The Unintended Consequences of Tort Reform: Rent Seeking in New York State s Structured Settlements Statutes

Transcription

1 The Ueded Cosequeces of Tor Reform: Re Seeg ew Yor Sae s Srucured Selemes Saues Publshed Joural of Foresc Ecoomcs, Volume 3 o, Wer 2 By Lawrece M. Spzma* Professor of Ecoomcs Mahar Hall Sae Uversy of ew Yor A Oswego Oswego, ew Yor Fax spzma@oswego.edu ad Elzabeh Due Schm Asssa Professor of Ecoomcs Mahar Hall Sae Uversy of ew Yor A Oswego Oswego, ew Yor edue@oswego.edu *The auhors acowledge he sghful commes of four aoymous referees as well as Fred Floss for hs commes o a earler verso of hs paper preseed a he Easer Ecoomc Assocao aual meeg Washgo D.C. 2. Ay remag errors are, of course, he sole resposbly of he auhors.

2 Page: 2 Absrac The Ueded Cosequeces of Tor Reform: Re Seeg ew Yor Saes Srucured Selemes Saue The or reform moveme of he 98's has resuled 9 perce of all saes passg some ype of or reform. ew Yor Sae, aempg o reduce lgao coss uder he spr of or reform, revsed s srucured seleme procedures for large jury verdcs by proscrbg a specfc dscoug procedure. Ths procedure dffers from radoal dscoug mehods used by foresc ecoomss. Whle or reform s eded o reduce he amou of lgao here may be ueded cosequeces resulg a crease or lgao. Ths paper shows ha ew Yor's or reform, CPLR Arcles 5-A ad 5-B, has a overall pro-plaff bas ha creaes ecoomc re. Ths bas, ad he cofuso surroudg, s cosse wh he crease or flgs ew Yor Sae bewee 988 ad 996.

3 Page: 3 The Ueded Cosequeces of Tor Reform: Re Seeg ew Yor Saes Srucured Selemes Saues I. Iroduco Dscoug fuure ecoomc losses was oe of several legal ssues addressed by he sae or reform moveme of he 98s ad 99s ad s also a mpora research opc amog Foresc Ecoomss. Durg hs perod, almos 9 perce of he saes eaced some ype of or reform legslao (Schm e al 997). These reforms clude laws affecg he assgme of lably, he sze of poeal damage awards, ad allowable mehods for dscoug fuure ecoomc ad o-ecoomc losses. Mos or reforms were respose o creasg lably surace coss ad herefore desged o reduce he amou of lgao (Vscus e al 993). However, some or reform laws may have had he oppose effec. Saes madag specfc dscoug procedures may dsor seleme egoaos ad acually crease he ceve o go o ral. ew Yor Sae's provsos for srucured judgemes, whle passed as or reform he 98s he hope of reducg lgao coss, acually overcompesae he plaff, resulg awards up o hree mes hgher ha he prese value of he jury award (Woloff ad Haushe 995, Lambros ad Harmo 995). 2 The purpose of hs paper s o exame he ueded The hree ma foresc jourals; Joural of Foresc Ecoomcs, Lgao Ecoomcs Dges, ad Joural of Legal Ecoomcs, devoe approxmaely 6% (from her frs ssues hrough 999) of her publcao les o deermg he proper dscou rae. 2 Boh sudes mae he comparso bewee 5-B ad acual prese value losses usg selecve bu geeral examples. Lambros ad Harmo prese he ecoomc loss for e cases focusg o base year eargs ad

4 Page: 4 cosequeces of ew Yor Sae s Arcles 5-A ad 5-B or reform by demosrag how he pos-verdc dscoug requremes resul a pro-plaff bas ad how hs bas mpacs seleme egoaos. We argue ha ew Yor Sae ("YS") or reform, raher ha reducg or lgao, may have corbued o a crease lgao ad lgao coss. Iflaoary pressures of he lae 97's ad early 98's precpaed dramac creases he cos of lably surace, whch led o he or reform moveme of he 98's (Vscus, Zechauser, Bor, ad Blacma 993). Vscus e al, po ou ha geeral lably premums rose from $3. bllo 975 o 9. bllo 988. Sae legslaures, cludg hose ew Yor, respoded o ese lobbyg by he surace ad medcal dusry by passg or reforms eded o reduce surace coss. 3 A jured pary's decso o brg a legal clam depeds o he coss ad beefs assocaed wh ha clam (Coer ad Rubfeld 989); or reforms chage he polcal ad ecoomc calculus of wheher cases are pursued ad/or resolved pror o ral. Several auhors have examed he emprcal l bewee or reform ad he decso o fle. Schm, Browe, ad Lee (997) fd a egave relaoshp bewee per capa clam flgs ad cera ypes of or reforms. Browe ad Puez (998) coclude ha accde vcms wll o see a legal remedy as ofe as hey dd pror o sae or reforms. eher of hese papers specfcally address srucured selemes. remag wor-lfe expecacy of he plaff ad wage flao ad he dscou rae. Woloff ad Haushe use hree represeave cases o llusrae he overvaluaos of 5-B. Whle s possble o develop cera facs ha mgh aler he overall oucome of hese wo sudes, everheless her geeraly appears o be correc, as we wll prove hs paper. 3 Vscus e al (993) dscuss hs ssue o a aoal level. Segel 985, 986 dscusses he polcal process volved passage of he ew law ew Yor Sae. Hs commeares sae ha he ew srucured judgmes were eaced prmarly as a beef o surace compaes. The cour Bermeo e al v Aae ad ew Yor Cy Healh ad Hospals Corporao, 24 A.D. 2d 235; 67.Y.S. 2d 727; 998.Y. also sae ha hs was he prmary reaso for srucurg selemes.

5 Page: 5 Sae or reforms may have ueded cosequeces, parcular, overcompesag plaffs relave o he rue prese value of fuure damages. Such reforms may resul a crease, raher ha a decrease lgao. Rolado Pelaez (995) exames he error rae of he Pesylvaa offse rule, whch requres a zero real eres rae dscoug all damages. Pelaez clams ha he smplcy of hs rule comes a he prce of overcompesag he plaff bewee 3% ad %, depedg o growh raes ad wor-lfe expecaces. The sze of seleme offers ad he decso o fle Pesylvaa s affeced by he overcompesao. YS law has a uque feaure ha he cour, o he jury, reduces fuure damages o prese value. 4 Cosequely, ecoomss persoal jury ad wrogful deah cases are permed o esfy oly o he omal value of he fuure loss sream (cludg a flao faco, o s prese value. All damage awards excess of $25, mus be srucured by he judge uder he gudeles of Cvl Pracce Law & Rules (CPLR) Arcles 5(a) (CPLR ) ad 5(b) (CPLR ) Sce arcles 5-A ad 5-B are decal laguage, we wll refer o hem as 5-B for he remader of hs paper Le he Pesylvaa offse rule, YS Arcle 5-B dscous fuure damages such a way ha produces a overall pro-plaff bas. 7 Pesylvaa's falure o accou for he me value of moey s somewha offse by he rule s smplcy because boh he plaff ad defeda ca easly recogze he bas ad adjus her seleme sraeges accordgly. Thus, seleme offers may chage, bu he ceve o proceed o ral wll o. The smplcy of Pesylvaa's offse rule s o mrrored YS. Chef Judge Kaye of he Cour of Appeals of 4 As of July 3, 986 he jury awards he full amou whou reducg o prese value (CPLR 4[f]). Also see YPJI suppleme PJI2: 32.3, p 499 for a dscusso o he evoluo of prese value ew Yor Sae. 5 Called Perodc Payme of Judgmes Medcal ad Deal Malpracce Acos. 6 I 986 he legslaure expaded arcle 5-A (whch deal wh Medcal Malpracce) o clude all persoal jury ad wrogful deah acos. I was decal o 5-A ad s called Arcle 5-B ( Perodc Payme of Judgmes Persoal Ijury, Ijury o Propery ad Wrogful Deah Acos ).

6 Page: 6 ew Yor summarzed four dffere case descrpos of 5-B by sayg "The srucured judgme provsos have deservedly bee labeled ' crcuous', 'vexg', 'as every Judge's ghmare', 'ad a bes *** ambguous' whch ca lead o explcable resuls.'" 8 Arcle 5-B creases he lelhood ha he plaff ad he defeda base her seleme decsos o dffere assessmes of poeal damages. Ths asymmery may aler he ceves o accep a seleme or proceed o ral. Sce he passage of Arcle 5-B, YS expereced he secod larges crease he coury of or cases per, populao bewee 988 ad I 996 he umber of repored ors per, was 462.6, up from , a crease of over 56%. Calfora, o he oher had, expereced a 45% reduco ors durg he same perod. Why dd YS experece such a dramac crease? We sugges hs crease s cosse wh legslao ha s boh flawed ad msudersood. Whle he geeral purpose of or reform s o reduce lgao coss, we coed ha YS's 5-B requremes crease he complexy ad herefore coss of lgao by requrg cumbersome pos-verdc egoaos. Ul he legslaure rewres he law a beer udersadg of he 5-B bas by he plaffs ad defedas may resul more effce seleme egoaos, whch wll reduce flgs over he log ru. The res of hs paper s orgazed as follows: seco II brefly descrbes he srucurg of judgemes uder 5-B, seco III documes he pro-plaff bas here 5-7 There are also several poeal defese bas ha may exs uder 5-B whch we dscuss seco II A. 8 Brya v ew Yor Cy Healh ad Hospal Corporao, Deprade e. al. v ew Yor Cy Healh & Hospal Corporao, o. 24, o. 37, decded July, Cour Sascs Projec 997. Lambros & Harmo (995) ad McKea (995) have made her ow recommedaos of wha ca be doe. YS Seae Bll o. 2277, sposored by Seaor Dale Voler (R-Buffalo) ad Assemblyma Joe Morelle (D- Rochese sees o place furher lms o or awards. I amog oher hgs sees o cap o-ecoomc damages a $25, whle lmg lawyers' cogecy fees. See Bowde (999) for a more complee dscusso of he Voler-Morelle bll.

7 Page: 7 B, seco IV dscusses he mpac of 5-B o he decso o fle ad o pos-verdc egoaos, ad seco V cocludes. II. Srucured Selemes uder YS 5-B A. A Overvew of 5-B Uder 5-B pas damages are pad as a lump sum. Oe-hrd of hs amou s pad o he aorey wh he plaff recevg wo-hrds. The jury award of omal fuure damages s bfurcaed uder 5-B. The frs $25, of fuure damages are pad as a lump sum payme. Two hrds of hs lump sum payme goes o he plaff ad oe hrd goes o he plaff's aorey. The mehodology for deermg he aorey's fee o he srucured poro of he remag verdc s as follows. The frs $25, lump sum payme s subraced from he oal omal fuure loss proporo o he damage losses each caegory (eargs, household servces, medcal coss, pa ad sufferg ec.) as awarded by he jury. 2 Afer subracg each caegores proporo of he $25, he remag fuure loss assocaed wh each caegory of loss (eargs, household servces, medcal, pa ad sufferg ec) s he dvded by he umber of years specfed by he jury o ge he aual average loss. The saue requres ha he omal amou of fuure pa ad sufferg mus be pad ou wh a maxmum of years. These aual average paymes are creased by 4% per year for he ere loss perod. The augmeed average aual paymes are dscoued o he prese. Oe- 2 For example, f earg losses are 6% of he oal award he 6% of $25, or $5, would be deduced from he award for fuure los eargs. If los household produco s % of he verdc award he % of he $25, or $25, s deduced from he fuure award for household produco ec.

8 Page: 8 hrd of hs amou s pad o he aorey as a lump sum payme. 3 The remag 2/3 of hs amou (for each caegory of loss) s he dvded by he umber of years he plaff s o receve he award. A auy wh he correspodg payme ad me srucure s he purchased for he plaff. Whle he srucure of 5-B coas several pro-plaff provsos, here are provsos of 5-B ha beef he defeda as well. 4 Rccard (996) dscusses how ermag he payme for o-ecoomc losses upo he deah of credor reduces he value of he srucured judgme o he plaff. Addoally, 5-B furher resrcs he collaeral source rule ad perms offses agas fuure plaff recoveres. Woloff ad Haushe (995A) dscuss why he mpac of hese provsos s lely o be small. Frs, f o-ecoomc losses are zero or mmal he ermao does o mae a dfferece. Secod, he -year maxmum o pa ad sufferg lms he probably of he plaff dyg pror o recevg all o-ecoomc damages. Furhermore, as he plaff approaches years of pos-verdc survval, he remag damages are worh creasgly less real erms. Ths s also rue for oher oecoomc losses. However, f he plaff should de suddely he hs par of he saue ca be a sgfca offse o he pro-plaff bases especally f here are large o-ecoomc damages awarded by he jury. Addoally, he Rohrg mehodology has a bas favor of he defeda because provdes for double dscoug. Ths occurs because he fal auy paymes are based o values already dscoued uder 5-B gudeles, o he omal fuure losses. Whle s 3 The fal sep of he srucured payme has proved o be coroversal wh evolvg erpreaos of he saue by he cours. Rohrg v Cy of agara Falls, 84.Y. 2d 6; 638.E. 2d mehodology appears o have whsood several challeges o appeal ad s he de-faco mehodology for calculag he aoreys fees ad srucurg he balace of fuure losses. 4 The laes case o hs ssue s Brya (999) where Chef Judge Kaye dscusses how defedas beef by "payg a judgme perodc sallmes..." She also dscusses how soppg cera paymes upo deah of he plaff may also beef he defeda.

9 Page: 9 possble, s hghly ulely ha he pro-defese bas wll be sgfca eough o offse he pro-plaff bas. Ths s so because he pro-plaff bas exss every suao whle he defese offses may or may o ever come o play. The coflcg pro-plaff ad prodefeda aspec of 5-B furher adds o uceray over damage awards, whch affec pos-ral seleme egoaos. B. Imposg he 5-B srucure Afer he Verdc Par of he 5-B srucurg process volves ese pos verdc egoaos by boh sdes, ofe hrough her ecoomss or a auy compay represeave. 5 The jury's verdc ses he omal value of he award, whch mus he be srucured by he judge uder 5-B ad relaed case law. However, pror o he judge seg he srucure, he wo pares o he dspue may reach a agreeme o her ow, whch may coss of a lump-sum payme, a alerave seleme srucure or a combao of boh. If he pares agree o sele he here s o eed for he cour o srucure he seleme uder arcle 5-B. 6 The egoaos revolve aroud deermg a approprae dscou rae ad he srucure of he auy. The egoaos become parcularly acrmoous whe ryg o deerme he fal srucure of he award. The acrmoy occurs because he plaff, whle eresed he acual srucure ad s fal dollar value pror o dscoug, s also eresed he cos o he carrer of purchasg he auy. The defeda, o he oher had, s eresed mosly s cos of purchasg he auy corac. 5 These pos-verdc egoaos occur afer a verdc for he plaff. Clearly f he case reached a jury verdc he pre-ral egoaos, o maer how ese, dd o resul a seleme. Ths paper s oly cocered wh he pos-verdc egoaos ha occur bewee he plaff ad defeda uder he wachful eye of he judge. 6 See Bermeo v. ew Yor cy Healh ad Hospal Corporao.

10 Page: The plaff's aoreys wll always ell her cles he acual o-dscoued value of he award, sce cles wll be pleased wh he larger umbers. The defeda's ma cocer s s cos of purchasg he auy. The plaff's aorey should also recogze he cos aspec because ha essece s wha he damages are o he orfeasor. Ay cos savgs realzed by he defeda he auy mare 7 are o shared wh he plaff, so he plaff was hese savgs pu bac o purchasg a larger auy. The defeda would be wllg o have a srucure ha pays more o he plaff f he cos s lower. The plaff's aorey was o mae sure ha he srucure s favorable o he cle whle cosg he defeda he maxmum amou. Thus, whle he srucure s mpora o he jured cle, he coss assocaed wh purchasg he auy are also mpora. Because of he complexy of 5-B, he dspuas ofe have dffere auy srucures ad cosequely dffere cos esmaes. The dspuas (eve uder he wachful eye of he judge) fd dffcul o recocle her umbers due o dffere erpreaos of he saue ad uceray abou he mg of damages, hece he pos-verdc bargag. 8 egoag he proper dscou rae was smplfed uder he frs ew Yor case dealg wh he ew legslao, Urs v Sussma 9. Judge Gammerma Urs chose he dscou rae based o he pas recommedaos of oher ecoomss. 2 Ths rulg se a precede, mag easer for 7 See Romas ad Floss (999) for a full dscusso of he eres rae swch bewee he auy rae of reur ad he dscou rae ad how ha may crease or decrease wha he plaff wll receve. 8 Cours sce Brya (999) sll fd 5-B o be a "daug as". Coye v Era ea al. Supreme Cour of ew Yor, assau Couy 73.Y.S.sd 869; 999 December 9. I addo o deermg he dscou rae he Coye case had o deerme wha cosued fuure damages. 9 Urs v. Sussma 54.Y.S2d 96 (Sup.989) 2 Gammerma polled 9 ecooms o he dscou raes hey have used prevously. He he chose 7.5%, whch was he upper rage of he 6 o 8% repored by he surveyed ecoomss. Judge Peer Tom wrg he opo of he Appellae dvso Bermeo e al v Aae ad ew Yor Cy Healh ad Hospals Corporao sad he followg: "he beer par of cauo would be for a cour o elc he pares' respecve posos as o wha rae would be approprae, afer whch he cour could mae s ow deermao."

11 Page: pares fuure cases o reach a agreeme o he proper dscou rae, bu dd o solve ay of he problems of agreeg o a srucure. C. How Does 5-B Wor? To udersad why boh plaffs ad defedas may derve dffere pos- verdc values for he 5-B srucured payme, s ecessary o udersad how 5-B affecs he sze of a jury's award. The madaed srucure of awards uder 5-B resuls a prese value calculao ha sysemacally dffers from he rue ecoomc value of he losses. A plaff eers lgao wh he expecao of recoverg damages, bu he 5-B srucure flaes hese damages over ad above he rue prese value of he jury award. Resources used o sasfy he jury award have opporuy coss. 5-B s a legslaed srucure, whch resuls a prese value calculao ha s greaer ha he acual prese value of he jury award. The dfferece bewee he opporuy cos ad he 5-B award ca be cosdered ecoomc re. Re seeg behavor by boh he plaff ad defeda occurs durg he pos verdc egoaos requred by 5-B ad s dscussed Seco IV of hs paper. To demosrae why a pro-plaff bas exss uder 5-B, le us cosder a persoal jury award cossg of los wages, los household servces, ad pa ad sufferg. A prese value calculao s used o cover he fuure loss sream o a equvale value oday s dollars. 2 Defe he followg varables: W = base year salary

12 Page: 2 g H h = salary growh rae = base year cos of provdg los household servces = growh rae of household servce coss P = base year pa ad sufferg losses 22 π T r = aual flao rae = expeced wor-lfe = lfe expecacy = dscou rae As prevously saed, ecoomss ew Yor ca esfy oly o omal losses ad o o prese value. However, hey are allowed o prese evdece o flao adjusmes o he jury. A jury deermes pa ad sufferg awards. ormally, prese value calculaos for los wages would be dscoued over he expeced wor-lfe of he jured pary, whle los household servces ad pa ad sufferg awards are dscoued over he plaff's lfeexpecacy. The prese value of he fuure loss sream s gve by T T W ( g) H ( h) P ( = ( = ( = ( () π) Correcly esmag he prese value depeds o a accurae esmae of fuure losses, choosg he proper dscou rae, ad dscoug over he approprae me perod. Oce a jury reaches a verdc ew Yor, he judge srucures he seleme o comply wh 5-B, whch madaes ha he prese value of fuure losses be calculaed a maer 2 Pas damages, hose curred bewee he dae of jury ad he dae of he award, are pad ou as a lump sum, so ha he 5-B srucures apply oly o fuure damages. 22 There are alerave vews as o how pa ad sufferg losses are calculaed. The vew used here s ha pa ad sufferg losses are measured aually ad crease a he rae of flao. Aoher vew s ha pa ad sufferg losses are a lump sum award, or are awarded over a me perod shorer ha he lfe expecacy of he plaff.

13 Page: 3 sgfcaly dffere from equao (). 23 Uder 5-B he frs $25, of he jury's verdc for fuure losses s pad ou as a lump sum. The remag omal ecoomc damages are dvded by he expeced wor-lfe for a average aual loss, whch s augmeed by a aual growh rae of 4%: Toal omal ecoomc damages,, are gve by T (2) = W ( g) H ( h). 24 = = Uder 5-B, he prese value of equao (2) s calculaed as 25 (3) 25, ( 25,) (.4). = ( omal pa ad sufferg losses are averaged over a maxmum of e years. 26 The pa ad sufferg award made by he jury s: T (4) P = = P ( π) Uder 5-B he prese value of he pa ad sufferg award s calculaed as We assume ha he jury verdc s based o he esmoy of he ecooms ad cludes a flao facor. 24 Oher compoes of losses such as medcal care, lfe care ec. ca be added o hs equao as requred by he jury verdc. 25 We assume ha omal ecoomc damages exceed $25,. For smplcy, we also assume ha =T, whch acually udersaes he pro-plaff bas. 26 The jury may have a verdc for pa ad sufferg less ha years bu he saue requres a maxmum of years or he umber of years se by he jury, whch ever s lower. We assume ha he lfe expecacy of he plaff exceeds e years. 27 If he plaff des pror o payg ou he fuure pa ad sufferg award he he defeda does o have o pay ha poro of he award. Ths s a defese bas ha oly occurs upo deah of he plaff wh e years. If here s formao o sugges ha he plaff may o lve o years he her aorey wll ceraly facor ha o her pos-verdc egoaos sraegy. Ths may also occur durg he acual ral ryg o ge he sympahy of he jury so ha hey fro load mos of he pa ad sufferg eher uder pas pa or sufferg or for less ha years. Uless here s formao ha would be brough ou a ral ha he plaffs lfe wll be sgfcaly shoreed hs defese bas would be purely radom. For hs o offse oally he plaff bas he jury award would have o be sewed almos erely o fuure pa ad sufferg, somehg a plaff aorey (owg ha a shore lfe spa has occurred) wll ae exra care a ral o expla o he jury.

14 Page: 4 (5) ( P )(.4) = ( Combg (3) ad (5), he prese value of he oal fuure loss sream uder 5-B s calculaed as (6) ( P )(.4) ( 25,) 25, (.4). = ( = ( Equao (6) s always larger ha equao (), resulg a pro-plaff bas. 28 There s a posve relaoshp bewee he sze of he bas ad he sze of he award, ad P, he legh of he dscoug perods, ad T, ad he dscou rae, r. The complexy of equao (6) ca lead o he plaff, defeda ad judge calculag dffere values for he prese value of he loss. 29 Because of hese dffere values, s o clear how much of he bas he judge wll allocae o he plaff. Ths uceray provdes he movao for pos-verdc egoaos, where boh sdes are ecouraged o agree o a dscou rae ad he o agree o a acual srucured payme. Eve f he verdc s reduced upo appeal, he 5-B srucurg mus be compleed. 3 Durg he 5-B egoaos, he plaff sees o maxmze s share of he bas, whle he defeda sees o mmze he re wll pay. I essece, he 5-B srucure esablshes a upper rage for he damage award whle radoal prese value calculaos esablsh a lower rage. Kowg ha he 5-B bas exss 28 We wll prove hs seco IV. 29 I s very possble ha oe of he dspuas ow how o mae he 5-B calculaos or for ha maer have ay experece a all large verdc awards. 3 The ral cour Bermeo v Aae e al ad ew Yor Cy Healh ad Hospal Corporao reduced a jury verdc of $45 mllo damages ad he Appellae Cour creased he ral cours reduco, bu sll requred o be srucured uder 5-B.

15 Page: 5 ad udersadg why occurs becomes crucal egoag he sze of he re he plaff wll receve. III. Reasos for he Arcle 5-B Pro-Plaff Bas The source of he 5-B bas s o ecessarly a correc dscou rae, bu a correc dscoug procedure. There are four major reasos why 5-B overesmaes he loss: () The $25, lump sum payme, (2) smoohg he fuure loss sream over he expeced wor-lfe, (3) double adjusme for flao, ad (4) averagg he pa ad sufferg award over a eyear perod. A. Lump Sum Payme Arcle 5-B requres ha he frs $25, of fuure losses be pad as a lump sum. ormally, calculag prese value, dscoug sars afer he frs perod ad coues o he wor-lfe expecacy or lfe expecacy of he plaff, wh each perod s losses mulpled by a smaller dscou facor. Uder 5-B, he frs $25, s o dscoued. The remag fuure losses are he dscoued. Ay provsos for a lump sum payme ha exceeds frs perod losses wll always resul a upward bas he prese value calculao. Cosder a sream of paymes over perods:,, 2,...,, where > ad < The prese value of hs payme sream s gve by

16 Page: 6 (7) ( =. ow suppose some of he paymes are pad ou as a lump sum (L) he frs perod, wh he remag paymes dscoued as equao (6). The prese value of hs lump sum payme sream s gve by (8) L, where L = ad <. ( = = Equao (7) s always less ha equao (8) for ay posve dscou rae. 3 The bas due o he lump sum payme depeds o he amou of damages, he wor-lfe expecacy, ad he dscou rae, bu ca ypcally accou for 5 o % of he oal 5-B bas (Woloff ad Haushe 995). B. Smoohg he Fuure Loss Sream The secod source of pro-plaff bas s due o he smoohg of fuure losses over he wor-lfe expecacy. Arcle 5-B madaes ha afer deducg he lump sum payme, he remag fuure losses are dvded by he umber of years recommeded by he jury (usually he ecooms s esmae of wor-lfe expecacy) o calculae he average loss per year. Cosequely, hs smoohes he me pah of fuure losses. However, he acual loss sream 3 Ths s rue because for ay >, > L (. = = (, ad herefore L = > > = = ( = ( = ( = (. Thus

17 Page: 7 creases over me, so replacg wh he smoohed loss sream overesmaes he prese value of he acual losses. I ca be show ha he prese value for ay mul-perod, creasg loss sream wll be less ha he prese value of he smoohed loss sream: (9) < where = ( (. 32 = = = The equaly equao 9 occurs because he acual loss sream (he umeraor he lef-had sde of equao 9) creases over me, sarg wh a base year loss ad rsg accordg o some growh rae, as equao (2). Thus, pror o dscoug, losses curred he laer perods are acually greaer ha he average aual loss, ad losses curred he earler perods are smaller ha he average aual loss. Also, dscoug fuure losses pus more wegh o earler losses ad less wegh o laer losses. Therefore, f he acual loss sream s replaced wh a loss sream equal o he average aual loss (he umeraor he rgh had sde of equao 9), more wegh s gve o aual losses ha are oo hgh (relave o he acual loss hose perods) ad less wegh s gve o aual losses ha are oo low. The resul s a overesmao of he prese value of he acual loss sream. Smoohg fuure losses accous for 2-26% of he oal overesmao (Woloff ad Haushe 995). If he jury specfes a me perod less ha he acual wor-lfe expecacy he he overesmao would be eve larger. C. Double Adjusme for Iflao 32 See he appedx A for he proof of hs saeme.

18 Page: 8 The hrd source of he pro-plaff bas occurs whe he smoohed loss sream s augmeed by he madaed 4% growh rae afer he jury verdc. Sce he ecooms esfed a ral o omal fuure losses ad cluded a flao facor for boh los wages ad household servces (See equao 2), cludg aoher 4% crease as madaed by he saue resuls a double adjusme for flao. Presumably, he jury also maes he ecessary adjusmes based o he flao esmoy whle delberag he pa ad sufferg award (see equao 4). Thus he prese value of he ecoomc damage award becomes: () (.4) where = ( = = Gve he equaly equao (9), ca furher be show ha equao () creases he pos-verdc plaff bas. 33 Wolhoff ad Haushe (995) arbue 46 o 5% of he oal bas o he double flao adjusme. The legslaure's movao for cludg a 4% pos verdc adjusme s uclear. However, hs adjusme may have bee perceved as a advaage by surace compaes, gve he hgh level of flao durg he lae 97s ad early 98s. 34 I appears ha lobbyss ad lawmaers dd o udersad ha 4% would be addo o he ecooms's flao esmoy of ecoomc damages a ral. Legal challegers o adjusg he srucured seleme by 4% coue o arse. Mos recely, Schulz v. Harrso Radaor, 9 Y2d 3 (997) he defeda red o dsallow 33 For all =,2,3,,, (.4) >. ( ( Therefore (.4) > > = ( = ( = (.

19 Page: 9 he plaff's esmoy abou flao, argug ha a 4% pos verdc flaoary adjusme would double compesae he plaff. The Cour of Appeals (he hghes cour YS), cg pas precede, ruled ha ecoomc esmoy abou flao s allowed, as s he pos verdc 4% madaed adjusme. The Schulz cour refused o chage he commo law ad lef he 4% ssue o be resolved by he legslaure. The same cour Brya (999) aga upheld he 4% adjusme. The cour correcly dscussed how he 4% adjusme proecs he plaff from a rse prces over me. However, he cour dd o appear o recogze ha exper esmoy abou flao durg he ral (o whch he jury bases s verdc) provded he same proeco. Thus he double adjusme for flao remas. I boh Schulz ad Brya, he cour's aemp o jusfy he 4% adjusme o ecoomc grouds demosraed a msudersadg of he rue ssue. 35 D. Pa ad Sufferg A fourh source for he pro-plaff bas occurs whe he me perod for damages due o pa ad sufferg s rucaed o a maxmum of e years. Jures ypcally mae a award based o her belef ha pa ad sufferg wll rema for he lfe expecacy of he plaff. If o, hey se he me frame for whch pa ad sufferg wll be compesaed. However, uder 5-B, he ere award for pa ad sufferg s pad ou over a maxmum of e years (or less, f he jury so srucs). Reducg he umber of years over whch he fuure loss s dscoued 34 I has bee suggesed, parly Romas ad Floss 999, ha he 4% adjusme s acually a rs premum o compesae he plaff for locg a srucure for fuure paymes, somehg ha would o occur he absece of jury. However he Schulz ad Brya cours clearly vew he 4% as a flao adjusme. 35 The Schulz cour sad he 4% adjusme "does o provde addoal compesao for a plaff above ad beyod damages awarded; raher, esures ha he passage of me wll o devalue he award because of a geeral

20 Page: 2 overesmaes he prese value of he loss. The followg equaos show ha he smoohed ad rucaed loss sream wll always resul a larger prese value. Equao (9) allows us o sae ha T T π () < where P = I s also rue ha = P ( ( ) = P T ( T = P ( π). T P T (2) < = ( = 9 P ( where T >. 36 The sze of hs overesmao depeds o he lfe expecacy ad he sze of he award for pa ad sufferg. Wolhoff ad Haushe (995) arbue 6 o 24% of he oal bas o he e-year lmao for pa ad sufferg losses. IV. The Effec of 5-B o Seleme egoaos Seco III esablshes ha he 5-B srucure always resuls a pro-plaff bas relave o he acual prese value of he fuure loss sream of he srucured seleme. 37 The proplaff bas aloe creases he poeal beefs of lgao for he plaff, hus creasg oal lgao (whch cludes cases seled pror o beg fled ad cases fled). 38 Therefore eve f he proporo of cases gog o ral says he same, here s sll a crease he umber of flgs due o he oal crease lgao. I addo, 5-B may crease o oly rse prces for goods ad servces..." The cour dd o recogze ha he ecooms already esfed o flao a ral. 36 See appedx B for he proof of hs saeme. 37 Whle hs may be offse by he defese bases dscussed seco II, he pro-plaff bas wll always exs he 5-B srucure. 38 Lee, Browe ad Schm (994) docume he posve relaoshp bewee he expeced reur of a or flg ad he umber of or flgs.

21 Page: 2 he oal amou of lgao YS, bu also he proporo of lgao proceedg o ral by alerg boh he formao srucures amog he dspuas ad plaff audes owards rs. A. 5-B ad he Percepo of Bas If boh pares udersad he ecoomc cosequeces of 5-B, he dspua's pre-ral seleme offers wll rse hus faclag seleme, 39 whle he decso o go o ral remas uaffeced. 4 However, f he dspuas do o equally udersad he ecoomc mplcaos of 5-B, he probably of a ral creases. The ey ssue s o he acual sze of he 5-B bas, bu he percepo by each sde of wheher a bas exss a all (Schm e al 997). Gve he complexy ad msudersadg surroudg 5-B, s o surprsg ha each sde has dffere percepos of he sze of he bas. We coed ha plaffs were beer formed abou he 5-B bas he years mmedaely followg he legslao. Wh dffere percepos abou he mplcaos of 5- B, plaffs' expeced damage awards crease, mag hem more wllg o go o ral ha he defedas, who may o expec ay chage expeced losses due o 5-B. 4 Sce boh pares have dffere damage values assocaed wh he or, preral agreeme s less lely, resulg more cases beg fled for ral. Ths s also cosse wh wha has happeed YS bewee 988 ad 996. There are several reasos why defedas, who are ofe surace compaes, have bee less lely o udersad he ecoomc cosequeces, or exsece, of he pro-plaff bas he 39 Boh pares, havg full formao, ow he sze of he plaff bas assocaed wh 5-B hus hey mae he ecessary upward adjusmes her seleme egoaos. 4 Sper (994) saes ha symmercally formed pares wll sele her dspue raher ha go o cour.

22 Page: 22 years mmedaely followg he eacme of 5-B. Sce he 5-B reform was a drec resul of lobbyg by he surace dusry ha dusry erroeously hough would be beefcal o hem. 42 Gve he ermao of cera paymes upo deah ad he offse allowed uder he collaeral source rule he defedas may have assumed a frs ha 5-B s always favorable o hem. Ths lobbyg effor was due o a 2% rse geeral lably premums bewee 984 ad Ths perod accoued for mos of he growh lably premums sce 975. Large jury awards durg hs me perod were parly o blame for hs crease. Severe surace raog followed, resulg may frms fdg dffcul o acqure lably surace a ay prce. Ths growh of surace coss coceraed a wo-year perod alog wh hgh raes of flao led o a geeral pac demadg or reform. I respose o hs ad uder pressure from he surace dusry, ew Yor Sae passed Arcles 5-A /5-B whou udersadg her ecoomc cosequeces. 44 Aoher reaso why he surace dusry dd o boher o exame more deal he ecoomc cosequeces of 5-B was due o he sudde crease profably 988. I s lely ha surace compaes arbued hs reur o profably parly o sae or reforms, cludg YS 5-B. The surace dusry expereced huge operag losses 985 due o clam losses exceedg premums. (Vscus e al 993) Profably was resored 988 as loss 4 Lee, Browe ad Schm (994) dscuss he posve relaoshp bewee he expeced reur of a or flg ad he umber of or flgs. 42 The surace dusry ceraly would o lobby for a law ha would be cosly for hem. 43 Vscus e al (993) also docume a 56% crease medcal malpracce surace from ad aoher 26% from 985 o 986. Eve auomoble premums, whch hsorcally do o rse much, creased by 25.6% from 985 o Vscus e al spea o he geeral lac of udersadg by sae legslaors he followg saeme "Sae respoded wh reform legslao, despe a geeral lac of udersadg abou how he exsg lably rules affeced surace mares or how specfc reforms mgh chage her performace." p. 66.

23 Page: 23 raos dropped by 4 perceage pos or more (Vscuss e al 993). 45 The surace dusry lely arbued s creasg profably o s successful lobbyg for or reform. Why dd ecoomss form surace compaes abou he pro-plaff bas here he 5-B srucure? Defedas (surace compaes) hsorcally have bee reluca o hre ecoomss, sce her esmoy abou damages esablshes a lower boud for he facal award ad may be see as a admsso of lably (Spzma 995). Sce defedas radoally do o use ecoomc expers 46 hey are o as well formed abou he cosequeces of 5-B. Plaffs do rouely use ecoomss o esmae ecoomc damages ad aalyze srucured selemes, ad are hus more lely o be formed abou he pro-plaff bas of 5- B. Ecoomc expers would o oly mae he ormal prese value calculao bu hey would also ry o mae he 5-B calculaos. If defedas do o rea a ecooms durg he acual preral egoaos hey may o be formed abou he sze or eve he exsece of he 5-B bas. Also, he lac of deposos YS or lgao perpeuaes he msformao. All of he above resuls defedas o recogzg he 5-B plaff bas, hus mag lower seleme offers, whle re-seeg plaffs, recogzg ha he bas exss, rejec defeda offers because of a greaer expeced pos-verdc reur. Therefore fewer preral seleme offers wll be acceped, ad he rae of flgs, as YS daa sugges, creases. I s lely ha hs msformao decled as cour rulgs clarfed he applcao ad cosequeces of 5-B srucurg. 45 Loss raos of less ha oe dcae profably whle a loss rao greaer ha oe dcaes losses. Vscus e all (993) show ha, wh respec o medcal malpracce, surers we from payg $.5 for every $ premums o payg 58.4 ces clams for every $ premums. Geeral lably showed a loss rao of drop o Ths appears o be chagg as more ad more defedas use ecoomss (eve f o dsclosed as expers) o aalyze he ecoomcs of he lawsu.

24 Page: 24 B. The Role of 5-B Alerg he Audes of he Plaff Towards Rs The expecao of pos-ral damage awards exceedg he acual prese value of fuure losses may duce a loery mealy amog re-seeg plaffs. Thus, hey may be more wllg o rs a ral eve f he defedas crease her seleme offers (Schm e al 997). If a plaff s ormally rs averse;.e., wllg o accep a offer below he expeced damage award order o avod he uceray of a ral, he he loery mealy maes he plaff less rs averse, effecvely creasg he mmum seleme a plaff wll accep. Thus, eve f he defeda recogzes he 5-B bas ad maes a larger pre-ral offer, he plaff may rejec he offer because 5-B alers o oly he expeced sze of he award, bu he plaff s aude owards rs. C. 5-B ad Pos-verdc egoao Oe of he more eresg aspecs of YS law s he pos-verdc bargag process. Pos-verdc egoaos ofe lead o "horse radg" a a level beyod pre-ral egoaos. Sce he jury verdc s o dscoued (ecoomss cao esfy o dscoued values) bu sead requres pos verdc srucurg uder 5-B, he wo sdes mus agree o he dscou rae (he easy par of he egoaos) ad he srucure of he auy o be purchased. If he wo sdes cao agree o a srucure, he law requres he judge o srucure he seleme uder 5- B. Boh sdes volved he 5-B egoaos see o maxmze her beefs or mmze her coss subjec o he jury s verdc. egoag he srucure, uder he wachful eyes of he

25 Page: 25 judge, ofe leads o boh sdes reachg a agreeme whou he judge mposg a 5-B srucure. The very exsece of he pos-verdc egoaos, however, s a esame o he uceray abou he proper applcao of a 5-B srucure. If he law were clear abou how o esablsh he srucure, he he ceve o egoae would be grealy reduced. The defeda has a eres egoag a pos-verdc seleme ha mmzes he pro-plaff bas a 5-B srucured seleme whle maxmzg ay pro-defeda bases. If he defedas do o acually ow he sze of he bas, wha may be perceved as rasgece egoaos may fac jus be ecoomc gorace. The plaff s ceve s o maxmze he re, where he re s he hgher prese value ha comes from a mposed srucure, raher ha acual damages. The egoaos occur uder he uceray of he judge's ably o correcly srucure he seleme f he dspuas fal o agree. 47 If plaffs "hold ou" for a 5-B award, hey may receve subsaally less due o judcal error. The uceray of how he judge, he absece of agreeme, wll dsrbue he 5-B bas leaves room for egoaos. By requrg subsaal pos-verdc egoao, 5-B creases he complexy ad herefore he coss of lgao. Pos-verdc egoaos resul a furher expedure of resources o lgao ad are ye aoher source of effcecy. Usually, whe a jury verdc s reached (uless he verdc s appealed), he case s closed qucly. However, uder 5-B he case ca be draw ou loger by addg he pos-verdc process. I s o he bes eres of he defeda o sall o hs maer however, because he saue also has a 9% eres pealy ha s added o he damage award afer a jury verdc s gve. or s advaageous for a serously jured plaff who 47 Whle s rue ha he saues ad case law provdes he judge wh a mehod of calculag he damages uder 5-B, as recely as 999 he cours have had o resolve ssues abou he correc applcao of he saue. (See Brya)

26 Page: 26 cao wor. Ths effcecy loss s aoher ueded cosequece of 5-B. The pos-verdc process uder 5-B appears o usurp he basc e of or reform. D. The Role of he Judge The role of he judge a 5-B hearg s uque because, he absece of he dspuas agreeg o a srucure, he judge mus mae he fal ecoomc calculaos. If judges ow he bas of 5-B, why do' hey se ew precedes ad rule such a maer ha he bas s reduced? The mos obvous reaso s ha hey beleve he legslave brach ad o he judcal brach should chage he law. 48 Tabarro ad Hellad (999) sugges aoher reaso why a judge wll o wa o aler he bas of 5-B. Sce ral judges are eleced ew Yor ad ral lawyers are bg corbuors for her reeleco, judges have a ceve o redsrbue wealh from he defeda, whch s usually a ou of sae compay, o he plaff, who s a poeal voer ad corbuor. Tabarro ad Hellad call hs cosuecy servce he "local voer" effec. 49 Wh hs md, here s o ceve for a judge o mae ew law by ryg o overur he legslaed 5-B bas. However here s sll a ceve for he judge o ge boh sdes o agree o a pos-verdc seleme ad avodg mag he calculaos o her ow. The judge ca expede agreeme he egoag process, because ofe boh sdes have appeared before he same judge ad may do so aga he fuure. Thus, boh sdes recogze he exeral cos of rasgece fro of he judge her 5-B egoaos. 48 Judges may, however, adverely reduce he bas f he judge correcly calculaes he losses ad he dspuas do o ow he correc calculaos. 49 Ther hypohess s ha hs effec causes awards o be larger saes where he judcary s eleced raher ha appoed.

27 Page: 27 Udersadg hs cour room calculus, whle also ryg o avod mag he 5-B calculaos herself, he judge uses moral suaso o faclae a agreeme o he srucure. V. Cocluso Ths paper dscusses he ueded cosequeces of he 5-B as oe eleme of or reform ew Yor Sae. Sae or reforms of he 98s were passed wh he eo of reducg he cos of lgao ad he umber of flgs. YS or reform has had ueded cosequeces whch have creased he expeced reur of flg a clam for he plaff whle addg o he legh ad complexy of pos-verdc egoaos for boh pares. Ths s cosse wh dramac crease he rae of flg he years followg he passage of 5-B Provsos for a srucured seleme uder 5-B resul a pro-plaff bas o damage awards, creag a dsceve for plaffs o sele for acual damages. The complexy of he saues sugges ha hese mplcaos are o well udersood by he dspuas, resulg dfferg assessmes of expeced damage awards ha dscourage pre-ral selemes ad leghe pos-verdc egoaos. Emprcal research o he affec of 5-B or or flgs wll clarfy wheher hese reforms have acually creased he cos of lgao. However, such research would requre daa o pre- ad pos-ral selemes. Much of ha daa s propreary ad hus uavalable, ad ay subse of avalable daa may o be represeave. Whle some may beleve ha he cours have resolved all he major ssues of srucurg uder 5-B, he Brya cour self does o agree: As a cocludg observao, we oe ha as CPLR arcles 5-A ad 5-B reach her 5h aversary, havg geeraed a good deal of frusrao ad lgao, would perhaps be a opporue me for he Legslaure o revew hese eacme's o assure ha, acual operao, hey are meeg her

28 Page: 28 objecves ad o--as submssos before us urge- egregously overcompesao or udercompesag ayoe. 5 Ul he legslaure rewres he law 5, a beer udersadg of he 5-B bas amog plaffs, defedas, ad judges may resul more effce seleme egoaos ad possbly reduce he flg rae ew Yor. Refereces Argee, Sephae L From Verdc o Judgme: The Evoluo, Cofuso ad Reformao of CPLR Arcles 5-A ad 5-B. Buffalo Law Revew 4: Bowde, Mchael M "ew Yor s he Laes Balegroud Over Tor Reform." Lawyers Weely USA, May 3, 99 LWUSA 428. Browe, Mar J., ad Puelz, Rober The Effec of Legal Rules o he Value of Ecoomc ad o-ecoomc Damages ad he Decso o Fle. Worg Paper, Uversy of Wscos-Madso. 5 Brya v. ew Yor Cy Healh ad Hospals Corp. 5 See Bowde (999).

29 Page: 29 Coer, Rober, ad Rubfeld, D Ecoomc Aalyss of Legal Dspues. Joural of Ecoomc Leraure 27: Cour Sascs Projec, 997. Sae Cour Caseload Sascs, 996, aoal Ceer for Sae Cours. Lambros, James, ad Harmo, Osar R Plaff Bas he CPLR 5-A/5-B Saues. Albay Law Revew 59: Lee Ha-Duc, Browe Mar J. ad Schm Joa T, 994, "How Does Jo ad Several Tor Reform Affec he Rae of Tor Flgs? Evdece from he Sae Cours", The Joural of Rs ad Isurace, vol. 6. o 2, McKea Parc J., 995 "A Four Percde Soluo o he "Judge's ghmare," ew Yor Sae Bar Joural, Ja. a 44. Rccard, Ahoy H, 996, "A Reply o Professors Woloff & Haushe o The Ecoomcs of Srucured Judgmes Uder CPLR Arcle 5-B," Buffalo Law Revew Fall, Vol. 44-3: Rolado, Pelaez Pesylvaa's Offse Rule: Faasy Masqueradg as Ecoomcs. Joural of Legal Ecoomcs 5(3): -6. Romas Thomas, Floss Frederc, 999 "Srucured Selemes ad he Ieres Rae Swch" Joural of Foresc Ecoomcs 2(), Schm, Joa T.; Browe, Mar J.; Lee, Ha Duc The Effec of Sae Tor Reforms o Clam Flgs. Rs Maageme ad Isurace Revew : -7. Segel, Davd D "Pracce Commeares The 985 Amedme Creag he Srucured Payme", McKey's Cos Laws of Y, CPLR arcle 5-A, 53, 75. Segel, Davd D "The 986 Amedme Exedg he Srucured Judgme o Tor Acos Geerally", McKey's Cos Laws of Y, CPLR arcle 5-B, 54, 726. Sper, K "Preral Bargag ad he Desg of Fee-Shfg Rules." Rad Joural of Ecoomcs 25: Spzma, Lawrece M The Defese Ecooms s Role Lgao Seleme egoaos. Joural of Legal Ecoomcs 5(2): Tabarro Alexader ad Hellad Erc, 999. "Cour Polcs: The Polcal Ecoomy of Tor Awards", The Joural of Law ad Ecoomcs, Vol. LII () (P.) Aprl,

30 Page: 3 Vscus, W. Kp; Zechauser, Rchard J.; Bor, Parca; Blacmo, Gle The Effec of 98s Tor Reform Legslao o Geeral Lably ad Medcal Malpracce Isurace. Joural of Rs ad Uceray 6: Woloff, Mchael, ad Haushe, Erc The Ecoomcs of Srucured Judgmes Uder CPLR Arcle 5-B. Buffalo Law Revew 43: Woloff, Mchael, ad Haushe, Erc. 995A. "Rush o Judgme: The (Ds)Iceve o Sele Persoal Ijury Cases" ew Yor Sae Bar Joural March/Aprl Bermeo e al v Aae ad ew Yor Cy Healh ad Hospals Corporao, 24 A.D. 2d 235; 67.Y.S. 2d 727; 998.Y. App.Dv. Lexs 443. Brya v ew Yor Cy Healh ad Hospal Corporao, Deprade e. al. v ew Yor Cy Healh & Hospal Corporao, o. 24. o. 37, 93.Y.2d 592;76.E. 2d 84; Lexs 432; 695.Y.S.2d 39 decded July, 999, Cour of Appeals of ew Yor. Cvl Pracce Law & Rules (CPLR) Arcles 5(a) (CPLR ) 985 ad 5(b) (CPLR ) 986 Coye v Era e al. Idex o /94 Supreme Cour of ew Yor, assau Couy 73.Y.S.2d 869; 999.Y. Msc. Lexs 65. Rohrg v Cy of agara Falls, o 7 Cour of Appeals of ew Yor, 84.Y. 2d 6; 638.E. 2d , Lexs 427; 64.Y.S.2d 74 Schulz v. Harrso Radaor, 9 Y2d 3 (997) Urs v. Sussma 54.Y.S2d 96 (Sup.989)

31 Page: 3 Appedx A Prove (A) < = ( = ( where (A2), r >, >, > ad = = Proof by mahemacal duco sep : Show ha (A) s rue for = 2( r < ( < r < 2 < 2 ( 2 ( ( whch s assumed (A2) Thus, (A) s rue for =. sep 2: Assume ha (A) s rue for = s rue (A3) < = ( = ( Mulplyg hrough by ()(, (A3) becomes (A3a) ( ) ( < ( = = =

32 Page: 32 sep 3: Show ha (A) s rue for = < = = rue s ( ( < = = ( ( ( ( (A4) = = = = < ( ( ( 2) ( ( 2) ( The frs erms o each sde of (A4) may be rewre as follows: () = = = = ( ( ) ( ( 2) ( ad () = = = = = = ( ( ( Subsug () ad () o (A4), (A5) ( ( ( ( 2) ( ( ( ) ( < = = = = = = (A) s rue f ad oly f (A5) s rue. Sce (A3) s assumed o be rue, hs mples ha he frs erm o he LHS of (A5) s less ha he frs erm o he RHS of (A5). Thus (A) s rue f he remag equaly holds, or

33 Page: 33 (A6) ( ( 2) ( = <. ( ( = (A6) s rue f ad oly f = = (A7) ( ) < ( ( ) = oe ha he LHS of (A7) = whe =, so (A7) s equvale o = = ( ) < ( ( ) Uder he assumpos (A2), ( ) for all < (. Thus, he equaly (A7) holds ad (A) s rue. Prove T P T (B) < where = ( = ( 9 P (B2) r >, T > ad P = = Proof by mahemacal duco T sep : Show ha (B) s rue for T =. P ( π) Appedx B P < = ( = ( < 9 P = ( = ( 9 s rue

34 Page: 34 ( 9 < 9 = ( (B3) < = ( = ( 9 Uder (B2), < for all =,,2, ( ( Thus, (B3) s rue ad herefore (B) s rue for T =. sep 2: Assume he (B) s rue for T =. P (B4) < = ( = ( 9 P sep 3: Show ha (B) s rue for T =. P ( ) (B5) < = ( = ( < 9 = ( = ( 9 P s rue from (B4) we assume ha < = ( = ( so (B5) s rue f 9 (B6) ( < 9 = ( 9 or < = ( = ( 9 Uder (B2), < for all =,,2, ( ( Thus, (B6) s rue, (B5) s rue for T =, ad (B) s rue.