JUSTIFYING IMPRISONMENT: ON THE OPTIMALITY OF EXCESSIVELY COSTLY PUNISHMENT

Transcription

1 JUSTIFYING IMPRISONMENT: ON THE OPTIMALITY OF EXCESSIVELY COSTLY PUNISHMENT Abraham L. Wickelgre * Federal Trade Commissio awickelgre@ftc.gov May 2002 Abstract: The crimial puishmet literature has focused o justifyig o-maximal puishmets ad the use of o-moetary sactios. It has ot addressed why imprisomet, rather tha cheaper forms of corporal puishmet, should be the domiat type of o-moetary sactios. David Friedma (1999) recetly hypothesized that, because covicts lack political ifluece, it is desirable to make puishmet more costly tha ecessary to prevet policy makers from excessively puishig covicts. This paper explicitly models this hypothesis ad uses simulatios to determie uder what circumstaces this hypothesis justifies usig imprisomet rather tha cheaper o-moetary sactios. * I am grateful to Matt Ellma, Steve Levitt, ad Joel Schrag for commets. The views expressed i this paper are those of the author ad do ot ecessarily represet the views of the Federal Trade Commissio or ay idividual Commissioer. All errors are my ow.

2 I. Itroductio I Gary Becker s (1968) semial article o the ecoomics of crime ad puishmet, he argued that the most efficiet way to deter crimials was through metig out very large puishmets with very small probability. Sice this leads to some paradoxical results (billio dollar fies for jaywalkig, eforced oce a milleium) ad does ot reflect the form of actual puishmets, may ecoomists have refied Becker s theory. Much of this literature aims to explai why we do ot see maximal puishmets ad very small detectio rates (Polisky ad Shavell 1979; Poser 1985; Shavell 1987a; Malik 1990; Miceli 1990; Miceli 1991; Polisky ad Shavell 2000b; Garoupa 2001), as Becker's theory suggests is optimal, ad why fies do ot play the large role i puishig crimials that Becker argues they should (Polisky ad Shavell 1984; Chu ad Jiag 1993; Levitt 1997). See Garoupa (1997) ad Polisky ad Shavell (2000a) for excellet surveys of this literature. While these refiemets do provide covicig justificatios for why we do ot always use maximal puishmets ad why we use o-moetary sactios, they do ot justify why imprisomet should be (by far) the domiat form of o-moetary sactios. Thus, this paper is ot about the choice betwee fies ad o-moetary sactios. Istead, it ivestigates the optimality of imprisomet i cases where o-moetary sactios, i some form, are optimal (for the reasos give by the authors listed above). From the stadpoit of deterrece theory, the use of imprisomet, as opposed to cheaper forms of o-moetary sactios, is quite puzzlig. If there are two differet puishmet mechaisms that impose idetical expected costs o covicts, yet oe is cheaper for society to impose tha the other, it seems that the cheaper oe would always be preferable (D. Friedma 1999). Cosider, for example, a five-year priso setece. Surely, there is some level of corporal puishmet 1 (or probability of executio) that is equivaletly upleasat (thus, providig the same level of deterrece) as a five-year 1 Though, techically, corporal puishmet icludes imprisomet, i this paper I will use it to refer to oimprisomet forms of corporal puishmet (e.g. whippig or caig). 1

3 priso setece but would substatially cheaper. 2 Why, the, do we cotiue to use imprisomet ad almost ever corporal puishmet (the death pealty beig the exceptio, ad it is ever used probabilistically to puish less severe crimes)? Could this possibly be optimal? I a recet paper, David Friedma (1999) iformally proposes a aswer to this puzzle. 3 He argues that the policy makers who determie the magitude of puishmets for covicts do so to maximize their ow utility fuctio rather tha that of a beevolet social plaer. How this utility fuctio weights the welfare of each member of society depeds o the power or ifluece of the iterest groups of which a idividual is a member (he appeals to Becker s (1983) formal model of this situatio). Because covicts have little or o power or ifluece, the policy maker gives them very little weight i his objective fuctio. (Admittedly, there are groups that represet the iterests of covicts, e.g, the ACLU or Amesty Iteratioal. As George Bush demostrated i the 1988 presidetial campaig, however, garerig the support of these groups, beig a card-carryig member of the ACLU, may do a politicia more harm tha good.) As a result, if puishig covicts were early costless, the policy maker would puish excessively. By icreasig the cost of puishmet to the rest of society, however, a social plaer ca iduce the policy maker to choose puishmets that are less excessive (though, give the cost, puishmet magitudes will still be socially excessive due to the reduced weight put o covict welfare). Imprisomet serves this fuctio. This justificatio ecessarily assumes that while policy makers ca say the pealty for robbery is 1 year or 5 years or 20 years i priso, the form of the puishmet is costraied, perhaps by costitutioal prohibitio agaist cruel ad uusual puishmet. Thus, it is importat to ote that this is a ormative, rather tha a positive, aalysis (hece, the title is Justifyig Imprisomet rather tha Explaiig Imprisomet ). That is, Friedma s hypothesis argues that costitutioal or other costraits that force policy makers to use imprisomet rather tha cheaper forms of corporal 2 Poser (1985) argues that afflictive puishmets that impose a equivalet disutility as a five-year priso setece would be so severe as to risk death or permaetly ijure the covict. This does ot explai, however, why such puishmets are welfare-reducig vis-à-vis imprisomet. Moreover, covicts ru these risks i priso as well. 3 I iformally discuss other possible aswers to this puzzle, ad why they are ot totally covicig, i sectio II. 2

4 puishmet are optimal but does ot explai why they exist. Of course, most other aalysis of optimal puishmets are ormative i ature i that they justify why puishmets should or should ot be maximal or whe moetary versus o-moetary sactios are optimal. This hypothesis, however, brigs up several additioal questios. Is it ecessarily the case that the social beefit (for a social plaer who weights all idividuals equally) of more optimal puishmet magitudes is worth the added social cost of more costly puishmet? Does our curret use of imprisomet come at all close to balacig the costs of costly puishmet with the beefit of more optimal puishmet magitudes? How much must the policy maker discout the welfare of covicts relative to the rest of society to make costly puishmet optimal? I this paper, I formally model Friedma s hypothesis ad solve it via simulatio to aswer these questios, alog with some others. I particular, I show that if the distributio of beefits from crime is uiform, the it is ever optimal to impose costly puishmet. For desity fuctios, such as the ormal or the logistic, that get very small for very large beefits of crime, however, costly puishmet ca be optimal whe the policy maker (the aget who determies the magitude of the puishmet) discouts the utility of the covicts sufficietly. For parameter values that are roughly cosistet with available data, costly puishmet is almost certaily optimal whe the weight the policy maker attaches to covicts is o larger tha five percet of the weight he gives to victims. Ad costly puishmet could be optimal whe this weight is as high as 15 to 20 percet. Give the lack of political power of covicts, it is probably reasoable to assume that their weight i the objective fuctio of policy makers is substatially less tha five percet. If this is i fact the case, just how costly should puishmet be? For illustrative puposes, I cosider the optimal puishmet cost whe the weight policy makers give to covicts is oe percet of the weight give to everyoe else. I this case, I show that the cost of puishmet to society should optimally be somewhere betwee 5 percet ad 25 percet of the cost to the covict (depedig o the values of other parameters). Sice that the average cost of imprisoig a covict for a year is about $20,100 (Departmet of Justice 1995), the simulatios i the paper suggest that curret cost of puishmet is probably ot too far from the optimal level. I additio to validatig our use of imprisomet i geeral, the simulatios also justify the way 3

5 priso coditios vary across offeses. I show that whe costly puishmet is optimal, the optimal cost is greater the less damagig the offese. Thus, while those covicted of lesser offeses should receive reduced puishmets, the form of this reductio should ot just be reduced time i priso but icer priso coditios as well. This is cosistet with curret practice (Poser 1985), ad (as I will discuss further below) icosistet with the icapacitatio theory of imprisomet. I additio to Friedma s paper, there are a few other papers that discuss the effects of a selfiterested policy maker o optimal puishmets. Garoupa ad Klerma (2002) discuss how a retseekig govermet will eforce the crimial law. This paper is primarily cocered with puishmet by fies. Garoupa ad Klerma (2001) cosider how the possibility of corrupt eforcers affects the optimal use of o-moetary sactios. While related i spirit, their paper assumes that o-moetary sactios take the form of imprisomet ad does ot discuss the tradeoff betwee imprisomet ad other forms of o-moetary sactios. Most closely related is a paper by Hylto ad Khaa (2001). This paper iformally justifies pro-defedat crimial procedure as a meas of costraiig law eforcers ad policymakers from usig the crimial law to advace persoal goals. They also metio Friedma s hypothesis that iefficiet puishmets ca serve this ed as well. Like the curret paper, the focus of their aalysis is ormative, ot positive, i that their aalysis justifies these practices from a social welfare stadpoit rather tha explaiig why they exist. Ulike my paper, however, they do ot formally aalyze uder what circumstaces (if ever) the beefits of iefficiet puishmets outweigh the costs. The rest of the paper is orgaized as follows. Sectio II iformally discusses some alterative explaatios for why imprisomet might be superior to less costly forms of o-moetary sactios. Sectio III describes the model of a social plaer choosig the form of puishmet kowig that a policy maker will the choose its magitude. Sectio III also cotais the aalysis of the case of a uiform distributio of beefits from crime. Sectio IV explais how the simulatios were doe for the cases of a ormal ad a logistic distributio of gais from crime. Sectio V summarizes ad discusses the results of the simulatios. I Sectio VI, I discuss calibratio issues raised i the simulatios. Sectio VII cocludes. Appedix A cotais the proof of Propositio 1. Appedix B cotais a more detailed descriptio of how the simulatios were doe. 4

6 II. ALTERNATIVE EXPLANATIONS FOR IMPRISONMENT Oe reaso for rigorously aalyzig Friedma s hypothesis is that, while there are may other possible justificatios for usig imprisomet istead of cheaper forms of o-moetary sactios, oe of them provide a totally covicig justificatio for why it is almost ever optimal to use other forms of o-moetary sactios (other tha capital puishmet). Oe explaatio is that we just do ot have a taste for corporal puishmet; society views it as ihumae or ufair (Miceli 1991 ad Polisky ad Shavell 2000b discuss fairess ad puishmet). This, however, just assumes away the puzzle. Moreover, sice corporal puishmet eed ot impose ay greater utility loss o the covict tha imprisomet, why should a puishmet that could make everyoe (weakly) better off be viewed as ihumae or ufair? Similarly, the fear of false positives caot justify imprisomet sice oe could always use less severe corporal puishmets. With false positives, however, imprisomet has the advatage that it is easier to reverse tha corporal puishmet if ew iformatio reveals that a covict is iocet. This certaily justifies usig imprisomet ad delayig corporal puishmet util after all a defedat s appeals have bee exhausted. The pro-defedat bias i our crimial justice system (Hylto ad Khaa 2001), however, already greatly reduces the risk of false positives. I fact, as Hylto ad Khaa (2001) demostrate, it is already hard to justify this bias based solely o the risk of false covictios. Thus, it is seems ulikely that false covictios are both prevalet eough ad likely eough to be reversed after may years i priso, that it is worth icarceratig huge umbers of covicts for log periods (at great expese) after they have exhausted all possible appeals based o the tiy probability that a covictio might be reversed (ad, eve the, there is o guaratee the freed covict is iocet). Certaily, this must be true for covicts that have bee icarcerated for a great may years. I fact, for reversal of false covictios to justify imprisomet, it would also have to be the case that the social cost savigs from edig a small umber of prisoer s seteces early could ot be more cheaply achieved by usig corporal puishmet ad spedig more resources o improvig the accuracy of trials. Some argue that prisos ca serve to rehabilitate crimials. This must mea that time i priso either reduces the taste for crime or icreases a prisoer s lawful opportuities. How effective 5

7 rehabilitatio is, ultimately, is a empirical questio, though the high recidivism rate suggests that efforts at rehabilitatio may ot be worth the cost of imprisomet. This is ot too surprisig sice by the time covicts are i priso, society has already icurred substatial expese (e.g., public schools) i the effort to make people value lawful behavior ad ehace their lawful opportuities. Such efforts, before a perso has already suk a great deal of ivestmet ito crime-specific huma capital, are likely to be more effective (though maybe ot as well targeted) tha priso rehabilitatio. Eve if rehabilitatio is effective, it ca justify imprisomet oly if it is eough more effective tha larger, cheaper, puishmets are at reducig crime to justify its costs. Moreover, rehabilitatio caot justify very log priso terms that icarcerate covicts log after the poit at which they are likely to commit more crimes. A more covicig explaatio is icapacitatio. Eve with corporal puishmet, deterrece may be costly eough that we will ot wat to deter all crime (or, alteratively, there are some people who just are impossible to deter), so there will always be crime. By usig imprisomet, we keep those crimials off the street so they caot commit further crimes (see Shavell 1987b for a model of optimal icapacitatio). Certaily, the icapacitatio argumet must be part of the explaatio for the use of imprisomet, but it caot be all of it. First, it is widely kow that for may offeses (may violet crimes, for example), almost all offeders are relatively youg. Yet, we still impriso people for such crimes past the age where there is ay possibility they will commit such crimes i the future. Surely, there ca be little, if ay, icapacitatio beefit to keepig a 75 year old covicted rapist i priso. Why ot release him at 45 ad make up the deterrece lost by addig i corporal puishmet? Better yet, why ot just release him at 45 ad sped less moey o priso ameities? This brigs up aother reaso why icapacitatio caot fully justify our puishmet policies: we sped a great deal more moey o prisos tha is ecessary to icapacitate crimials. Oe could claim that the optimal legth of priso terms for icapacitatio is greater tha the legth ecessary for deterrece (whe priso ameities are miimal), thus we have to sped moey to meet both goals with the same setece. This argumet has two flaws. First, it is icosistet with the fact that we ofte icapacitate covicts log past the poit where there is ay reasoable chace they will commit further crimes. Secod, evidece idicates that icreased priso terms reduce crimes primarily through 6

8 deterrece for property crimes, while icapacitatio is more importat for violet crime (Levitt 1998). This suggests that each year i priso for property crime should be more upleasat tha each year i priso for violet crime (sice it is less ecessary to icrease priso ameities to balace icapacitatio ad deterrece affects for property crime). This is the exact opposite of curret practice (Poser 1985). Thus, the extreme paucity of the use of corporal puishmet for covicts alog with the large amout spet o priso ameities still requires justificatio. Moreover, i the Uited States, costitutioal prohibitio agaist cruel ad uusual puishmet (eforced by judiciary, the brach of govermet least subject to iterest group ifluece) plays a key role i requirig such spedig. Almost every state has bee the defedat i at least oe litigatio allegig ucostitutioal overcrowdig (Levitt 1996). Because the simulatios demostrate that it ca be optimal to force policy makers to use costly puishmets, they provide a explaatio for this costitutioal itervetio that icapacitatio caot. III. THE MODEL There is a populatio where each perso ca choose to commit a crime or ot. If a perso commits a crime she receives a beefit from doig so valued at v, where v is draw from a distributio with a cumulative distributio fuctio H, with a associated probability desity fuctio h. For each crime that is committed, there is a exogeous probability, π c, that someoe will be caught ad covicted for that crime. The perso covicted for a crime, however, eed ot be guilty. The probability of beig covicted of a crime as a result of committig it is give by π π c (some covictios could be of iocet people). If covicted, a perso is puished by a puishmet amout z, where z is measured i terms of the utility loss to the covict. Whe a covict is puished by z, this costs society a amout αz. The form of the puishmet determies the value of α. For example, the use of fies collected without cost would yield α=-1, society gais from the puishmet via the receipt of fies. Note, however, that I rule out the use of fies i this model (I require a 0) sice, as discussed above, there are already covicig explaatio for why fies are ot always optimal, ad I wat to aalyze the optimality of imprisomet versus other forms of o-moetary sactios. The fact that fies are sometimes used together with costly puishmet, however, is ot icosistet with the claim that costly puishmet may 7

9 be preferable to costless puishmet. Because the social welfare fuctio is o-liear (see below), the tradeoff betwee the greater cost of more costly puishmet ad its beefits i terms of reduced overpuishmet eed ot be idetical at all levels of puishmet cost. (I fact, give that there is o social cost to excessive puishmet with fies, sice society gais exactly what the covict loses, the tradeoff is ot a issue for fies.) For corporal puishmet α will be ear zero. For imprisomet, however, α will be (o-trivially) positive, ad it will be larger the icer the priso coditios. I assume that through combiatios of corporal puishmet ad imprisomet or by varyig priso coditios that there is a puishmet form that correspods to ay (o-egative) magitude for a (that is, I assume that ay a 0 is i the feasible set for a). Every perso, the, will commit a crime if ad oly if v> π z. 4 Notice that the possibility of wrogful covictio has o direct impact o a idividual's decisio to commit a crime. Eve if iocet people get covicted quite ofte, ay give perso is just oe of a very large umber of iocet people who could be wrogfully covicted, so ay effect of wrogful covictio o the icetive to commit a crime will be extremely small. Moreover, as was poited out by Schrag ad Scotchmer (1994), crime opportuities ca be either exclusive or o-exclusive. If a crime opportuity is exclusive, the if oe does ot commit this crime, o oe will. Thus, ot committig the crime does ot icrease oe's probability of beig wrogfully covicted of a crime. I additio to this ratioal crime, however, I assume there is a fractio, κ, of the populatio that will commit a crime regardless of the costs or beefits (ote, I allow these people to commit a additioal crime if it is ratioal for them to do so, i.e. if v> π z). Oe ca thik of κ as represetig some combiatio of irratioal ad accidetal crimials ad people who are very isesitive to ay sort of puishmet. Of course, sice oe ca set κ =0, the model does ot require that such people exist. I the simulatios below, however, I fid that, without a positive κ, the geerated crime rate is far below the actual Uited States crime rate ad costly puishmet is almost ever optimal. Thus, although a small amout irratioal or accidetal crime or puishmet isesitive people is ecessary to justify costly puishmet, it is also ecessary to geerate realistic crime rates. This is ot surprisig sice without it the 4 I assume that for each perso there is oly oe crime that gives her this value v. 8

10 puishmet magitude could be set at ifiity, deterrig all crime without ay cocer for overpuishmet sice o puishmet would ever be imposed. 5 So log as the beefits from crime very rarely exceed the cost, this would be optimal (this poit is related to that made by Kaplow (1990)). 6 The total crime rate, the, is give by (( 1 κ )(1 H ( π z) + κ ). 7 The harm from crime is give by a costat d. 8 I assume that the magitude of z is chose by a policy maker, who takes α as give, to maximize his objective fuctio. The social plaer chooses the form of the puishmet, which determies α, before the policy maker chooses z. Because I rule out fies, the social plaer is restricted to puishmet forms that geerate o-egative α. (Notice that I take the probability of covictio as give ad igore the social cost of obtaiig it. Sice the social cost of puishmet is liear i magitude, this assumptio has o affect o the results. The policy maker will choose to icrease the probability of covictio util the margial cost of that equals the margial cost of puishmet, which is idepedet of the magitude. Thus, the policy maker ca sequetially optimize over the probability of covictio ad the the 5 Oe ca also thik of k as represetig situatios where society mistakely believes a crime as bee committed ad thus puishes someoe who could ot have bee deterred ex ate (because he did ot commit a crime). Because the simulatios below idicate k makes up the bulk of the crime rate, however, this iterpretatio is ot very realistic. 6 Of course, k>0 also suggests that icapacitatio must be a importat reaso for imprisomet. Noetheless, as I argued i sectio II, it does ot explai why we sped moey o priso ameities or why we icarcerate people past the poit where they are dagerous (Shavell 1987b). 7 Notice that because udeterrable crimials ca commit both a ratioal ad a irratioal crime, the crime rate could exceed 1 if o oe is deterred from ratioal crime. While this poses o problem for the aalysis, oe could boud the crime rate at 1 by assumig that udeterrable crimials caot commit a ratioal crime. This would make the crime rate (( 1 κ )(1 H ( π z) + κ ). 8 The assumptio that the harm from crime is costat is made for coveiece, but for may crimes is probably roughly accurate. Eve where it is ot, oe ca thik of d as the average cost of a particular crime. Give that the crimial ad the courts probably do ot have much if ay additioal kowledge about the particular cost of crime to the victim, this assumptio should ot make ay differece. The assumptio that the beefits of crime vary is based o the idea that people s morality ad opportuity costs vary, ad it is justified by the fact that some people commit crimes while others do ot. 9

11 magitude of puishmet. Sice the key variable of iterest is the expected puishmet (this paper is ot about the tradeoff betwee probability ad magitude of puishmet), there is o loss of geerality by assumig a exogeous covictio rate.) The social plaer chooses α, takig ito accout how this will affect the policy maker s choice of z, to maximize her objective fuctio. Notice that the social plaer acts as a Stackleberg leader i this game. Oe ca thik of the social plaer as the oe who sets up the costitutioal restrictios o what is allowable puishmet (e.g. rulig out cruel ad uusual puishmet, as i the Uited States). The objective fuctios of the social plaer ad the policy maker are as follows: 9 U U sp pm B = vh( v) dv (1 H ( π z) + κ )[ d + π z(1 + α)] π f B = θ vh( v) dv (1 H ( π z) + κ )[ d + π z( θ + α)] π f c c (1) (2) The social plaer weights the welfare of all citizes equally. Her objective fuctio is split ito two mai terms, the beefits less the cost of crime per perso. The first term represets the expected utility gai from crime, where B is the maximum possible gai from crime (this could be ifiite). The itegral rages from π z to B sice oly those whose beefits from crime exceed the expected sactio commit a (ratioal) crime. (I igore ay beefits received by those who commit irratioal crimes, but this has o affect o the results sice this just itroduces a costat ito the welfare fuctio). The ext term is the social cost of crime. As discussed above, ( 1 H ( π z) + κ ) is the crime rate. Each crime creates a social loss of d to the victim ad a loss to the covict, if she is caught, which happes with probability π c, of z. Whe puishmet is costly, a covicted crimial also costs society αz. The policy maker s utility fuctio is idetical to that of the social plaer s except that he oly weights the gais ad losses to the crimials ad covicts by θ<1. Notice that I am assumig that the policy maker applies this reduced weight to covicts eve if they are iocet (though, this assumptio is 9 There is o term i either utility fuctio reflectig ay gai from irratioal crimial activity. Sice I assume irratioal crimials are uaffected by puishmet, this does ot affect the results either here or i the simulatios below. 10

12 ot critical to the aalysis). As discussed i the itroductio, this reduced weight is due to the fact that crimials ad covicts (eve iocet oes) have very little political power. They ted to be poor, disorgaized, ad ot very politically active (felos typically lose the right to vote i the Uited States, though eve with the right to vote, they still would probably have disproportioately little ifluece). Sice the policy maker, ulike the social plaer, cares about beig re-elected, he gives the welfare of covicts ad crimials very little weight. For ay give α, the policy maker chooses z to maximize U pm. The first order coditio for this maximizatio is give below: 10 ( α + θ ) π (1 H ( π z) + κ π zh( π z)) + h( π z) π ( d θπ z) = 0 (3) c The first term represets the policy maker s direct cost from puishig covicted defedats more, the secod is the gai from fallig crime rates. A margial chage i z results i chage i the total amout of puishmet of π ( 1 H ( π z) + κ π zh( π z)). Existig crimes, 1 H ( π z) + κ, are puished a little c more, but the drop i crime results i a total loss i puishmet for those crimes that are o loger committed, π zh( π z). Of course, all of this puishmet oly occurs whe a crime results i a covictio, which happes with probability π c. The policy maker s margial cost from a icrease i puishmet is ( α + θ ) sice extra puishmet produces a social loss of α ad costs the covict uity (sice I measure puishmet i terms of cost to the covict) whose welfare the policy maker weights by θ. The policy maker s margial gai from icreased z due to fallig crime rates is h( π z) π ( d θπ z) because icreasig z reduces crime rates by h( π z) π. Reducig crime by oe uit beefits the victim by a amout d, but it also elimiates the gai that the margial crimial receives from crime, which is π z sice the margial crimial is the oe whose beefit from crime just equals expected puishmet, which the policy maker weights by θ. 10 Whether the secod order coditio is satisfied will deped o the distributio for h. As Kaplow (1990) poits out, corer solutios are ot ureasoable. I explicitly address this issue below whe I cosider various distributios. At this poit, I assume that the secod order coditios are satisfied for expositioal purposes. I do ot make this assumptio for ay of the results below. 11

13 Equatio (3) implicitly defies a fuctio z*(α), the policy maker s optimal choice of z give α. The social plaer, whe maximizig (1), must take ito accout how her choice of α will affect the policy maker s choice of z. Thus, her first order coditio for α is the followig: π z * ( α)(1 H ( π z * ( α)) + κ ) + c z * ( α){ ( α + 1) π c (1 H ( π z * ( α)) + κ π z * ( α) h( π z * ( α)) + (4) h( π z * ( α)) π ( d π z * ( α)} = 0 The first lie of (4) represets the direct effect of icreasig α, icreasig the cost of puishmet, which is always egative. The secod two lies represet the affect of α o the policy maker s choice of z times the margial affect of z o the policy maker s welfare. If θ =1, the term i the curly brackets is idetical to the policy maker s first order coditio ad thus is zero. It is oly because θ <1, the policy maker weights covicts welfare less tha does the social plaer, that the curly bracket term is egative. Whe this iequality is satisfied, the social plaer might wat to make puishmet more costly tha ecessary. By icreasig α, the social plaer iduces the policy maker to choose a smaller z. 11 At the margi, reducig z icreases social welfare due to the policy maker s icetive to over-puish. I geeral, there will be more tha oe α that solves (4). Sice the social plaer is restricted to α 0, however, costly puishmet will icrease social welfare if ad oly if the left-had side of (4) is positive at α=0. This is ot ecessarily the case. I fact, as Propositio 1 establishes, whe the beefits of crime 11 To verify that z * ( α) < 0 totally differetiate (3) with respect to α ad solve for z * ( α) : πc(1 H( πz) + κ πzh( πz)) z* ( α) = π {( α + θ) π [2h( π z) + π zh ( π z)] + π [ h ( π z)( d θπ z) h( π z) θ]} c The deomiator is egative wheever (3) describes the policy maker s choice of f (it is the secod order coditio). While the policy maker s objective fuctio will ot be everywhere cocave, for most distributios that are decreasig ad very small at extreme values it should be cocave at the optimal z (the expected puishmet should be large eough that h is very small ad h is egative). The umerator is positive uless puishmets are so high that eve a policy maker who discouts the welfare of covicts relative to victims gais from risig crime rates, d < θπ z. 12

14 are uiformly distributed, the cost of costly puishmet is always too great to justify the beeficial icetives it gives the policy maker uless the fractio of irratioal crimials is quite large. Ad eve the, costly puishmet is ever actually imposed sice the policy maker chooses ot to puish crime at all. Propositio 1 Whe the beefits of crime are uiformly distributed betwee A ad B, the social plaer π ( B d ) always prefers costless to costly puishmet, i.e. α>0 is ever optimal, uless κ > 2 ad A<0 π ( B A) A + B π [ d ] or κ > 2 ad A 0. Whe costly puishmet is optimal, the policy maker sets z=0, crimes Bπ c are ever puished. Whe crimial beefits are uiformly distributed, icreasig the puishmet at ay level deters the same umber of crimes. The larger the puishmet, however, the fewer covicts there are to puish, thus the lower the cost of imposig that puishmet. Thus, the et beefit to a policy maker of icreasig the puishmet is icreasig i the size of the puishmet util that puishmet is sufficiet to deter all ratioal crime. As a result, the policy maker either sets the puishmet high eough to deter all ratioal crime or does ot puish crime at all, if the social cost of puishmet, (α+θ), is large eough (Kaplow (1990) was the first to poit out that extremes of o-moetary puishmet will ofte be optimal). Kowig this, the social plaer has two optios. She ca set α=0, iducig the policy maker to deter all crime at the lowest possible cost, or she ca make α large eough that the policy maker does ot puish at all (thus, the exact level of α does ot matter). The former is optimal uless κ, the fractio of populatio that commits crimes regardless of the puishmet, is quite large or differece betwee the harm from crime ad the average beefit from crime is very small or the justice system does a very poor job of covictig the guilty party. Also, otice that the level of κ i Propositio 1 ecessary to make costly puishmet optimal icreases as the judicial system gets more accurate, that is as π apporaches π c. The more accurate the c 13

15 judicial system, the less likely costly puishmet will be optimal. I the simulatios below, I show that this effect holds for other distributios of crimial beefits, but is quite small. Propositio 1 demostrates that if crimial beefits are uiformly distributed, Friedma s hypothesis caot justify the actual impositio of costly puishmet. With such a distributio, costly puishmet is a very blut a istrumet. As a result, the social plaer is will ot to use it to ifluece the puishmet magitude decisios of the policy maker (uless it prefers that crime go upuished). A uiform distributio of crimial beefits, however, is almost certaily ot a reasoable descriptio of ay populatio. If the distributio of crimial beefits is decliig above some poit, the the policy maker s objective fuctio will be cocave i z so log as z is large eough. This makes a iterior solutio to the policy maker s choice of z possible. Whe the solutio to z is iterior, small chages i α will iduce small chages i z. That is, costly puishmet is o loger such a blut istrumet. Thus, it is possible that the social plaer might wat to make puishmet more costly tha ecessary to ifluece the policy maker s choice of puishmet. Ufortuately, for more realistic distributios of crimial beefits, it is ot possible to solve the model aalytically. Therefore, I use umerical simulatios to study these cases. IV. THE SIMULATIONS I perform two differet types of simulatios. The first set of simulatios determie for what values of θ the social plaer prefers a positive α to α=0. That is, they aswer the questio: how much does the policy maker have to discout the welfare of covicts to iduce the social plaer to make puishmet costly? I the secod set, usig a small value of θ, the simulatios determie how costly the social plaer wats to make puishmet for differet parameter values. Thus, the goal of the secod set of simulatios is to determie possible optimal values of α. I these simulatios, I fix θ=.01 to illustrate how costly puishmet should be whe differece betwee the policy maker s ad the social plaer s objective fuctios is close to its maximum. Appedix B explais how the simulatios are doe. As much as possible, parameters for the simulatios were chose to accord with existig data o crimial activity ad covictio rates i the Uited States. Where differeces i the parameter values have a meaigful affect o the results, I did simulatios for such a wide rage of possible values that 14

16 (where at least some calibratio is possible) the rage of results i the simulatios provides reasoable bouds for the true values. The Federal Bureau of Ivestigatio (FBI) reports that the clearace rate for violet crime i the Uited States is 49 percet, ad for property crimes it is 17 percet (Federal Bureau of Ivestigatio 1999). The clearace rate is the percetage of offeses cleared by a arrest. Oce charged with a crime, the covictio rate is about 60 percet for violet crimes ad about 76 percet for property crimes (Departmet of Justice 1998c). Thus,.3 for violet crimes (.49(.6)) ad about.13 for property crimes (.17(.76)) are reasoable values for π c. 12 I all the simulatios, I did oe for π c =.3 ad oe with π c =.13. Not surprisigly give the first order coditios (see footote 21) this differece did ot make a differece i the results. Thus, all the results I report i Sectio IV will be for π c =.3, the violet crime case. How π, the et icrease i the probability of beig covicted of a crime whe oe commits a crime, will differ from π c, the overall probability of beig covicted of a crime, depeds o the accuracy of the crimial justice system. Ufortuately, there is little data available o that issue. For the violet crime case, I looked at values of π that varied from.297 (about oe percet of all covicts are iocet) dow to.230 (over 23 percet of all covicts are iocet). The values for κ, the fractio of the populatio that commits a crime o matter how large the expected puishmet is (I call these people irratioal crimials), are also drive by crimial justice statistics. As metioed earlier, a positive κ is ecessary for costly puishmet to be optimal. Otherwise, 12 Of course, these umbers are based o the umber of offeses reported to the police. Sice aywhere from a third to two-thirds of crimes are ot reported to police (Departmet of Justice 1998c, Levitt 1998), these covictio probabilities are too small. That said, chages i the covictio probability, π c, will ot sigificatly affect the results. By examiig the policy maker s first order coditio for z, oe ca see that so log as the relatioship betwee π c ad π is fixed, chages i π c will affect the size of z, but will ot affect the size of the expected puishmet, π z. By ispectig the social plaer s first order coditio, it is apparet that, so log as π z*(α) is uchaged, chages i π c (agai, so log as it is a fixed multiple of π ) will ot chage the optimal α. This ituitio is cofirmed by the simulatio results for the two differet values of π c that were used. 15

17 excessive puishmets will ever be imposed because the policy maker will set z>b, deterrig all crime. So log as the umber crimes with v>d is small eough, the social loss from deterrig efficiet crimes will be dwarfed by the reduced puishmet costs. With a positive κ, however, excessive puishmets will be imposed, icreasig the divergece betwee the social plaer s ad the policy maker s icetives. Thus, by choosig κ so that the model geerates a empirically reasoable crime rate, I ca determie whether it is plausible that κ could be large eough to justify costly puishmet. Accordig to the FBI s Uiform Crime Reports, i the Uited States there are about 4.6 reported crimes per 100 people (Federal Bureau of Ivestigatio 1999). This figure, however, is almost certaily a uderestimate of the true figure, due to ureported crime ad other factors, maybe by as much as a factor of three (Departmet of Justice 1998d; DiIulio 1996; Levitt 1998). Thus, the actual crime rate is probably somewhere betwee 8 ad 14 percet. 13 The model i this paper, however, geerates a crimial rate rather tha a crime rate. To get such a estimate, I use the fact that about five percet of the Uited States populatio will sped time i priso i their lifetime (Departmet of Justice 1998a). Of course, sice people have more tha oe time i their lives to commit crimes, the actual umber of crimials at ay poit i time must be well uder five percet (assumig that almost all crimials sped time i priso at some poit, though certaily ot for every offese). If the crimial rate, the, is about oe percet, this would imply that people have approximately five idepedet draws from the distributio that determies crimial activity i their lifetime (to geerate the five percet imprisomet figure) ad each crimial commits betwee 8 ad 14 differet crimes a year. While I have o data to support the first cojecture, there is some evidece that the secod cojecture is roughly accurate. Prisoer surveys idicate that the media prisoer reports ivolvemet i 12 to 15 crimes per year, though that umber may overestimate the umber of differet crimes committed due to multiple perpetrators (Levitt 1998). Thus, a oe percet crimial rate seems about right. I the simulatios, it turs out that irratioal crimials represet from Citig victimizatio surveys, Levitt (1998a) estimates that oly 38% of crimes are reported, implyig a crime rate of just over 12%. 16

18 to 99 percet of crimials. 14 Thus, to get a crimial rate close to the actual oe, I use κ=.01 i most of the simulatios, though I also do may simulatios with κ=.005 ad.02. While havig such a large fractio of crimials be irratioal may seem implausible, it is equally implausible that it is optimal to set puishmets so low that oe percet of the populatio ratioally decides to commit crimes. Of course, the curret method of puishmet, imprisomet, may be costly eough that deterrig more crimes by icreasig setece legth is ot efficiet. But this just gets us back to the puzzle of why we use a method of puishmet so costly that it is optimal leave so may crimes udeterred? If Friedma s hypothesis is the aswer to this puzzle, without cotradictig the existece of high crime rates, a very large fractio of existig crimials (albeit a very small fractio of the populatio as a whole) must ot be deterred by icreases i the magitude of puishmet. 15 This is ot to suggest that these crimials are udeterrable. Surely, most of these people would ot commit crimes if they faced executio with certaity. Oe (somewhat) plausible explaatio for the existece of these crimials is that there are people who are so overcofidet about their ability to escape apprehesio or covictio 16 that their estimate of the cost of crime is oly trivially affected by icreases i puishmet. 14 The fact that irratioal crimials make up such a large fractio of the total crimial populatio does ot mea that a ecoomic model of puishmet is iappropriate. The overwhelmig majority of the populatio make the decisio about whether to commit a crime or ot based o a ratioal compariso of costs ad beefits, thus puishmet remais a importat deterret. Moreover, the agets decidig o the form ad magitude of puishmet are ratioal actors. 15 Almost ay plausible explaatio for the use of imprisomet, however, will require a explaatio for why it is optimal to have high crime rates rather tha supplemet costly imprisomet with some cheaper puishmets. Eve the argumet that humaitaria behavior has become more affordable as society has grow richer oly explais why we use imprisomet rather tha corporal puishmet ow but we did ot i the past. It does ot explai why we would prefer higher crime rates to some use of corporal puishmet ow that we have grow richer. I fact, for violet crime, oe would expect that as health ad life expectacies improve that prefereces for less violet crime would grow stroger. 16 The pervasiveess of overcofidece is a very well supported psychological pheomeo, see, for example, Debot ad Thaler (1995). 17

19 I the simulatio results, the absolute amout for the d parameter, the damages suffered by the victims of crime, affects the results oly through its relatioship with the distributio of crimial beefits. Therefore, eve if I could estimate the true harm from crime, such a estimate is useless without a correspodig estimate of the distributio of beefits. Thus, I set the harm from crime so that a give fractio of crimes has beefits that exceeded the harm (I call these efficiet crimes, though they are oly efficiet give o puishmet). The three differet values I looked at were harms such that oe i 740 crimes are efficiet (the lowest harm level), harms such that oe i about 32,000 crimes are efficiet (the medium harm level), ad harms such that about oe i 3.5 millio crimes are efficiet (the highest harm level). These umbers correspod to levels of d that are three, four, ad five stadard deviatios away from the mea beefit of crime whe crimial beefits are ormally distributed. Of course, ay compariso of the beefits ad harms from crimial activity are problematic sice it requires iterpersoal utility comparisos. That said, it is reasoable to assume that i some, albeit very small, fractio of crimes the crimial does gai more tha the harm s/he causes. The three levels of efficiet crimes I postulate are meat to capture poits close to the reasoable extreme possibilities ad a poit i the middle. Oe ca also thik of these three poits as represetig differet classes of crimes. The lowest harm level, or the highest fractio of efficiet crimes, might be roughly appropriate for mior offeses. The highest harm level, or the lowest fractio of efficiet crimes, might be appropriate for the most serious crimes. 17 For the distributio of beefits from crime, I did simulatios that used both a ormal ad a logistic distributio (which looks like the ormal but with fatter tails). Give that there is o iformatio o the potetial gai from crime for those people do ot commit crimes, ad available iformatio about 17 While it might be hard to imagie that the gai from murder is ever greater tha the harm, it is ot impossible. Of course I have o data o the matter, but it seems plausible that maybe oe out of every 3.5 millio murders ivolves a case where the murder might save lives o balace. These murders may ot be idetifiable, eve ex post, so there will be o legal rule that excuses them. 18

20 the gais for those who do commit crimes is extremely icomplete, 18 I chose these two distributios because (i additio to beig commoly used statistical distributios) both have the property that the desity of crimial beefits falls as beefits icrease whe beefits exceed the mea (ulike the uiform distributio). Both of these distributios are uiquely defied by their mea ad variace. I ra simulatios with a variety of differet mea ad variace combiatios for both the ormal ad logistic distributios. Whe d is a fixed umber of stadard deviatios away from the mea gai from crime, the results of the simulatios for each distributio deped o mea ad stadard deviatio of crimial beefits oly through their ratio. For most specificatios, simulatios were doe with the mea to stadard deviatio ratio varyig from egative four to positive four. 19 V. RESULTS Whe the beefits of crime are distributed ormally or logistically, a social plaer will prefer to make puishmet costly rather tha costless so log as the policy maker s weightig of the welfare of covicts, θ, is sufficietly small. Exactly how small θ has to be depeds o the value of the other parameters i the model ad whether crimial beefits are distributed accordig to the ormal or logistic distributio. For ay give level of the other parameters, i both the ormal ad the logistic case, the cutoff level of θ, which I will call θˆ, such that for all θ < ˆ θ costly puishmet is optimal, is a decreasig fuctio of the ratio of the mea gai from crime to its stadard deviatio. The reaso for this is that this ratio measures the fractio of people who will ot commit a crime eve if there is o puishmet. The more such people there are, i.e. the more stadard deviatios the mea gai from crime is below zero, the less costly it will be to deter crime, eve with costly puishmet. 18 While there is some data o the cost of crime to victims, this does t tell us how the gai to the crimial differs from the loss to the victim. Eve stole goods are likely to be worth less to the crimial (stole goods do ot sell at retail price) tha to the victim. Crimials also have umeasured effort costs ad some may eve have psychic costs (or, maybe, beefits) from committig crimes. 19 I the ormal case, I sometimes had to limit the (egative) rage of the mea-stadard deviatio ratio to esure that o average crime is iefficiet. 19

21 The followig table summarizes the iformatio the simulatios provide about the effect of various parameters ad choice of crimial beefits distributio o how much the policy maker must discout the welfare of covicts, θˆ, ad o the optimal cost of puishmet, α. Summary of Results Critical Weight o Covict Optimal Cost of Puishmet, α Welfare,θˆ Mea-S.D. Ratio, m/σ Decreasig Decreasig Harm from Crime, d Fractio of Irratioal Crimials, κ Normal rather tha Logistic Distributio Decreasig Greater Effect at Smaller m/σ Icreasig Greater Effect at Smaller m/σ Smaller for Larger m/σ Larger for Smaller m/σ Decreasig Greater Effect at Smaller m/σ Icreasig Greater Effect at Smaller m/σ Smaller for Larger m/σ Larger for Smaller m/σ For ay level of this mea-stadard deviatio ratio, θˆ is larger the less harmful is crime (or, more precisely, the greater the fractio of efficiet crimes). This is ot surprisig. The less harmful is crime the less welfare is lost from reduced deterrece. Sice the reaso the social plaer might use costly puishmet is to reduce the magitude of the puishmet chose by the policy maker, costly puishmet ecessarily reduces deterrece, ad icreases crime. The less harmful this is, the smaller is this egative effect o social welfare. Before presetig the figures depictig the results for specific simulatios, I will make a few geeral commets about all the figures that I will preset i this sectio. First, ote that all parameters will take stadard values i all the figures uless otherwise specified. The stadard values are π c =. 3, π =.297, κ =.01, d = m + 5σ (ormal case) or d = m σ (logistic case), where m is the mea gai from crime ad σ is the stadard deviatio. The harm from crime i the logistic case is far higher tha i the ormal case. This is to hold costat the fractio of efficiet crimes (crimes where the 20

22 crimial beefits more tha the victim suffers). Thus, the three values of harm that I use for the logistic case are d = m σ, d = m σ, ad d = m σ. These correspod to the same umber of efficiet crimes, oe i 3.5 millio, 32,000, ad 740 respectively, as do d = m + 5 σ, d = m + 4σ, ad d = m + 3σ i the ormal case. As metioed above, i the ormal case, the curves will ot exted as far to the left, ito the egative m/σ rage, to esure that d is positive. This may also occur i other istaces because the simulatios were uable to coverge o a solutio, usually this happeed i the ormal case whe m/σ was very small. Figure 1 presets the simulatio results whe the crimial beefits are ormally distributed for the three differet levels of damages from crime. {Figure 1 Here} Whe the mea value of crime is positive ad the stadard deviatio low, the policy maker must give the welfare of covicts less tha 10 percet of the weight he gives the welfare of victims for costly puishmet to be optimal. Whe the mea gai from crime is egative, however, costly puishmet ca be optimal for much larger θ, eve greater tha 15 percet, provided the stadard deviatio is quite small. Give the relative political power of covicts, it is hard to imagie that θ is aywhere ear that large. Thus, particularly if the mea gai from crime is egative, 20 these simulatios provide support for Friedma s justificatio for costly puishmet. Chagig the value of other parameters does ot chage the qualitative ature of the results. As discussed above, usig a value for the covictio probability, π c, that is more appropriate for property crime (or takes ito accout ureported crime better) has oly a trivial affect o the results. Reducig the accuracy of the justice system, chagig π from.297 to.230, icreasesθˆ, but, agai, the affect is extremely small. 20 The mea gai from crime will be egative whe there is a commo morality that is widely eough held that most people would t commit a crime eve if there is o possibility of puishmet, or, at least, govermet imposed puishmet. While for some relatively mior offeses, e.g. speedig, this is seems ulikely to be the case, for may other offeses (particularly more serious oes) it is ot implausible. The mea gai from crime will also be egative for may crimes that require effort give that for may citizes that effort will have a high shadow price due to the value of their lawful opportuities (I thak Joel Schrag for poitig this out). 21