Taxes and the present value assessment of economic losses in personal injury litigation: Comment 1

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1 Taxs and h prsn valu assssmn of conomc losss n prsonal njury lgaon: Commn 1 Sco Glbr Economcs Dparmn Souhrn Illnos Unvrsy Carbondal Carbondal, IL mal: offc phon: (618) cll phon: (618) , fax: (618) Augus 8, For hlpful commns on h prsn work I hank parcpans a h annual naonal mng of h Amrcan Acadmy of Economc and Fnancal Exprs (Las Vgas, 2012).

2 Taxs and h prsn valu assssmn of conomc losss n prsonal njury lgaon: Commn Absrac Andrson and Barbr (2010) provd a rcn dscusson of ax ffcs on conomc damags, for fornsc conomss and smlar xprs who supply h cours wh opnons on conomc damags. Andrson and Barbrs papr flls a vod n h fornsc conomcs lraur, by offrng a formal hory of how ax consdraons can mpac conomc damags. In h prsn work I pon ou a lmaon of hs hory - va a counr-xampl, and dscuss condons undr whch whch h hory sms o hold approxmaly. Kywords: ncom sram; prsn valu; ax; prsonal njury; wrongful dah 2

3 I. Inroducon In cour cass nvolvng prsonal njury and wrongful dah, conomc damags ypcally h prsn valu of fuur ncoms whch would hav bn arnd f no for h njury or dah. For h ypcal U.S. workr, s naural for hr ncom o grow ovr mos of hr worklf du o nflaon and ncrass n producvy. Whn valung a growng ncom sram, h fornsc conoms (FE) has varous facors o consdr, ncludng h ncluson or xcluson of ax -- n h workr s wags and also n nrs ncom assocad wh h nvsmn fund rqurd o rplca los fuur wags. Th FE drmns whhr or no o nclud ax or no dpndng on lgal rqurmns of h cour n whch h cas s fld, and on h FE s opnon abou h rasonablnss of pr-ax vs. pos-ax valus. In cass whr h ramn of ax s subjc o h FE s choc, s mporan o b abl o xplan h consquncs of such chocs o h cour. Dos h ncluson of ax nd o ras conomc damag smas, or no? Economc hory s a ponally valuabl rsourc for fornsc conomss who wsh o b abl o answr hs quson confdnly a ral and n dposon. In rms of conomc hory, h ffc of ax on h prsn valu of a growng ncom sram has no bn addrssd n h gnral conomcs lraur. Ovr h las 25 yars fornsc conomss hav publshd rsarch n journals spcalzd o hr fld, ncludng h Journal of Fornsc Economcs and h Journal of Lgal Economcs. In boh journals, a numbr of rsarch paprs hav dscussd ax and prsn valu. In h Journal of Fornsc Economcs, hs ncluds som arly work by Goodwn and Paul (1988) and Ccka (1989), and a spcal ssu n 1994 (volum 7, numbr 3) dvod o h subjc of ax and conomc awards. In h Journal of Lgal Economcs, h rcn work of Gary Andrson and Jol Barbr (2010) maks a numbr of conrbuons o h lraur, ncludng a dscusson of ax ffcs on

4 h prsn valu of srvcs and mdcal car. I also survys h xsng lraur, and poss a horcal rlaonshp bwn ax and h prsn valu of growng ncom. Andrson and Barbr (2010) pos ha ax has a ngav ffc on h prsn valu of growng ncom whn h numbr fuur pay prods s small, and has a posv ffc whn h numbr of prods s larg. Thy also provd a mahmacal formula for h brakvn pon -- h numbr of prods a whch h ax ffc swchs from ngav o posv. Ths horcal work s nsghful and xnds h scop of h fornsc conomcs lraur. Th prsn work rconsdrs h ffc of ax on h prsn valu of growng ncom. I show by counr-xampl ha h Andrson and Barbr (2010) ax hory dos no gnrally hold n h mahmacal form n whch hy sa, and I dnfy h sourc of rror. I also provd xampls whr hr brakvn formula s narly corrc, gvng som hns for fuur rsarch on h ffcs of ax on conomc damags. II. Andrson and Barbr (2010) Andrson and Barbr (2010), hncforh abbrvad AB, analyz h ffc of ax on h prsn valu of arnngs, undr h assumpon ha arnngs grow a a consan ra g ovr m: (2.1) E E0 g (1 ) wh E 0 h bas arnngs arnd n prod 0. Andrson and Barbr pos an nrsng hory abou how ax affcs h prsn valu of arnngs. I brfly summarz hr rsuls n Scon II.A, hn xamn hm n mor dal n Scons II.B-II.C. 2

5 A. Tax Effc Andrson and Barbr (2010) us h followng noaon and formulas for h prsn valu of arnngs, wh and whou ax: (2.2) E (1 )(1 g) 1 g ( r(1 ) g) 1 r(1 ) 0 arnngsafrax 1 (2.3) arnngswhouax E (1 ) 1 0 g g 1 ( r g) 1r whr r s h nrs ra on a (rsklss) bond, s h ax ra on arnngs, and s h ax ra on nrs. In rms of ax ffcs, AB pos a brakvn pon, such ha ax has a ngav ffc on prsn valu whn h numbr of arnngs prods s lss han som valu whch I wll call *, and has a posv ffc on prsn valu whn h numbr of prods xcds *. In ohr words, hy pos ha arnngsafrax s smallr han arnngswhouax whn s small, and ha h rvrs rlaonshp holds whn s larg. Thy dnfy h brakvn pon va h followng quaon: (2.4) rd (1 r) whr D s h consan: 3

6 (2.5) 1 g (1 r) 1 r D ( r g) 1 g 1 1 r Accordng o AB, for smallr h lf-hand sd of (2.4) s largr han h rgh-hand sd, and h afr-ax prsn valu of arnngs s lss han h whou-ax prsn valu. For largr, h lf-hand sd s smallr han h rgh-hand sd, and h afr-ax prsn valu of arnngs s grar han h whou-ax prsn valu. B. (Counr)Exampl Andrson and Barbr provd an xampl o llusra hr mahmacal hory. Thy s h ax ra on arnngs qual o ha on nrs, wh o h bfor-ax ra of nrs, wh g 0.1 r 0.05, and hy s h growh ra on arnngs qual. Thy fnd ha h brakvn condon (2.4) holds wh D 21 and ha hs valu of D s achvd a 40. Thy conclud ha, for h gvn valus of gr,, and, adjusmn for ax wll rsul n a dcras n h prsn valu of loss wh h nroducon of axs n suaons whr 40, and wll ncras h prsn valu of loss n suaons whr 40. Examnng h prsn valu formulas (2.2)-(2.3) and h dfnon (2.5) of h consan D, som roubl arss. Snc h dffrnc r g quals 0, h whou-ax prsn valu (2.3) and h formula (2.5) for D ar undfnd. hr can b a fn quany, as boh nvolv dvson by zro. Also, hy can no b rlably assgnd h valu plus nfny (+ ) or mnus nfny (- ), as boh valus mrg n h lm as w l r g convrg o zro from abov or, alrnavly, from blow. Consqunly, s 4

7 no possbl o chck h brakvn condon (2.4) n Exampl 1 va hs formulas, or o apply hs condon o h drmnaon of ax ffcs. Whl formulas and (2.3) and (2.5) do no apply n hs xampl, Andrson and Barbr provd alrnav formulas ha do, hs bng: (2.6) and: arnngswhouax 1 E0 (1 g) (1 r) (2.7) D 1 1 E0(1 g) (1 r) E0(1 g) (1 r) whch ar wll-dfnd for any posv valus of g, r apply h AB brakvn condon (2.4) o g:, and E 0. To chck h posd valu -21 for D, I (2.8) (1 r) D r 21 whch machs hr rsul. W can also chck h posd valu 40 for h numbr of fuur work prods. Gvn h valu for D, I solv for by applyng h formula (2.7) wh oal offs ( g r ): 5

8 (2.9) (2.10) (2.11) D ( 1) / Th valu * of ha sasfs h brakvn condon (2.4) whn s hn: (2.12) 1 r * 2 Wh r 0.05, (2.12) ylds * 41, whch s narly h sam as h valu 40 mnond by Andrson and Barbr. Turnng now o h posd ax ffcs, I compu prsn valus and rpor hm n Tabl 1, wh bas arnngs E 0 s qual o 1. A ach horzon, prsn valu whou ax quals du o oal offs. A 39, whou ax s largr han wh ax, hnc ax lowrs prsn valu. Th sam s ru a 40 hrough 42, whras Andrson and Barbr suggs ha for largr han 40 (or 41) ax should ras prsn valu. A 43 ax rass, hnc f hr s a brakvn valu of a whch h ax ffc gos from ngav o posv, s 42 or 43, whch s nconssn wh AB's brakvn condon (2.4). 6

9 C. Ohr Evdnc To furhr xamn h workably of h brakvn condon (2.4) for ax ffcs, consdr som mor xampls. As arlr, l hr b oal offs bwn h growh ra on arnngs and h nrs ra, wh g r 0.05, and l h ax ra on arnngs b qual o h ax ra on nrs. Th brakvn valu * of h arnngs horzon, whch I drvd arlr, s agan * 41 hr bcaus r has no changd. Tabl 2 rpors prsn valus wh and whou ax, analogous o Tabl 1, bu wh a lowr ax ra: As shown, ax lowrs prsn valu a 39,40,41, and rass a 42,43,44, whch s conssn wh Andrson and Barbrs' posd ax ffcs. A possbl xplanaon s ha hr hory s applcabl whn ax ras ar suffcnly small, bu no for all ax ras. As a furhr chck, I rpor n Tabl 3 rsuls for a hgh ax ra: 0.5. Hr h dscrpancy bwn posd ax ffcs and acual ffcs s dramac. Tax lowrs prsn valus a 39,,40,...,51, and rass prsn valus for 52,53,54, hnc f hr s a brakvn valu of mus b bwn 51 and 52, rahr han h valu * 41. Basd on h forgong xampls, a possbly s ha hr s a brakvn valu for whch ncrass wh h ax ra. In suppor of hs da, Tabl 4 rpors brakvn possbls for a h ax ras 0.01,0.05,0.1,0.2,...,0.5. A ach sad valu of, ax lowrs prsn valu for narby smallr valus of, and rass prsn valu for narby largr valus of. As ndcad, hs candda brakvn valus ar ncrasng n h ax ra. 7

10 III. Fornscs To xplan h parn of obsrvd dscrpancs bwn acual ax ffcs and hos posulad by Andrson and Barbr (2010), l's consdr h argumns undrlyng hm. Th crux of h mar s h brakvn condon (2.4), whch for AB rprsns a balancng of offsng ffcs: on h lf-hand sd of (2.4) s a loss of prsn valu assocad wh ax on ncom, whl on h rgh-hand sd s a gan n prsn valu assocad wh ax on nrs. Th valdy of hs balancng ac s no obvous, and nvolvs som rasonng n rms of lascs and also h consan D. Andrson and Barbr rfr o D as duraon, bu f duraon rfrs o m hn should b posv, or a las non-ngav, whras Andrson and Barbrs' D s always ngav. As a frs sp n rconsdrng h brakpon condon (2.4) I dscuss h concp of duraon, hn urn o lascs and h margnal analyss of offsng ax ffcs. A. Duraon Andrson and Barbr dnfy h consan D as duraon n h sns of Macaulay (1938). To g a sns of wha Macaulay mans by duraon n hs work, hr s a quo -- Macaulay (1938, Chapr 2, pag 48): ow, f prsn valu wghng b usd, h 'duraon' of a bond s an avrag of h duraons of h spara sngl paymn loans no whch h bond may b brokn up. To calcula hs avrag h duraon of ach ndvdual sngl paymn loan mus b wghd n proporon o h sz of h ndvdual loan; n ohr words, by h rao of h prsn valu of h ndvdual fuur paymn o h sum of all h prsn valus, whch s, of cours, h prc pad for h bond. For a rsklss bond payng a coupon amoun I n fuur prods 1, 2,...,, and rurnng a fac valu F n prod, wh gross yld R1 r, Macaulay compus hs duraon D as follows: 8

11 (3.1) I 2I I F 2 D R R R R I I I F 2 R R R R Ths D s a wghd avrag of h das 1, 2,...,, and as such s valu ls bwn h valus 1 and, a posv valu. Whl Macaulay (1938) focuss on bonds, hs concp of duraon -- as a prsn-valu wghd avrag of paymn das -- s applcabl o any known posv ncom sram ovr fuur das 1,2,..., : (3.2) D 1 wh oal prsn valu and h prsn valu of h arnngs arrvng n yar. For addonal dscusson of Macaulay's duraon, s Wl (1973). Appld o h bond xampl (3.1), h gnral duraon formula (3.2) has h followng componns: (3.3) I I I F R R 2 R R (3.4) 1 I R (3.5) I R 2 2 and so on, unl: I (3.6) 1 1 R 9

12 (3.7) I F R For a gnral arnngs sram E,..., 1 E, h (pr-ax) prsn valu of ncom arrvng n fuur prod s: (3.8) and Macaulay's duraon s: E (1 r ) (3.9) D 1 1 E (1 r) E (1 r) Macaulay (1938, pag 51) llusras bond duraon by compung for a vary of bonds wh dffrn ylds and yars o maury. In Tabl 5 I rpor duraon for growng ncom srams, wh D compud va formula (3.9) and arnngs growh ra g As n Macaulay's work, all duraon valus ar posv. In h las row of Tabl 5 I rpor rsuls n h lm cas whr horzon approachs nfny. For h cass g r s asy o show ha D as. For h cas g r I us Andrson and Barbrs' D formula (2.5), mulpld ms -1. John Hcks (1939, pag 186) ndpndnly dvlops h duraon formula (3.9), whch h calls h avrag prod of h ncom sram. Agan, duraon s a masur of m, hnc s non-ngav. Popular sofwar for fnancal calculaons, ncludng Mcrosof Excl 2010 and Wolfram Mahmaca 8, mplmn duraon n hs rms. For xampl, consdr a 5 yar bond ha pays no coupon and has 10

13 an nrs ra of 10 prcn. Is only paymn s a h nd of 5 yars, so Macaulay's duraon s D 5 n accordanc wh h followng Excl funcon: =duraon(01/01/10,1/1/2015,0,0.1,1). Could h rrors gnrad by Andrson and Barbrs' posulad ax ffcs b du o a wrong sgn -- ngav rahr han posv -- for hr Macaulay duraon formula? Th answr urns ou o b no, h rason bng ha f w rplac Andrson and Barbrs' consan D wh s ngav D n h brakvn formula (2.4), w g a nw formula whch can' work unlss h ngav sgn on h rghhand sd s slf rvrsd -- ohrws h posv valu on h lf-hand sd can no mach a posv sgn on h rgh-hand sd -- whch hn rvrs o h orgnal formula. Dsp hs fac, s hlpful o hav som undrsandng of Macaulay's duraon, as plays an ngral par n Andrson and Barbrs' basc argumn movang h brakvn condon (2.4). Ths argumn nvolvs lascs, and as shown n h nx scon h argumn s no qu rgh. I dos howvr conan usful lmns, and o apprca hs fully wll b hlpful o no hr ha, accordng o Hcks (1939), h duraon formula (3.9) can also b nrprd as h lascy of prsn valu wh rspc o h dscoun facor 1/ R. Tha s, wh prsn valu xprssd as: (3.10) 1 R E and s lascy dfnd as: (3.11) R R 11

14 Hcks shows ha lascy (3.11) s gvn by h duraon formula (3.9). B. Elascy and Margnal Analyss Consdr now h margnal analyss of ax ffcs. If h ngav ffc of ncom ax jus offss h posv ffc of nrs ax on prsn valu hn, a h margn, h chang d n prsn valu quals 0: (3.12) d d d 0 Suppos, morovr, ha h arnngs ax ra grows a h sam nsananous ra as dos h ax ra on nrs: (3.13) d d Thn w can hn nrpr h margnal condon (3.12) on prsn valu n rms of lascs: (3.14) Th lascy of (afr-ax) wh rspc o h arnngs ax ra s: (3.15) 1 12

15 For a small ax ra on arnngs, w can approxma hs lascy as follows: (3.16) On h ohr hand, h lascy of wh rspc o h nrs ax ra s: (3.17) (3.18) (1 r(1 )) (1 r(1 )) r 1 r(1 ) 1 r(1 ) (1 r(1 )) (3.19) r D 1 r(1 ) afrax wh D h varan of Macaulay-Hcks duraon D afrax basd on afr-ax arnngs and nrs: (3.20) D afrax 1 1 (1 ) E (1 (1 ) r) (1 ) E (1 (1 ) r) Th sp (3.19) follows from Hcks' lascy rsul mnond arlr. If h ax ras ar clos o zro hn afr-ax duraon s abou h sam as bfor-ax duraon: 13

16 (3.21) Dafrax D Applyng h small-ax approxmaons (3.16) and (3.21), h balancng condon (3.12) s approxmaly: (3.22) r D 1 r whch s h Andrson and Barbr brakvn condon (2.4) rsad as a small-ax approxmaon. A mnus (-) sgn appars on h rgh-hand sd of (2.4) bu no (3.22). Ths dffrnc n sgn rflcs h dffrnc bwn h Andrson and Barbr consan D and Macaulay-Hcks duraon D. Th upsho of hs analycs s ha Andrson and Barbrs mahmacal formulaon of a brakpon - for sgnng ax ffcs sms o b a workabl approxmaon whn h rlvan ax ra s suffcnly small. If so should b possbl o confrm hr mahmacal hory as a locally vald, n h nghborhood of a zro ax ra, hough a rgorous confrmaon of hs sor xcds h prsn scop. IV. Concludng Rmarks Th prsn work rconsdrs h ffc of ax on h prsn valu of fuur growng ncom srams, by dscussng a mahmacal hory proposd by Gary Andrson and Jol Barbr (2010). Ths auhors pos ha an ncras n h ax ra on boh arnngs and nrs ncom wll lowr h prsn valu of ncom srams f h work prod s suffcnly shor, bu wll ras prsn valu f h work prod s suffcnly long. Morovr, hy propos an xac brakvn pon a formula for h lngh of work prod a whch h ax ffc swchs from ngav o posv. Th prsn rsarch pons ou 14

17 ha Andrson and Barbrs brakvn formula s no gnrally vald, bu dos sm o b a good approxmaon whn h ax ra s suffcnly small. Fuur rsarch should b drcd o h quson of whhr or no Andrson and Barbrs gnral concpon of ax ffcs (on h prsn valu of growng ncom srams) s vald: s hr a brakvn pon a whch ax ffcs swch from ngav o posv, as h work prod lnghns? Whl a gnral formula for such a brakpon s unknown, h small-ax approxmaon skchd n h prsn work provds a nav sarng pon. For addd prspcv on ax ffcs, I llusra som possbls n Fgurs 1 and 2. Each fgur s a plo of whou-ax and afr-ax prsn valu of arnngs srams, for arnngs horzons 1,2,...,100. In Fgur 1 I s h arnngs growh ra and nrs ra ach qual o 0.05, and h ax ra qual o 0.5, as n h las xampl dscussd n Scon II.C and furhr dscrbd n Tabl 3. Hr h ax ra s hgh, bu h Fgur suppors h vdnc n Tabl 3 of a possbl brakpon n ax ffcs a 51 yars of fuur worklf. In Fgur 2 I s g r 0.03 and 0.2, whch mgh mor closly mach currn conomc condons. Hr h apparn brakpon s a 75 yars of worklf, and gvn ha vry fw popl work mor han 75 yars, ax lowrs h prsn valu of ncom srams for mos vryon n hs xampl. Boh Fgurs 1 and 2 suggs ha, whl ax ffcs may swch from ngav o posv as h work prod lnghns, whn h ax ra s no small h m o wa for h swch may b vry long, n whch cas h ffc of ax on prsn valu may ssnally b ngav. 15

18 Rfrncs Andrson, Gary A., and Jol R. Barbr (2010). Taxs and h prsn valu assssmn of conomc losss n prsonal njury lgaon, Journal of Lgal Economcs 17, Ccka, Jams (1989). Th consdraon of progrssv axs n prsn valu calculaons for prsonal njury and wrongful dah cass: Commn. Journal of Fornsc Economcs 2(2), 3-6. Goodwn, Randall and Chrs Paul (1988). Th consdraon of progrssv axs n prsn valu calculaons for prsonal njury and wrongful dah cass. Journal of Fornsc Economcs 1(1), Hcks, John R. (1939). Valu and Capal (Oxford: Clarndon Prss). Macaulay, Frdrck R. (1938). Som horcal problms suggsd by h movmns of nrs ras, bond ylds, and sock prcs n h Und Sas snc 1856 (Columba Unvrsy Prss for h aonal Burau of Economc Rsarch). Wl, Roman L. (1973). Macaulay's duraon: An apprcaon. Journal of Busnss 46(4),

19 Tabl 1: Prsn valus, Tax Ra = 10 prcn whou ax wh ax Tabl 2: Prsn valus, Tax Ra = 1 prcn whou ax wh ax

20 Tabl 3: Prsn valus, Tax Ra = 50 prcn whou ax wh ax Tabl 4: Brakvn Possbls au brakpon possbly

21 Tabl 5: Duraon horzon r=4% r=5% r=6% nfny nfny nfny

22 _TAX _OTAX Fgur 1: Prsn Valu of Incom Sram ( r g.05,.5 ) 20

23 _TAX _OTAX Fgur 2: Prsn Valu of Incom Sram, ( r g.03,.20 ) 21

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