How To Solve A Law Of The Land Problem

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1 Enforcement n Prvate vs. Publc Externaltes Hakan Inal Department of Economcs, School of Busness and L. Douglas Wlder School of Government and Publc Affars Vrgna Commonwealth Unversty hnal@vcu.edu November 13, 2009 Draft Abstract Law and enforcement polcy s among the key elements of a cvl socety that ensures the achevement of a hgher socal welfare. In ths paper, I study socally optmal law and enforcement polcy makng under two dfferent envronments. In the frst envronment, prvate externaltes, an actvty a person engages harms equally lkely everyone n the socety. In the second envronment, publc externaltes, t harms the whole socety. I show that socal welfare functon of these two problems are the same under certan condtons. Polnsky and Shavell [4] show that the optmal level of punshment n equlbrum s such that expected level of punshment s less than the harm t causes. I generalze ther result to publc and prvate externalty envronments where all agents are ether rsk neutral or rsk averse wth respect to uncertantes n harms they face. On the other hand, by allowng prvate and publc externalty acts n the same envronment, I show that even though contrbuton of agents to the publc harm s greater than harm they may cause by choosng prvate externaltes, the punshment level of a prvate externalty may be greater than the punshment level of publc externalty f agents are suffcently rsk averse, whch s dfferent from a result n Shavell [5]. Ths result shows that the dstncton between prvate and publc externaltes s mportant. 1 Introducton In cvl socetes, there are some rules for the protecton of both ndvduals and socety s nterests aganst ndvduals or groups actons harmng those 1

2 nterests. A partcular way of mplementng these rules s through deterrence by punshment. These rules, as well as the ways and levels of punshment for rule volaton are usually formed n varous ways lke copyng and adaptng them from another socety or comng up wth new ways and forms through common sense n a partcular socety. In economcs, begnnng wth Becker [1], several papers have been wrtten suggestng how to choose the level of punshment and the level of detecton so that t would be optmal from the law maker s pont of vew. Becker [1] says that the am can be mnmzng socal cost, whch conssts of harms caused by agents, costs of detecton, and the costs of punshment to the crmnals less gans of crmnals. Indvduals and ther choces are mplctly explaned, and they get utlty only from consumpton, and face uncertanty only because of the possblty of gettng caught. He shows that the probablty of detectng the crme should be set to mnmum possble and the level of punshment should be as hgh as possble, for example total wealth of an ndvdual. Hs result s based on the rsk neutralty of agents. In Polnsky and Shavell [3], they study rsk averse agents as well as rsk neutral agents. In ther model, an agent derves utlty from hs consumpton. If he does not engage n the externalty creatng actvty then hs consumpton conssts of hs wealth remanng from taxes and nsurance premum whch covers all losses due to externaltes caused by ndvduals actons n the socety. If he engages n the actvty then hs consumpton, n addton to taxes and nsurance premum pad, wll ncrease wth hs gan from the actvty and wll decrease wth fne pad unless he does not get caught. Externaltes created s felt equally lkely by everyone n the socety but f agents are rsk neutral then premum pad can also be seen as the rsk they bear due to possble harms caused by others n the socety. Snce Polnsky and Shavell [3] use contnuum (measure one) of people, t s possble to see ther model as the one n whch the whole socety faces externaltes wthout any uncertanty, and the level of harm they face s gven by the harm a sngle actvty causes, and the rato of people engagng n the actvty. The objectve of the law maker n Polnsky and Shavell [3] s to maxmze the total expected utlty of ndvduals n the socety 1. They show that, f ndvduals are rsk neutral t s optmal to set probablty of detecton to a mnmum level below whch t s not possble to detect any crme, and optmal to set the punshment level as hgh as possble whch s constraned by wealth of ndvduals. They have 1 Cooter and Ulen [2] (p. 443) suggest that the am of the law maker should be mnmzaton of the socal cost whch conssts of costs of protecton and the net harm caused whle crme s commtted. 2

3 two results when agents are rsk averse. They say that these results explan why the prevous result s not realstc. The frst one s that, probablty of detectng should be set to 1 when ndvduals are rsk averse, and cost of catchng s suffcently small. The second one s that, t may not be optmal to set low probablty to detectng, and settng the punshment levels very hgh even f cost of catchng s very large when ndvduals are rsk averse. In Polnsky and Shavell [4], the objectve of the law maker s to maxmze total expected utlty of ndvduals n the socety. The expected utlty of an agent conssts of gan from engagng n the actvty, expected loss due to the possblty of gettng caught, expected loss due to the possblty of beng a vctm of a crme, and the per capta cost of enforcement. If the agent does not engage n the actvty, then the expected utlty wll not nclude the gan from engagng n the actvty, and the expected punshment. They show that n equlbrum, expected punshment for actvty wll always be less than the harm caused by t. In ths paper I defne publc and prvate externalty envronments formally, and fnd what utlty functons of agents n these envronments become under certan assumptons (these assumptons greatly smplfes utlty functons, and make t possble to analyze models, and derve results). I also show that these two envronments are equvalent from socal planner s pont of vew. 3

4 2 The Sngle-Act Model A sgnfcant reason for socetes to have laws and enforcement s that people n the socety fear from possble negatve externaltes that may be caused by others n the socety. Laws exst, of course together wth the enforcement, to acheve a socally optmal state of the world by decreasng the possblty of ndvduals beng hurt,.e. by deterrng agents from takng certan actons. An agent s utlty functon s u (c, a, h ) where c R S + s the consumpton plan of the ndvdual. A state s S s determned by the set of all offendersufferer pars and the set of all offenders captured. a A = {a 0, a} 2 s the acton taken by agent. Agent can ether choose acton a that gves pleasure to hm/her and causes negatve externalty to others (to ether one person or to all socety, dependng on the type of the acton), or choose acton a 0 that does not change anybody s utlty. The externalty faces s denoted by a random varable h. Each realzaton of h determnes the group of people harmng (takng acton a aganst) agent. For any agent, the set of people takng acton a he/she faces,.e. the set of people of who are lkely to hurt agent, s denoted by N a and s defned as N a = N a \{} where N a s the set of people who are takng acton a n the socety. If the acton s publc externalty then h s the same for all agents as everyone suffers from the same externalty. There s a sngle consumpton good, and t s assumed that utlty s an ncreasng functon of consumpton. Consumpton of an agent consst of hs ntal endowment, and transfers made to hm less per capta cost of enforcement f he dd not take any llegal acton. Otherwse, f the agent takes an llegal acton and he gets caught then hs consumpton s the amount of endowment, and transfers made to hm remanng from the punshment, and per capta cost of enforcement. If the agent takes an llegal acton and he does not get caught then he consumes hs endowment and transfers made to hm. 2.1 Prvate Externaltes Prvate externalty s a negatve externalty that affects any and only one person. Property crme s a good example for that knd of externalty. DWI s another good example. In both of these examples, at any pont n tme, although offender threatens every ndvdual n the socety, t wll possbly 2 A stands for the set of all possble actons. The set of llegal actons wll be chosen by the law maker, and t s a subset of A. Note that I asssume that an ndvdual can choose only one acton at a tme, as t has always been done n the lterature, so multple actons are not allowed. An acton s llegal f and only f ts punshment s postve. 4

5 hurt only one ndvdual. If actons agents take cause prvate externaltes, then agent j wll offend agent by takng acton a wth gven probablty 3 π j. Note that N π j = 1 for any j. So the probablty that agent wll be offended by a group of R Na people s β(r, Na) = π j l Na \R(1 π l ). If Na s the set of offenders agent faces, and r of them offend agent, then α(p, r, k) = p k (1 p) r k( r k) s the probablty that any k of these r people wll be captured, and punshed. Revenue from punshments of these k agents then wll be transferred to agent. So c (k,0) stands for the consumpton level of agent when he/she gets caught for takng the acton a. If the agent takes the acton a 0 or takes the acton a but he/she does not get caught then hs/her consumpton s c (k,1). Note that k stands for the transfers from those k people who are caught after hurtng agent. Hs budget constrant s c (k,0) + f w c(p) + kf f he takes the acton a, and gets caught, and t s c (k,1) w c(p) + kf f he takes acton a 0, or he takes acton a and does not get caught. In both cases agent gets compensated by k crmnals punshments, kf as these are the only crmnals caught among those who hurt agent. Each agent choosng acton a wll cause harm e. So, gven Na, the set of offenders, other than agent hmself, R e s the total externalty faces when people n R Na offend hm. The subset of offenders R agent faces, the set of agents among R caught, and whether agent caught f he took acton a determne the state agent faces. The expected utlty functon of agent wll then be Eu (c, a, h ) = p R N a + (1 p) R β(r, Na) α(p, R, k)u (c (k,0), a, e R ) R N a k=0 R β(r, Na) α(p, R, k)u (c (k,1), a, e R ) Note that f agent takes acton a 0 then the expected utlty functon above wll become Eu (c, a, h ) = as c (k,0) R N a = c (k,1) for all k. R k=0 k=0 (1) β(r, Na) α(p, R, k)u (c (k,1), a, e R ) (2) 3 The dstrbuton of crme s to be determned endogeneously accordng to ratos of ndvduals who choose actons. Note that the uncertanty presented here s endogeneous. It may be an nterestng problem and may be worth studyng decson makng under ths type of uncertanty. 5

6 2.2 Publc Externaltes Publc externalty s a negatve externalty that affects everybody n the socety. Ar polluton s a good example for publc externalty. If actons agents take cause publc externaltes then each one of these agents can cause a harm level of e P, whch wll affect everybody. So the harm level agent faces s h = e P N a, e.g. total level of harm. Let α(p, Na, r) = p r (1 p) N a r( Na ) r denote the probablty that any r of the Na people commtng crme wll be caught, and punshed. Revenue from punshment of an offender s transferred to every other agent n the socety equally. Agent s budget constrant s c (r,0) + f w c(p) + rf N 1 f he takes the acton a, and gets caught, and t s c (r,1) w c(p) + rf N 1 f he takes acton a 0, or he takes acton a, and does not get caught. In both cases agent gets compensated by r crmnals punshments, rf N 1 as these are the only crmnals caught who hurt agent together wth the rest of the socety. Agent takes N a, the number of people commtng crme,.e. takng acton a, other than agent hmself, as gven. w s the wealth of the agent, p s the objectve probablty 4 of gettng caught f an llegal acton s taken, c(p) s the per capta cost of detecton wth probablty p, and f s the punshment 8 for acton a. The punshment f R + s postve f the acton a s consdered as llegal. 4 In a more general model, the probablty of detecton may depend on the acton taken, and t can be denoted by a functon p : A [0, 1]. 5 c(p) s assumed to be an ncreasng functon of p. 6 As the number of people s fxed througout the model, the cost of mantanng probablty of detecton p s also fxed. It may be possble to thnk that the cost of mantanng probablty p depends on the number of people both n the socety and those commtng acts. 7 Fnes crmnals pay are transferred to vctms. Ths s assumed as t makes analyss smpler. If fnes are consdered as a part of the cost of enforcement then the taxes wll be nondetermnstc when there are fnte number of people n the socety. If agents have addtvely separable utlty functons and they are rsk neutral n consumpton then ths won t affect the optmal polcy as fnes pad by offender wll be exactly offset by the gan of the ndvduals recevng the fne revenue. 8 The punshment can very well depend on the ncome of the agent. Then the constrant n law maker s problem wll become 0 f(w ) w where w s agent s ncome level. Whether the punshment should depend the level of ncome can be analyzed from the welfare and ncentve compatblty pont of vews. 6

7 The expected utlty functon of an agent s 9 10 Eu (c, a, h ) = E[u (c, a, e P N a )] = p N a r=0 + (1 p) α(p, N a, r)u (c (r,0), a, e P N a ) N a r=0 α(p, N a, r)u (c (r,1), a, e P N a ) (3) 2.3 Agent s Problem Gven the punshment level f, the probablty of detecton p, and the externalty level h agent faces, acton a and the consumpton plan c solve agent s problem whch s max Eu (c, a, h ) (4) a A,c R S + subject to the budget constrants for each state he faces. 2.4 Equlbrum Gven the punshment level and probablty of detecton, equlbrum s a sequence {c, a } N of allocatons and actons such that for each agent, (c, a ) s a soluton to hs/her problem defned above. The equlbrum utlty level of an agent s defned as v (p, f) = Eu (c, a, h ) where c,and a are solutons to agent s problem gven n Law Maker s Problem I assume that the am of the law maker s to choose the punshment level f for acton a, and a probablty of detecton p so that total expected utlty of ndvduals n the socety n equlbrum wll be maxmzed 11. The level 9 If the utlty functon s lnear and addtvely separable n ts arguments then the consumpton can be dscarded and the utlty functon can be redefned as a functon of only actons taken, the punshment level for that acton, and externalty of other people. 10 A more general model would be to consder not total harm caused by agents but actons of other agents separately. 11 It may be easer (or not?) to use polcy maker s problem n whch the am s to mnmze the cost of enforcement subject to a certan socal welfare 7

8 of punshment 12 f for acton a s constraned by the net wealth of agents, and the probablty of detecton p s n [0, 1]. max f,p N Eu (c, a, h ) =1 subject to 0 f w for all and 0 p 1 where w s the net wealth of ndvdual 13, and a s the best response of ndvdual to punshment level f, and probablty of detecton p, and the externalty he faces, h. It wll be assumed that f s gven through the paper. 12 Other types of punshments are possble too. The constrant on the wages can be relaxed by mplementng dfferent levels of punshment on avegare lfe expectancy, e.g. dfferent levels of restrctons on freedom. The am here would be not only punsh offenders but also to protect socety f the offender s seen as a threat to the socety. The tradeoff wll be solved through socal welfare maxmzaton. Nonmonetary based punshments may be used for farness purposes. For example, only gvng tcket to hgh speedng may create an ncentve for wealthy people to hgh speed. It s worth studyng ths problem. 13 Here t wll be equal to w c(p). 8

9 3 Separable Preferences In ths problem, gven the probablty of detecton and punshment level, each agent faces a choce among dfferent alternatves that causes externatltes to someone n the socety or to the whole socety. The am here s to determne the set of actons whch are to be treated as llegal and a socally optmal detecton and punshment levels for ths set of actons. An agent may choose not to choose any acton. For smplcty, and for now I assume that agents can choose only one acton 14 (other than not dong anythng), and ther utlty functons are addtvely separable and they are rsk neutral n consumpton. In the next secton, I wll show the dervaton of objectve functons used n Polnsky and Shavell [4] model. The man dfferences are that I consder fnte number of people model and that make dstngush between Prvate and Publc externaltes. 3.1 Prvate Externaltes Theorem 1. Assume that an agent has a utlty functon gven n 1, and hs utlty functon s addtvely separable and he s rsk neutral wth respect to uncertantes n consumpton. Then hs expected utlty functon becomes: Eu (c, a, h ) = v (a ) pf(a ) c(p) + pf π j E[h ] (5) j N a where v (a ) s the utlty gets by takng acton a, and v (a 0) = 0. Lemma 1. For any fnte set of probabltes {π j } j J, R J ( π j l J\R (1 π l )) = When there are more than one acton, and t can be assumed that the orderng of harms caused by these actons s the same among agents. In ths case, agents are generally deterred from more harmful actons to less harmful actons. These actons may also stand for dfferent levels of a certan acton, e.g. a factory that causes ar polluton. Ths s called margnal deterrence (See Stgler [6]). The am of the law maker would be to draw a lne between llegal and legal actons, and determnng levels of punshments and the level of detecton (f t s optmal to use the same enforcement for them) so that socal welfare wll be maxmzed. 9

10 Proof. For any k, k {1,.., J }, (1 π l )) =π jk So, R J R J ( ( π j π j l J\R l J\R (1 π l )) = ( π j R J\{j k } l (J\{j k })\R + (1 π jk ) = (1 π l )) ( π j R J\{j k } l (J\{j k })\R ( π j R J\{j k } l (J\{j k })\R (1 π l )). (1 π l )) ( π j R J\{j 1,j 2,..,j J 1 } l (J\{j 1,j 2,..,j J 1 })\R =π J + 1 π J = 1. (1 π l )) Lemma 2. For any fnte set of probabltes {π j } j J, (1 π l )) R = π j. j J Proof. For J = 1, R J Assume that for J = k 1, For J = k, R J ( π j l J\R ( R J R J π j l J\R ( π j ( π j (1 π l )) R =π j J l J\R l J\R (1 π l )) R = π 1. J (1 π l )) R = π j. j=1 ( π j \{j {j J } R J J } l J\R + (1 π j J ) =π j J + (1 π l )) R ( π j R J\{j J } l (J\{j J })\R ( π j R J\{j J } l (J\{j J })\R ( π j R J\{j J } l (J\{j J })\R J 1 =π j J + J = π j. j=1 j=1 π j (1 π l )) (1 π l )) R (1 π l )) R 10

11 ( n 1 r Proof of Theorem 1. Frst note that (x + y) n = n ( n r=0 r) x n r y r for any nteger n, and n ( n r=0 r) (1 p) n r p r r = pn n 1 ) r=0 (1 p) n 1 r p r. By substtuton of budget constrant, the expected utlty functon gven n 1, by addtve separablty, and rsk neutralty, becomes p R N a R β(r, Na) α(p, R, k)(w c(p) + kf f(a ) + v(a )) + (1 p) R N a R N a k=0 R β(r, Na) α(p, R, k)((w c(p) + kf + v(a )) E[h ] k=0 k=0 = R β(r, Na) α(p, R, k)(w c(p) + kf + v(a )) pf(a R ) β(r, Na) α(p, R, k) E[h ] =(w c(p) + v(a ) pf(a )) + R N a =(w c(p) + v(a ) pf(a )) + f R N a =(w c(p) + v(a ) pf(a )) + f R N a R N a R β(r, Na) α(p, R, k)(kf) E[h ] k=0 R β(r, Na) α(p, R, k)k E[h ] k=0 =(w c(p) + v(a ) pf(a )) + pf R 1 β(r, Na)p R α(p, R 1, k) E[h ] R N a k=0 β(r, N a) R E[h ] =(w c(p) + v(a ) pf(a )) + pf π j E[h ] j N a =(w c(p) + v(a ) pf(a )) + pf π j E[h ] k=0 v(a ) pf(a ) c(p) + pf j N a π j E[h ] j N a Rsk Neutral Agents Corollary 1. Assume that an agent has a utlty functon gven n 1, and hs utlty functon s addtvely separable and he s rsk neutral wth respect to uncertantes n consumpton and n harm he faces, the level of prvate externalty an agent can cause over another agent s e, and s the same for all agents. Then hs expected utlty functon becomes: Eu (c, a, h ) = v (a ) pf(a ) c(p) e π j + pf π j (6) j N a j N a 11

12 where v (a ) s the utlty gets by takng acton a, and v (a 0 ) = 0. Note that an agent, say, wll take the acton whch causes an externalty f and only f v (a) > pf(a) Rsk Averse Agents In prevous secton, I assumed that agents are rsk neutral both n consumpton and harm they face. Now, I wll study the case where agents are rsk averse n harms they face. Corollary 2. Assume that an agent has a utlty functon gven n 1, and hs utlty functon s addtvely separable and he s rsk neutral wth respecto to uncertantes n consumpton, rsk averse wth respect to uncertantes n harm he faces, and the level of prvate externalty an agent can cause over another agent s e, and s the same for all agents. Then hs expected utlty functon becomes: Eu (c, a, h ) = v (a ) pf(a ) c(p) + pf π j β(r, Na)h ( R e) (7) j N a R N a where v (a ) s the utlty gets by takng acton a, and v (a 0) = Publc Externaltes The model here s the same as the prevous one except that an acton taken by an ndvdual wll affect everyone n the socety. Theorem 1. Assume that an agent has a utlty functon gven n 3, and hs utlty functon s addtvely separable and he s rsk neutral n consumpton, the level of publc externalty an agent can cause s e P, and s the same for all agents. Then hs expected utlty functon becomes: Eu (c, a, h ) = v (a ) pf(a ) c(p) + N a N 1 pf h ( N a e P ) (8) where v (a ) s the utlty gets by takng acton a, v (a 0 ) = 0, and n equlbrum { Na {} f v > pf + (h (( N N a = a + 1)e P ) h ( Na e P )), otherwse. N a 12

13 Proof. Frst note that (x + y) n = n ( n ) r=0 r x n r y r for any nteger n, and n ( n r=0 r) (1 p) n r p r r = pn n 1 ( n 1 ) r=0 r (1 p) n 1 r p r. By substtuton of budget constrant, the expected utlty functon gven n 3, by addtve separablty, and rsk neutralty, becomes N a p r=0 α(p, N a, r)(w c(p) + N a + (1 p) N a = r=0 r=0 α(p, N a, r)(w c(p) + α(p, N a, r)(w c(p) + rf f(a) + v(a)) N 1 rf N 1 r=0 rf + v(a)) E[h] N 1 N a + v(a)) pf(a) N a =(w c(p) + v(a ) pf(a )) + α(p, Na, rf r)( N 1 ) E[h] =(w c(p) + v(a ) pf(a )) + r=0 Na 1 N 1 f α(p, Na, r)r E[h ] r=0 α(p, N a, r) E[h ] =(w c(p) + v(a ) pf(a )) + N a Na 1 N 1 pf α(p, Na 1, r) E[h ] r=0 =(w c(p) + v(a ) pf(a )) + N a pf E[h] N 1 v(a ) pf(a ) c(p) + pf N a N 1 E[h] Rsk Neutral Agents The model here s the same as the prevous one except that an acton taken by an ndvdual wll affect everyone n the socety. Theorem 3. Assume that an agent has a utlty functon gven n 3, and hs utlty functon s addtvely separable and he s rsk neutral wth respect to uncertantes n consumpton and n harm he faces, the level of publc externalty an agent can cause s e P, and s the same for all agents. Then hs expected utlty functon becomes: Eu (c, a, h ) = v (a ) pf(a ) c(p) N a e P + N a N 1 pf (9) where v (a ) s the utlty gets by takng acton a, v (a 0 ) = 0, and n equlbrum { Na {} f v > pf + e P, N a = otherwse. N a 13

14 Note that an agent, say, wll take the acton whch causes an externalty f and only f v (a) > pf(a) + e P. 3.3 Multplcty of Equlbra When agents are rsk averse wth respect to uncertantes n harms they face and rsk neutral wth respect to uncertantes n consumpton, multple equlbra may arse. Consder the followng example. Example: There are two agents wth dstnct valuatons and same convex harm functons. Ther valuatons are suffcently close,.e. v 2 v 1 < h(2) 2h(1). If v 2 (h(2e P ) h(e P )) pf < v 1 h(e P ) then n one equlbrum agent 1 s deterred whle agent 2 s an offender whereas n the other equlbrum agent 2 s deterred whle agent 1 s an offender. Ths happens because expected punshment s large enough to deter only one person and not large enough to deter everybody.convexty of harm functons guarantees that as expected level of punshment ncreases, number of offenders do not ncrease. Although an effcent enforcement polcy s unlkely to result n an neffcent equlbra, exstence of multple equlbra under certan polces s lkely to arse as long as valuatons of agents are suffcently close. It s possble though to have multple equlbra under an effcent polcy f there are agents wth dentcal valuatons. 4 Monotoncty of Harm Probabltes Ths secton explores how probablty of beng offended s affected by changes n the number of offenders n the socety and by changes n the number of offenders an agent faces. If the number of offenders n a socety ncreases then addton of a new offender wll affect all other agents. The number of states an agent faces wll ncrease because of the addton of ths new offender. On the other hand, probablty of each state (beng offended by a certan number of agents) wll decrease as for a gven number of offenders an agent faces, number of offenders who are not offendng that agent s ncreasng. It s shown below that, n general, effect of more number states an agent faces wll outweght the effect of lower probablty of facng each state, and ths wll cause an agent to face a hgher probablty of beng offended by a certan number of agents. Ths obvous result has an excepton. If there are hgh number of offenders n the socety, more precsely N 2, then an ncrease n the number offenders wll not affect any agent as those two effects wll cancel 14

15 each other and an agent wll face the same probablty of beng offended by a certan number of offenders. Theorem 4. Assume that n a prvate externaltes envronment number of offenders s N a, 0 N a N 2, N Let π(r, N a ) = ( 1 N 1 )r (1 1 N 1 )Na r( N a ) r denote the probablty that an agent wll be harmed by r of N a offenders. Then 1. π(0, N a ) > π(0, N a + 1). 2. For N a < N 2, π(r, N a ) < π(r, N a + 1) for r For N a = N 2, π(r, N a ) < π(r, N a + 1) for r > 1, and π(1, N a ) = π(1, N a + 1) Proof. If N = 2 then N a = 0 by assumpton. So π(0, N a) = 1 > π(0, N a + 1) = 0. Now assume that N > 2. Case 1. r = 0 For 0 r N a, π(r,na+1) π(0,n a+1) π(0,n a) = (1 1 N 1 ) < 1. Case 2.1 N a < N 2 N a < N 2 0 < Na+1 N 1 π(r,n a) = ( 1 N 1 )r (1 N 1 1 )Na+1 r ( Na+1 r ) ( 1 N 1 )r (1 N 1 1 )Na r ( Na r ) = (1 1 Na+1 )( N 1 N a+1 r ). So < 1. Ths mples that for r 1 r > Na+1 (N 1)r > N 1 N a + 1 (N 2)(N a + 1) > (N 1)(N a + 1) (N 1)r ( N 2 )( N a+1 ) > 1 N 1 N+a+1 r π(r, N a) < π(r, N a + 1). Case 2.2 N a = N 2 π(1,n a+1) π(1,n a) = 1, and for r > 1, π(r,na+1) π(r,n a) = (1 1 Na+1 )( ) = ( N 2 N 1 N 1 N a+1 r )( ) > 1. N 1 N 1 r The followng theorem shows that, n general, the probablty of beng offended by a certan number of agents s hgher than the probablty of beng offended by a bgger number of agents. There cases when ths s not true. Probabltes of beng not offended and beng offended by only one agent are equal when total number of offenders s very hgh, N 2 to be more precse. Moreover probablty of beng not offended s less than the probablty of beng offended by only one agent when total number of offenders s N 1. Theorem 5. Assume that n a prvate externaltes envronment number of offenders s N a, 0 N a N 1, and N Let π(r, N a ) = ( 1 N 1 )r (1 1 N 1 )Na r( N a ) r denote the probablty that an agent wll be harmed by r, r < N a, of N a offenders. Then 15 N a N 2 as the maxmum number of people that can hurt an agent s N N a N 1 as the maxmum number of people that can hurt an agent s N 1. 15

16 1. For N a < N 2, π(r + 1, N a ) < π(r, N a ) for r For N a = N 2, π(r + 1, N a ) < π(r, N a ) for r 1, and π(0, N a ) = π(1, N a ). 3. For N a = N 1, π(r + 1, N a ) < π(r, N a ) for r 1, and π(0, N a ) < π(1, N a ). Proof. If N = 2 then N a = 1 s the only case to consder, and π(0, 1) = 0 < π(1, 1) = 1. Now assume that N > 2. π(r+1,n a) π(r,n a) = ( 1 N 1 )( r+1)(1 N 1 1 )Na r 1 ( r+1) Na ( 1 N 1 )r (1 N 1 1 )Na r ( Na r ) = ( 1 N 1 ) (1 1 ( Na r N 1 ) r+1 ). So π(r + 1, Na) π(r, N a) (N a r) (N 2)(r + 1) Na (N 2) N 1 r. Ths mples that f N a < N 2 then π(r + 1, N a) < π(r, N a) for r 0. If N a = N 2 then π(r + 1, N a) < π(r, N a) for r 1, and π(0, N a) = π(1, N a). Fnally, f N a = N 1 then π(r + 1, N a) < π(r, N a) for r 1, and π(0, N a) < π(1, N a). 5 Socal Welfare In ths secton, I wll look at the socal welfare of the socety, and socally optmal level of enforcement. Wthout loss of generalty assume that f < j then v v j. For notatonal smplcty, I wll use f for f(a), a a 0, and v for v (a ), a a 0, and assume that f v N, where v N s the hghest level of valuaton of the acton n the socety. It s also assumed that an ndvdual wll not take any acton f he s ndfferent between takng and not takng t. Otherwse, socal planner s problem may not have a soluton. N c (p) s the total cost of enforcement, and the level of enforcement p s n [0, 1]. Indvduals wealth are dentcal,.e. w = w for all {1,.., N}. The punshment level f s gven, and t does not bnd budget constrant of ndvduals. Optmal value of probablty of detecton s denoted by p. Socal welfare of the socety s a functon of the level of enforcement, W (p) = N α Eu, N α = 1, α > 0 for all N. Eu s the equlbrum utlty level of agent. The followng two theorems are generalzatons of the man theorem n Polnsky and Shavell [4] where they show that the optmal level of punshment n equlbrum s such that expected level of punshment s less than the harm t causes. Theorem 6. Assume that each agent has an equal weght n a prvate externaltes envronment socal planner s problem, and that each agent wll hurt another agent wth equal probablty. Then the optmal level of enforcement p s such that N p f p f < N ( R N a N p f \{} β(r, N a N p f \ 16

17 {})h ( R e) R N a β(r, N a)h ( R e)) where N pf s the set of all people whose valuatons are equal to pf. Proof. The socal welfare functon W R (p) = α R Eu for a prvate externaltes problem for vj R pf < vj+1 R (f such j exsts) s W R (p) = α R ( c (p) E[h ] + pf π k ) N\N a k Na + α R (v R pf c (p) E[h ] + pf π k ) N a k Na = α R v R α R c (p) E[h ] pf α R + pf α R π k. N a N N a N Snce α = 1, and π N jk = 1 for all, j, k N, N 1 α R + α R N a N k N a π k = 0. k N a Then socal welfare becomes W R (p) = N a v R c (p) E[h ]. N N If 0 pf < v1 R then the socal welfare functon s W R (p) = N αr v R N αr c (p) E[h ], and t s W R (p) = N αr c (p) for vn R pf. So the socal welfare functon for any p [0, 1] s W R (p) = c (p) E[h ]. N N a v R N Now assume on the contrary that p f e. If p f v j for any j N then a slght decrease n p wll decrease the second part of the expresson wthout changng the frst one and ths wll lead to a ncrease n the socal welfare, a contradcton to optmalty of p. On the other hand, f p f = v j for some j N then a slght decrease n p wll ncrease the number of offenders, N a, and ncrease the frst sum, besdes the second sum, n the equaton above as p f e, causng the welfare to ncrease, a contradcton. Theorem 7. Assume that each agent has equal weght n a publc externaltes envronment socal planner s problem. Then the optmal level of enforcement N p f p f < N (h ( N a N p f e P ) h ( N a e P )). Proof. The socal welfare functon W P (p) = α P Eu for vj P pf + e P < vj+1 P (f such j exsts) s for a publc externaltes problem W P (p) = α P (v P pf) α P c (p) α P h ( N a e P ) + α P N a pf N a N N N = α P v P α P c (p) + pf Na pf α P α P h ( N a e P ) N a N N a N 17

18 As α = 1 for all N the socal welfare functon becomes N = c (p) h ( N a e P ). N N a v P N If 0 pf + e P < v1 P then the socal welfare functon s W P (p) = N =1 αp v P N αc(p) N h( Na ep ), and t s W P (p) = N αp c (p) for vn P pf + e P. So the socal welfare functon s W P (p) = c (p) h ( N a e P ). N N a v P N Now assume on the contrary that p f e. If p f v j for any j N then a slght decrease n p wll decrease the second part of the expresson wthout changng the frst one and ths wll lead to a ncrease n the socal welfare, a contradcton to optmalty of p. On the other hand, f p f = v j for some j N then a slght decrease n p wll ncrease the number of offenders, N a, and ncrease the frst sum, besdes the second sum, n the equaton above as p f e, causng the welfare to ncrease, a contradcton. V stands for a partton of N, and defned as for any V k V, for any, j N, V k and j V k f and only f v = v j, and for any k > l, V k and j V l f and only f v > v j. The set of all sets of possble offenders for varous deterrence levels s V = { j V 0 V } { }. So, for each K V, K = { N v > v for some v}, and the set of offenders s N a = { N v > pf} V where p s the enforcement level. Theorem 8. Socal planner s problems n a prvate externaltes problem wth externalty level e, and n a publc externaltes problem wth the externalty level e P are the same whenever the followng condtons are satsfed: 1. For all N, α R = α P, 2. For all K V, N αr k K\{} π k = K, 3. For all N, v R = v P e P, 4. For all K V, e = e P ( K K αp ). Proof. The socal welfare functon W R (p) = α R Eu for a prvate externaltes problem for vj R pf < vj+1 R (f such j exsts), N a = {j + 1, j + 2,..., }, s W R (p) = + j α R ( c (p) e π k + pf π k ) =1 N =j+1 = α R (v R N a α R (v R k N a k N a pf c (p) e π k + pf π k ) pf) k N a k N a N α R c (p) (e pf) N α R π k. k N a 18

19 If 0 pf < v1 R then the socal welfare functon s W R (p) = N αr (v R e) N αr c (p), and t s W R (p) = N αr c (p) for vn R pf. The socal welfare functon W P (p) = α P Eu for a publc externaltes problem for pf + e P < vj+1 P (f such j exsts), N a = {j + 1, j + 2,..., }, s v P j W P (p) = α P (v P pf) α P c (p) α P N a e P + N a N N N = α P (v P pf) α P c (p) (e P pf) Na N a N = α P (v P e P pf) α P c (p) + [e P α P N a N N a α P N a pf N a e P ] + pf Na If 0 pf + e P < v1 P then the socal welfare functon s W P (p) = N =1 αp (v P e P ) N αc(p), and t s W P (p) = N αp c (p) for vn P pf + e P. Hence, W P (p) = W R (p) for any p f condtons n the statement of the theorem are satsfed. Corollary 9. Socal planner s problems n a prvate externaltes problem wth externalty level e, and n a publc externaltes problem wth the externalty level e P are the same whenever the followng condtons are satsfed: 1. α R = α P j for all, j N, 2. For all K V, N αr k K\{} π k = K, 3. For all N, v R = v P e P, 4. e = e P ( 1). Lemma 3. For all K V N αr k K\{} π k = K f and only f for any V m V, N αr k V m \{} π k = V m. Proof. Suppose for any K V N αr k K\{} π k = K holds. Then for K = V V N αr k V V \{} π V V k =, and for K = V V V V 1 N αr k V V V V 1 \{} π k = V V V V 1. As N αr k V V V V 1 \{} π k = N αr k V V 1 \{} π k+ N αr k V V \{} π k = N αr k V V 1 \{} π V V V V V V k + = 1 mples that N αr k V V 1 \{} π V V 1 k =. Smlarly, for V m V, N αr k V m \{} π k = V m. If part of the proof s trval. Lemma 4. If for all k, for all, j, k, j k, π k = π kj then for any V m V, N αr k V m \{} π k = V m V m αr = V m, and for those agents who has dfferent valuatons from every other agents n the socety, α R = 1. 19

20 V m αr = V Proof. For all k, for all, j, k, j k, π k = π kj mples that π k = 1 for all N 1 k, N, k. For any V m V, N αr k V m \{} π k = N αr / V m k V π m k + N αr V m k V m \{} π k = N αr / V m V m + N 1 N αr V m V m 1 = V m N 1 N 1 N αr m 1 V N 1 N 1 m = N 1 V m N V α R = V. If has dffer- ent valuatons from every other agents n the socety then α R = 1 some V m V. as V m = {v } for Lemma 5. If for all k, for all, j, k, j k, π k = π jk = π k then for any V m V, N αr k V m \{} π k = V m V m αr π V m = N α π 1, and f every agent has a dstnct valuaton n the socety, α Rπ 1 = ( 1). Proof. For any V m V, N αr k V m \{} π k = N αr / V m k V π m k + N αr V m N αr / V m V m π + N αr V m ( V m 1)π = V m N αr π V α R m π = V m V m αr π V m = N απ 1. If for all V m V, V m = 1 then for all N,, and ths shows that α π = N απ 1. Addng over all N, N απ = 1 for all N, α π = 1 ( 1). 1 Note that W (v ) > W (pf) whenever v < pf < v +1. So, the planner has actually N + 1 alternatve expected punshment levels, ncludng the legalzaton of the acton, pf = 0, to choose from. If e v 1 then t s socally optmal to legalze that acton,.e. p = f = 0. If w < v j for some j then t s mpossble to deter any agent for j. So, w < v 1 mples that t s socally optmal to legalze that acton,.e. p = f = 0. Whenever all agents are dentcal then the optmal polcy s complete deterrence,.e. p f = v, f c( v w ) + v e; and t s optmal to legalze the acton,.e. p f = 0, f c( v w )+v > e. It s never optmal to deter everbody f v > Nh where v s the average of valuatons of agents nvolvng n the actvty. If there s a unt measure of agents n the socety, g(v) s the probablty densty of people who gan utlty v from the acton, and agents do not know ther types, and ex ante they are dentcal then the expected utlty of a representatve agent n a unt measure socety wll be k V m \{} π k = Eu(c, a, h) = v pf v vg(v)dv e pf g(v)dv c(p). (10) Ths s exactly the same objectve functon used n Polnsky and Shavell [4]. 20

21 Indvduals do not know ther types when the law and enforcement polcy s made but they learn ther types before they engage n any actvty. Theorem 10. Let p R be a soluton to socal planner s problems n a prvate externaltes problem wth externalty level e, and p P be a soluton n a publc externaltes problem wth the externalty level e P. Then p R p P f 1. for all N, α R = α P, 2. for all N, π k = 1 N 1, 3. for all, k N, v R = v P e P, 4. for all K V, e = e P ( 1), 5. agents are suffcently rsk averse. Proof. The socal welfare functon W R (p) = α R Eu for a prvate externaltes problem for vj R pf < vj+1 R (f such j exsts), N a = {j + 1, j + 2,..., }, s W R (p) = + j α R ( c (p) β(r, Na)h ( R e) + pf π k ) =1 N =j+1 = α R (v R N a α R (v R R N a pf c (p) pf) R N a N α R c (p) + pf N k N a β(r, Na)h ( R e) + pf π k ) α R k N a k N a π k N α R R N a β(r, N a)h ( R e). If 0 pf < v1 R then the socal welfare functon s W R (p) = N αr (v R R N β(r, N a)h a ( R e)) N αr c (p), and t s W R (p) = N αr c (p) for vn R pf. The socal welfare functon W P (p) = α P Eu for a publc externaltes problem for pf + e P < vj+1 P (f such j exsts), N a = {j + 1, j + 2,..., }, s v P j W P (p) = pf) α P N a pf α P (v P α P c (p) α P h ( N a e P ) + N a N N N = α P (v P pf) α P c (p) + pf Na α P h ( N a e P ) N a N N = α P (v P e P pf) α P c (p) α P h (( N a 1)e P ) + pf Na N a N N If 0 pf+e P < v1 P then the socal welfare functon s W P (p) = N =1 αp (v P h (e P )) N αc(p), and t s W P (p) = N αp c (p) for vn P pf + e P. Hence, W P (p) = W R (p) for any p f condtons n the statement of the theorem are satsfed. 21

22 References [1] Gary S. Becker. Crme and punshment. J. Pol. Econ., 76: , [2] Robert Cooter and Thomas Ulen. Law and Economcs. 3rd edton, [3] Mtchell Polnsky and Steven Shavell. The optmal tradeoff between the probablty and magntude of fnes. Amercan Econ. Rev., 69: , [4] Mtchell Polnsky and Steven Shavell. The optmal use of fnes and mprsonment. J. Pub. Econ., 24:89 99, [5] Steven Shavell. A note on margnal deterrence. Int. Rev. Law Econ., 12: , [6] George J. Stgler. The optmum enforcement of laws. J. Pol. Econ., 78: ,