A Theory of Tax Effects on Economic Damages. Scott Gilbert Southern Illinois University Carbondale. Comments? Please send to

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1 A Theory of Tax Effecs on Economic Damages Sco Gilber Souhern Illinois Universiy Carbondale Commens? Please send o gilbers@siu.edu ovember 29, 2012 Absrac This noe provides a heoreical saemen abou he effec of ax on he presen value of los income sreams. I consider he simple case of fla ax raes on earnings and ineres income. I approximae ax effecs via he insananeous rae of change - in presen value when he ax rae goes from zero o a small posiive number. In his seing I show ha presen value is lower before ax han afer ax when he earning sream is shor, wih he reverse oucome holding when he earnings sream is long. The swich poin, where he ax effec goes from negaive o posiive, depends on he heoreical model s inpus. I characerize he effec of inpus on his swich poin, and illusrae via an example of an injured railroad worker s claim of economic damages. Keywords: income sream; ax; presen value; or; personal injury; wrongful deah

2 1. Inroducion A common undersanding in he legal communiy is ha he ax effecs on personal injury damage awards are more plainiff issue when he plainiff is young, and more a defense issue when he plainiff is old. The reasoning is ha, for a young person wih a long fuure period of growing incomes, if awards are adjused for ax hen he posiive effecs of ineres ax inclusion come o ouweigh he negaive effecs of income ax inclusion, whereas he reverse is hough o hold for an older person. This undersanding is based on acual experience a rials, wih damage esimaes compared pre-ax and pos-ax, and so forms a sor of empirical rule. In erms of economic heory, Anderson and Barber (2010) ake up he ask of proving he validiy of he empirical rule of ax effecs on growing income. Gilber (2012) repors counerexamples o his heory, bu also describes a scenario where he logic migh work, namely when earnings are axed a a small (percenage) rae. The argumens in Gilber (2012) are informal, and he presen work aims o resae hem more formally and in greaer deph. In he remainder of his work, Secion 2 (Tax Effecs) presens he main resul abou ax effecs, and Secion 3 describes he swich-poin a which ax effecs go from negaive o posiive. Secions 4 and 5 characerize he link beween he heoreical model s inpus raes earnings growh and ineres on he swich-poin. Secion 6 applies he heory o an acual legal case involving an injured railroad worker, and Secion 7 concludes. Proofs of mahemaical resuls appear in he Appendix.

3 2. Tax Effecs As in Gilber (2012), consider a fuure pre-ax earnings sream E1, E 2,..., in fuure periods 1, 2,, growing a a consan rae g. In period 0, he presen value of he earnings sream is he lump sum of money which, when invesed a he risk-free ineres rae r, generaes he earnings sream. If neiher ineres nor earnings are axed hen presen value is before ax, while if boh ineres and earnings are axed a he same (consan) rae hen presen value is afer ax. The issue is wheher he inroducion of ax lowers or raises he presen value of earnings sreams. Gilber (2012) skeches an argumen in suppor of he idea ha ax lowers presen value for older workers (wih shorer anicipaed income sreams) while raising i for younger workers (wih longer anicipaed income sreams), provided ha he ax rae is close o zero. The following heorem provides a formal basis for ha claim. Theorem 1: Suppose ha a person suffers an income loss ha deprives hem of earnings ha would have grown a a consan posiive rae g over a finie worklife. Suppose also ha he ineres rae r is consan, and ha ax raes on earnings and ineres are equal. Then he effec of ax on presen value swiches from negaive o posiive a some horizon : for any T and all ax raes sufficienly small, he presen value of he earnings sream is lower afer ax han before ax when horizon, bu is higher afer ax when T. For an older worker, he earnings horizon is relaively shor, and Theorem 1 says ha he presen value of his worker s income sream is negaively impaced by he deducion of ax from earnings and 2

4 ineres. For a younger worker, Theorem 1 says he opposie. In boh cases, he resul is rue only for ax raes ha are sufficienly small. 3. Swich-Poin The heory in Secion 2 esablishes he exisence of an earnings horizon a which ax effecs on presen value swich from negaive (for shorer horizons) o posiive (for longer horizons). To be more explici abou his swich-poin, le me briefly summarize he logic underlying Theorem 1. As deailed in he Appendix, he insananeous effec of ax on presen value presen value ( PV ) is given by he derivaive of derivaive of PV wih respec o. Compuing his derivaive, and seing i equal o zero, esablishes a siuaion where he insananeous ax effec is zero as in equaion A.2 in he Appendix. This equaion can be usefully recas in erms of Macaulay duraion 1 D : (1) D 1 1 E (1 r) E (1 r) which measures he (weighed) average waiing ime unil fuure (pre-ax) earnings are received. In erms of D, A.14 in he Appendix: 2 is he closes ineger approximaion o he following equaion which is also equaion (2) 1 D 1 r 1 See Macaulay (1938) and also Hicks (1939). 2 If here are wo such bes approximaions hen hey are successive inegers. Le 3 be he smaller one.

5 A closed-form soluion o he swich-poin equaion (2) is no generally available, bu consider he case of oal offse, where earnings growh rae g equals he ineres rae r. Then duraion D equals ime horizon, in which case (2) has soluion: (3) 1 r 1 4. The role of earnings growh Having characerized he effec of ax on he presen value of growing income sreams, i is useful o consider how he rae g of earnings growh influences he resul. The value of g is poenially imporan. If here is no growh ( g 0 ) hen ax adjusmen lowers presen value see Samuelson (1964), regardless of he earning horizon. On he oher hand, wih growh ( g 0 )he effec of ax is mixed, lowering presen value a shor horizons, raising i a longer horizons, in he sense described by Theorem 1. One way o raionalize hese disparae resuls is o conjecure ha he swich-poin where ax effecs go from negaive o posiive, depends on he growh rae g and increases as g falls oward 0. The remainder of his secion is devoed o confirming his conjecure., To deermine he effec on of changes in g, I ll use he swich-poin equaion (2) in he previous Secion. In ha equaion, is he value of ha equaes duraion D o a funcion of he ineres rae. To see how g changes rae of earnings, as in Gilber (2011, equaion 8):, we can firs see how g changes D. To his end define he ne growh (4) 1 g h 1 1 r 4

6 Then Macaulay s duraion D akes he form: (5) D 1 1 (1 h) (1 h) I is hen possible o deermine he effec of changes in h and on D, as follows: Lemma 1: Duraion D is increasing in horizon and ne growh rae h. Reurning o he original problem, we can link a change in g o a change in h, a change in h o a change in D, and a change in D o a change in he soluion o he swich-poin equaion ( 2). I sae he resul as a heorem, wih deails in he Appendix. Theorem 2: Under he condiions of Theorem 1, he swich-poin earnings horizon earnings grow a a slower rae g. lenghens when For a worker wih low projeced earnings growh, Theorem 2 says ha ax adjusmen will have a negaive effec on presen value unless he earnings horizon is long. 3 Wih a higher growh rae, posiive ax effecs become more plausible. 5. The role of ineres Having linked earnings growh o he ax effec swich-poin, we can similarly aemp o link he ineres rae r o. To his end we can ry adap he logic of Secion 4, by showing ha a change in r 3 As earlier, ax effecs are assered only for ax raes τ ha can be considered sufficienly small ha Theorem 1 applies. 5

7 changes ne growh rae h, a change in h changes duraion D, and a change in D changes he soluion o he swich-poin equaion (2). Indeed, an increase in r clearly decreases h, and by Lemma 1 his decreases D, causing a drop in he lef-hand side of (2). Bu i also causes he righ-hand side of (2) o fall, in which case he ulimae effec on equaion-balancing is no eviden. Taking a differen roue, noe firs ha since duraion D is a (weighed) average of ime periods, i is no larger han he ime horizon. This fac, ogeher wih he swich-poin condiion (2), implies a bound on swich-poin in erms of he ineres rae: (6) 1 1 r 1 According o (6), if he ineres rae falls oward zero hen he swich-poin increases o infiniy. In oher words, if he ineres rae is low hen ax effecs on presen value are negaive excep a long horizons. The bound (6) on is sharp, as i is achieved as an equaliy in he case of oal offse beween earnings growh and ineres see (3). Wih oal offse, here is a negaive relaionship beween r and. The relaionship holds more generally, and can be shown by aking he derivaive of presen value PV wih respec o boh and r, hen signing i. I sae he resul as follows, wih proof provided in he Appendix. Theorem 3: Under he condiions of Theorem 1, he swich-poin earnings horizon ineres rae r. is decreasing in he 6

8 6. Example To illusrae he heory se forh in Secions 2 hrough 5, consider he following legal case. In he year 2005 a railroad worker was hur on he job in he sae of Illinois. The worker, a 48 year-old male, sued a railroad company for loss of fuure income caused by his injury on he job 4. A rial was held in he year As his is a railroad case i falls under FELA 5 rules, wih los earnings measured afer-ax. An economis served as exper winess and esimaed economic loss, and assumed an earnings growh rae g = 0.051, an ineres rae r = 0.066, earnings horizon = 17, ax rae = 12.23%, and base salary E0 = $61,061. To illusrae ax effecs, in Table 1 I show he presen value of he railroad worker s los fuure income sream on pre-ax and pos-ax bases, for earnings horizon ranging from 1 hrough 40 years, wih base salary normalized 6 o $1 and wih equal ax raes on earnings and ineres. For horizons 1 hrough 34 years, subracion of ax has a negaive effec on presen value. For horizon 35 and beyond, he reverse holds. According o Theorem 1, if he ax rae is sufficienly small hen ax effecs should swich signs a horizon defined via he breakeven condiion (2). To deermine he las wo columns show relevan inpus, including Macaulay-Hicks duraion D and he difference 1 D 1 r whose sign swiches a. According o he able, = 34. Theorem 1 predics ha ax will have negaive effecs 4 This case, in which I had no involvemen, was ried in he Firs Judicial Circui of he Illinois sae cour, deails available upon reques. 5 Federal Employers Liabiliy Ac. 6 Tha is, each normalized presen value in Table 1 mus be muliplied by acual base salary ($61,061) o ge acual presen value. 7

9 a horizons 1 hrough 34, posiive effecs a horizons 35 up o some poin T whose value remains unspecified. The heory is consisen wih he acual presen value resuls in Table 1. According o Theorem 2, a lowering of he earnings growh rae g should raise he swich-poin ime horizon. As a check, when I cu he value g in half, rises o 42. When I cu g in half again, rises o 52, consisen wih Theorem 2. Similarly, when he iniial g is doubled, and when g is quadrupled falls o 22. falls o 27, According o Theorem 3, he swich-poin should fall when he ineres rae rises. In he case a hand, when I cu he ineres rae in r half, he swich poin rises from 34 o 53, suggesing a negaive relaionship beween r and. Similarly, cuing r in half again raises o 85, hese being bigger changes han when I cu earnings growh g in half and in quarers. Doubling, and quadrupling, he saring value of r lowers increased earnings., firs o 21, hen o 13, hese being bigger changes han when I 7. Conclusion This work has provided a specific formal sense in which he empirical rule of ax effecs is righ: deducion of ax from earnings and ineres lowers he presen value of fuure income sreams for younger plainiffs, bu raises presen value for older plainiffs. The heory, based on a highly simplified ax model, is also hedged in wo ways, in erms of he earnings horizon and he ax rae. 8

10 The heory can, in principle, be generalized by making a global mahemaical comparison of wih-ax and wihou-ax presen values, raher han he local comparison (wih ax rae near 0) provided here. Along hese lines, an analysis in coninuous-ime would make clearer he iming of he swich-poin I have discussed. I leave his agenda o fuure research. 9

11 References Anderson, Gary A., and Joel R. Barber (2010). Taxes and he presen value assessmen of economic losses in personal injury liigaion, Journal of Legal Economics 17, Gilber, Sco (2012). Taxes and he presen value assessmen of economic losses in personal injury liigaion: Commen. Unpublished manuscrip, Souhern Illinois Universiy Carbondale. Hicks, John R. (1939). Value and Capial (Oxford: Clarendon Press). Macaulay, Frederick R. (1938). Some heoreical problems suggesed by he movemens of ineres raes, bond yields, and sock prices in he Unied Saes since 1856 (Columbia Universiy Press for he aional Bureau of Economic Research). Samuelson, Paul A. (1964). Tax deducibiliy of economic depreciaion o insure invarian valuaions. Journal of Poliical Economy 72,

12 APPEDIX Proof of Theorem 1 Wih fuure income growing a a consan posiive rae g, and wih ax on earnings and ineres each a rae he presen value of he fuure earning sream, compued in afer-ax erms, is as follows: (A.1) PV E0 1 (1 )(1 g) (1 (1 ) r) ormalizing base earnings E0 o equal 1, and inerpreing presen value as a funcion of he ax rae, he derivaive of presen value wih respec o he ax rae is: (1 g) r(1 )(1 g) (A.2) PV 1 (1 r(1 )) (1 r(1 )) 1 1 Evaluaing he derivaive a 0, and denoing he resul as PV, yields: (A.3) PV 1 r 1 g 1 1r 1r To describe he derivaive PV as i relaes o he ime horizon, define he consan: (A.4) # 1 r r 11

13 r In he derivaive formula (4.2), noe ha he erm 1 1 r is negaive for #, posiive for #, and zero for #. Consequenly, he derivaive PV is negaive a 1, falls as increases so long as # #, and increases wih y for all. While he derivaive PV sars ou negaive for small, i is no obvious wheher i ever reaches 0 or a posiive value when ges large. The answer depends on wheher he evenually-posiive erms in he sum appearing in (4.2) are sufficien o offse he iniially-negaive erms. To check, consider firs he case of oal offse, where g r. Compuing: (A.5) PV 1 r 1 1 r (A.6) 1 r r Here PV is posiive for all larger han #. If oal offse fails, meaning ha g r, hen wih some algebra we can rewrie PV as follows: (A.7) (A.8) r PV a a 1 r 1 1 r a 1 a a(1 a ) a 1 r 1 a 1 a 1 a 12

14 where a is he raio of pre-ax gross earnings growh o gross ineres rae: (A.9) 1 g a 1 r Rearranging erms in (A.8), and expressing he resul in erms of g and i, we have: g(1 g) 1 g 1 g r(1 g) 1 r (A.10) PV 2 ( r g) 1 r r g (1 r)( r g) i g If g value: r hen, for large, (A.10) implies ha PV is evenually posiive, converging o a posiive g(1 g) (A.11) lim PV y 2 ( r g) On he oher hand, if g i hen PV diverges o : (A.12) r(1 g) 1 g PV (1 r)( g r) 1 r hence PV is again posiive for sufficienly large. Given he behavior of he derivaive PV as he earnings horizon varies, for each income growh rae g and ineres rae i here mus be a hreshold value a y for, such ha he derivaive PV is 13

15 negaive for less han and posiive for greaer han one ha makes he derivaive PV as close o zero as possible. Ideally:. This hreshold value for is he (A.13) PV 0 In ligh of (A.3) and he definiion of Macaulay duraion D, we can rephrase he resricion (A.13) as follows: (A.14) 1 D 1 r The hreshold value mos nearly so. for he earnings horizon is he posiive ineger ha makes (4.13) rue, or To apply his characerizaion of he derivaive PV o he impac of small ax on he presen value of earnings sreams, noe ha presen value PV is a smooh funcion of he ax rae, and so admis a Taylor series approximaion in he neighborhood of 0 : (A.15) where (0) PV PV PV u (0) PV denoes before-ax presen value, PV is he derivaive of PV evaluaed a 0, and u is a remainder erm: (A.16) 2 u 2 PV ( ) 14

16 for some in he inerval [0, ], wih PV ( ) he second derivaive of PV wih respec o, evaluaed a. If we could bound he remainder u uniformly in hen we could apply a linear approximaion o P () wih accuracy ha is uniform in, and his would allow us o sign he effecs of (small) ax on he presen value of earnings sreams. However, o do so i would be necessary o bound he second derivaive PV ( ) uniformly in and also uniformly in for all beween 0 and. I will no aemp his ask here, bu insead noe ha for any given upper limi T on we can bound he second derivaive uniformly in for 1,2,..., T, as he second derivaive is coninuous for each. Wih an upper bound on he earnings horizon, linear approximaion o he presen value PV hen yields Theorem 1. Proof of Lemma 1 D is a weighed average of periods, a fac ha we can make more explici as follows: (A.17) D w 1 wih weighs w : (A.18) w (1 h) s1 (1 h) s 15

17 By consrucion, he weighs w,..., 1 w sum o 1. For larger, weigh is shifed oward laer periods, meaning ha w rises for larger and falls for smaller. To check his, consider he case 1 versus 2. In he former case w1 1. In he laer case, w h h h and he weigh w2 2 1 (1 ) / ((1 ) (1 ) ) is posiive, resuling in less weigh on smaller and more weigh on larger. The comparison generally of k versus k 1, for any couning number k, analogous. To deermine he effec of an increase in h on D, i suffices o show ha he increase shifs weigh from smaller o bigger. To his end, noe ha he raio of w in year y o w ws in any oher period is: s h (A.19) 1 w w s An increase h in herefore raises his raio if s, as was o be shown. Proof of Theorem 2 An increase in he earnings growh rae g raises he ne growh rae h. In urn, an increase in h raises duraion D by Lemma 1. If horizon had exacly saisfied he swich-poin equaion (3) in he ex before he increase in g, i can no longer do so afer he increase since he lef-hand side of (3) has risen bu he righ-hand side has no changed. To make he lef-hand side equal o or less han he righ-hand side, i suffices o lower D. By Lemma 1, we can do his by lowering up 16. In oher words, when g goes goes down, or is a leas non-decreasing. If no exacly saisfies (3), and we ake he value of ha mos nearly saisfies (3), we ge he same resul. o mean

18 Proof of Theorem 3 Differeniae (A.2) wih respec o r o ge: (A.20) wih componens A and B defined as: 2 PV r AB (1 g) (1 ) r (1 ) (1 r(1 )) 1 r(1 ) (A.21) A (A.22) B 1 2 (1 g) (1 ) (1 ) r (1 r(1 )) 1 r(1 ) (1 r(1 )) 2 Rearranging erms, simplifying he resul, and evaluaing a 0 yields: (A.23) 2 (1 g) ( 1) r PV 2 r 1 (1 r) (1 r) which is posiive. Therefore, an (incremenal) increase in r raises he value of he (firs) derivaive PV. Consider now he swich-poin ime horizon goes up hen so does PV, in which case, and suppose ha i solves equaion A.2 exacly. If r no longer solves (A.2). Also, noice ha when is evaluaed a 0, he resul (A.3) is increasing in. Therefore, hose smaller han PV will have 17

19 lower PV when evaluaed a 0. One of hese may solve (A.2) exacly, in which case his is he new (lower) value of. If no solves (A.2) exacly, hen need no change when r rises, bu remains a weakly decreasing funcion of r. 18

20 Table 1: Tax Effecs and Earnings Horizon horizon pv pos-ax pv pre-ax pv diff. D D (1 + 1/i)

21 20