone is detected. The disutility to a person who is punished is $d\cdot s $. After serving his sentence, he works during the rest of his youth, a length $1-s$. The cost to the state of maintaining $p$ is $c(p)$, where $ c' > 0 $ and $c'' > 0.$ There is also a social cost of jailing a person of $k \cdot s$, where $s$ is the length of the sentence and $ k > 0.$ We require that the government choose a sentence such that $s < \overline{s} <1$. This represents the need to limit sentences out of a sense of limited retributive justice (it is unfair to sentence someone to 10 years for shoplifting a radio) or the need for marginal deterrence (if someone is sentenced to 10 years for shoplifting a radio, they will commit the more profitable crime of robbery instead of shoplifting).\footnote{We could have different sentences for each period if we had different $\overline{s}$'s. Maybe we shoudl think about different values of $s$ too. } The state makes three choices about law enforcement: it chooses $ p,s$, and whether or not to disclose criminal records to employers (that is, to disclose whether a person was caught for a crime). In the second, maturity, all people work and nobody commits crimes. In youth, everyone is equally productive, with output $r_1$ and wage $w_1= r_1>0$. In maturity, crime-prone people are less productive than crime-averse people. The productivity parameter of a crime-prone person is $r_\beta$ and of a crime-averse person is $r_\gamma$ , where $ r_1 < r_\beta < r_\gamma .$ The interpretation of the assumption is that people who are disposed to commit crime-prone acts are people who have personal characteristics that would lead them to benefit from committing criminal acts—like social irresponsibility, low intelligence, poor impulse control, lack of respect for authority, etc.--- also have low productivity in adult jobs that require trust. A person's actual productivity, however, depends on his job as well as his parameter. There are two kinds of maturity jobs, good jobs and bad jobs. The productivity of a crime-averse person is $r_\gamma$ in a good job and $r_\gamma- x_\gamma$ in a bad job. The productivity of a crime-prone person is $r_\beta$ in a bad job and $r_\beta- x_\beta$ in a good job. Thus, $ x_\gamma$ and $x_\beta$ are the loss from matching a person to the wrong job. We assume that $r_\gamma> r_\beta$. We make no assumption on whether $r_\gamma- x_\gamma> r_\beta$; it could be that a crime-prone person is better at a crime-prone job than a crime-averse person would be. We assume that engaging in crime has no effect on a person's productivity (in contrast to Rasmusen (1996)). Denote the equilibrium wage of someone in a good job with a clean criminal record by $\overline{w}_\gamma$ and the wage with a conviction as $\underline{w}_\gamma$. Similarly, the wages for a bad job are $\overline{w}_\beta$ and $\underline{w}_\beta$. Note that these are independent of whether the worker is crime-averse or crime-prone, but if the quality of the worker is known, the crime-averse workers will never take bad jobs and the crime-prone workers will never be offered good jobs. The wage of an employee will equal his expected marginal product due to competition among employers. What employers know about the productivity of employees will be discussed below. Note, however, that since the productivity of a crime-prone worker is lower in a good job than in a bad job, and only crime-prone workers commit crimes, the equilibrium wage of a worker with a criminal record in a good job will always be lower than his wage in a bad job: $\underline{w}_\gamma = r_\beta- x_\beta <\underline{w}_\beta. $ \noindent \underline{Basic assumptions: social welfare. } Social welfare is (a) the sum of the benefits people derive from committing crimes and the net production of people on the labor market, less (b) the sum of the disutility suffered by people in jail ($d \cdot s$ for each person in jail $s$ years), the cost to state of jailing people ($k \cdot s$ for each person in jail $s $ years), the cost $c(p)$ of maintaining $p$ over the two periods, and the harm done by crime ($h$ for each person who commits the crime-prone act). Comment: this is an intuitively appealing social welfare measure. And, if we literally added up the utilities of people (and had them pay taxes to finance state activities), we'd see it is the sum of their utilities. \noindent \underline{ The Equilibrium under Full Information.}\\ Suppose whether people are crime-averse or crime-prone is public knowledge, but it is still difficult to detect crime. What is the optimal penalty? In this situation, in the second period, crime-prone people will get bad jobs at a wage of $w=r_\beta$ and crime-averse people will get good jobs at a wage of $w=r_\gamma$. If $b$ is high enough, a crime-prone person will commit a crime. Denote by $b^*$ the benefits to a crime-prone person just indifferent about committing a crime in period 1, which is determined by his expected loss from crime. Since the wage is unaffected by that decision, the cutoff benefit is determined by the disutility of prison and the lost income. In youth, a person's flow utility for the period from abstaining from crime whether is, since $w_1 = r_1$, \begin{equation} \label{e0} U^1(crime-prone, honest) = r_1, \end{equation} and his utility from crime is \begin{equation} \label{e0} U^1(crime-prone,criminal) = b+ (1-p)r_1 + p (1-s)r_1 - p\cdot s \cdot d = b+ r_1 -ps (r_1 +d) \end{equation} so \begin{equation} \label{e0} b^* = p\cdot s (r_1+ d) \end{equation} In the second period, a crime-averse person's utility is \begin{equation} \label{e0} U^2(crime-averse) = r_\gamma, \end{equation} and a crime-prone person's utility is \begin{equation} \label{e0} U^2(crime-prone) = r_\beta. \end{equation} Those who are deterred are those for whom $b \leq b^*$. crime-prone people with $b> b^*$ commit crimes; crime-prone people with $b \leq b^*$ do not. Thus, the fraction of crime-prone people deterred in period $1$ is $F(b^*) =b^*/h$ and the fraction of crime-prone people who commit crimes is $1 - F(b_t^*)= 1 - b^*/h$. The cost to victims and the cost to government of detection and punishment is, since there are $\beta$ crime-prone people in total, \begin{equation} \label{e0} \begin{array}{lll} U(victims, govt.) & = &\displaystyle - (1- \frac{b^*}{h})\beta h \\ && \\ && \displaystyle - (1- \frac{b^*}{h}) \beta p \cdot s \cdot k - c(p) \\ \end{array} \end{equation} The first line of the social welfare equation below consists of the utility of the crime-averse people in both periods plus the utility of the crime-prone people who commit and do not commit crimes in the first period. The second line consists of the utility of the crime-prone people in the second period, the disutility of victims of crime, and the disutility of the government. \begin{equation} \label{e0} \begin{array}{lll} Welfare &= & \displaystyle \gamma r_1 + \gamma r_\gamma + \left(\int_0^{b^*} r_1 f(b)db \right) \beta + \left(\int_{b^*}^h ( b+ r_1 -ps (r_1 +d) ) f(b)db \right) \beta \\ &&\\ && \displaystyle +\beta r_\beta - (1- \frac{b^*}{h})\beta h - (1- \frac{b^*}{h}) \beta p \cdot s \cdot k - c(p) \\ \end{array} \end{equation} This simplifies to \begin{equation} \label{e0} \begin{array}{lll} Welfare &= & \displaystyle \gamma r_1 + \gamma r_\gamma + \displaystyle \frac{b^*}{h} r_1 \beta + \frac{h^2}{2h} \beta - \frac{(b^*)^2}{2h} \beta + \left(1- \frac{b^*}{h}\right) (r_1- p\cdot s (r_1+ d) \beta \\ &&\\ &&\displaystyle + \beta r_\beta - \left(1- \frac{b^*}{h}\right) \beta h - \left(1- \frac{b^*}{h}\right) \beta p \cdot s \cdot k - c(p) \\ \end{array} \end{equation} We can differentiate social welfare to get the optimal $p$, using $ \frac{db^*}{dp } = s( r_1+d) $, \begin{equation} \begin{array}{lll} \frac{d\; welfare }{dp} & = & \displaystyle \frac{s( r_1+d) r_1\beta}{h} - \frac{2 b^*s(r_1 + d) \beta}{2h} - \frac{s( r_1+d) (r_1- p\cdot s (r_1+ d) ) \beta}{h}\\ & & \\ & &\displaystyle - \left(1- \frac{b^*}{h}\right) \cdot s (r_1+ d) \beta + \frac{ s( r_1+d) \beta h}{h} + \frac{ s( r_1+d) \beta psk }{h} -\left(1- \frac{b^*}{h}\right) \beta \cdot s \cdot k - c'(p) \\ & & \\ & & \\ & = & \displaystyle \frac{s( r_1+d) r_1\beta}{h} - \frac{ ps(r_1 + d)s(r_1 + d) \beta}{ h} -\frac{s (r_1+d) r_1\beta}{h} + \frac{s^2( r_1+d)^2p \beta}{h}\\ & & \\ & &\displaystyle - \left(1- \frac{ps(r_1 + d)}{h}\right) \cdot s (r_1+ d) \beta + s( r_1+d) \beta \\ & & \\ &&\displaystyle + \frac{ s( r_1+d) \beta psk }{h} -\left(1- \frac{ps(r_1 + d)}{h}\right) \beta \cdot s \cdot k - c'(p) \\ & & \\ & & \\ & = & \displaystyle \beta \Big[ - \left(1- \frac{ps(r_1 + d)}{h}\right) \cdot s (r_1+ d) + s( r_1+d) \\ & & \\ & & \displaystyle + \frac{ s( r_1+d) psk }{h} -\left(1- \frac{ps(r_1 + d)}{h}\right) \cdot s \cdot k \Big] - c'(p) \\ & & \\ & & \\ & = & \displaystyle \frac{\beta}{h} \Big[ - \left(h- ps(r_1 + d) \right) \cdot s (r_1+ d) + h s( r_1+d) \\ & & \\ & & + s( r_1+d) psk -\left(h- ps(r_1 + d) \right) \cdot s \cdot k \Big] - c'(p) \\ & & \\ & & \\ & = & \displaystyle \frac{\beta}{h} \Big[ ps^2(r_1 + d)^2 + s^2( r_1+d) p k - hsk + ps^2(r_1 + d) k \Big] - c'(p) \\ \end{array} \end{equation} That ends up rather a mess, but we can do comparative statics on $p^*$ using the implicit function theorem. Let's check the second derivative: \begin{equation} \begin{array}{lll} \frac{d^2\; welfare }{dp^2} & = & \displaystyle \frac{\beta}{h} \Big[ s^2(r_1 + d)^2 + s^2( r_1+d) k + s^2(r_1 + d) k \Big] - c''(p) \\ \end{array} \end{equation} Almost every term is positive, instead of the negative that we want for a concave maximand. So if $c''=0$, the government would choose enough punishment to stomp out crime completely. If $c''>0$ and is big enough, we'd get a smaller $p$ and some crime, and there might even be multiple equilibria. At any maximum, though, we'd have $\frac{d^2\; welfare }{dp^2} <0$ and we could use the implicit function theorem. Our only other parameters besides $s$ (dealt with later-- it turns out that $s^* = \overline{s}$) are $h, r_1, d, k$. \begin{equation} \begin{array}{lll} \frac{d^2\; welfare }{dp \cdot d h } & = & \displaystyle - \frac{\beta}{h^2} \Big[ ps^2(r_1 + d)^2 + s^2( r_1+d) p k - hsk + ps^2(r_1 + d) k \Big] - \frac{\beta}{h } \Big[ sk \Big] \\ & &\\ & = & \displaystyle - \frac{\beta}{h^2} \Big[ ps^2(r_1 + d)^2 + s^2( r_1+d) p k + ps^2(r_1 + d) k \Big] + \frac{\beta}{h^2} \Big[ - hsk \Big] - \frac{\beta}{h^2 } \Big[ hsk \Big] \\ && \\ & = & \displaystyle - \frac{\beta}{h^2} \Big[ ps^2(r_1 + d)^2 + s^2( r_1+d) p k + ps^2(r_1 + d) k \Big]<0 \\ \end{array} \end{equation} So $p$ goes up with $h$, since at an equilibrium $\frac{d^2\; welfare }{dp^2 } >0$ and \begin{equation} \displaystyle \frac{dp }{d h } = - \frac{\frac{d^2\; welfare }{dp \cdot d h }}{ \frac{d^2\; welfare }{dp^2 } } \end{equation} How about $r_1$ and $d$? \begin{equation} \begin{array}{lll} \frac{d^2\; welfare }{dp \cdot d r_1 } & = &\displaystyle \frac{\beta}{h} \Big[ ps^2 2(r_1 + d) + s^2 r_1 p k + ps^2(r_1 k \Big] >0 \\ \end{array} \end{equation} Thus, $\frac{dp }{d r_1 }<0$. The parameters $r_1$ and $d$ only enter in sum, so $d$ would have the same comparative statics. \begin{equation} \begin{array}{lll} \frac{d^2\; welfare }{dp \cdot d k } & = & \displaystyle \frac{\beta}{h} \Big[ s( r_1+d) ps -\left(h- ps(r_1 + d) \right) \cdot s \Big] \\ &&\\ & = & \displaystyle \frac{\beta s}{h} \Big[ ( r_1+d) ps - h+ ps(r_1 + d) \Big] \\ &&\\ & = & \displaystyle \frac{\beta s}{h} \Big[ ( r_1+d) ps - h+ ps(r_1 + d) \Big] \\ &&\\ & = & \displaystyle \frac{\beta s}{h} \Big[ b^* - h + b^* \Big] \\ \end{array} \end{equation} If $b^* > h/2$ then $\frac{d^2\; welfare }{dp \cdot d k }>0$ and $\frac{dp }{d k }<0$. What is happening is that if $b^*$ is big, then few crime-prone people are engaging in crime, and the effect of an increase in $p$ is to increase the number of them that go to prison, increasing prison costs. Thus, a higher prison per-unit cost $k$ leads to less detection being more attractive. If, on the other hand, $b^*$ is small enough that most crime-prone people are engaging in crime, the effect of an increase in $p$ is to reduce the number of prisoners, because increased deterrence more than makes up for the greater probability of detecting a criminal. In that case, an increase in $k$ makes high $p$ more desirable. \noindent Proposition: The optimal sentence is $s = \overline{s}$. \noindent Proof: The probability $p$ of detection enters welfare only via $b^*$, $ps$, and $c(p)$. Recall that $b^*= ps(d+ r_\beta)$, so $p$ only enters via $c(p)$ and the multiple $ps$. Suppose that the optimal level of $ps$ equals $X$. We can maintain $ps=X$ while reducing the cost $c(p)$ by shrinking $p$ and increasing $s$, so long as $s