The ages of extremal impact on life disparity caused by averting deaths

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1 Mx-Plnck-Institut für demogrfische Forschung Mx Plnck Institute for Demogrphic Reserch Konrd-Zuse-Strsse 1 D-1857 Rostock GERMANY Tel +49 () ; Fx +49 () ; MPIDR WORKING PAPER WP FEBRUARY 21 The ges of extreml impct on life disprity cused by verting deths Peter Wgner (wgner@demogr.mpg.de) This working pper hs been pproved for relese by: Jmes W. Vupel (jwv@demogr.mpg.de), Hed of the Lbortory of Survivl nd Longevity. Copyright is held by the uthors. Working ppers of the Mx Plnck Institute for Demogrphic Reserch receive only limited review. Views or opinions expressed in working ppers re ttributble to the uthors nd do not necessrily reflect those of the Institute.

2 The ges of extreml impct on life disprity cused by verting deths Peter Wgner Jnury 1, 21 Mx Plnck Institute for Demogrphic Reserch Konrd-Zuse-Strße Rostock, Germny e-mil: wgner@demogr.mpg.de Abstrct. This pper is concerned with sensitivity nlysis of life disprity with respect to chnges in mortlity rtes. Recently Zhng nd Vupel introduced threshold ge, such tht verting deths before tht ge reduces disprity, while verting deths fter tht ge increses disprity. We provide refinement to this result by chrcterizing the ges t which verting deths hs n extreml impct on life disprity. A procedure is given for pproching the threshold ge numericlly. The results re illustrted using dt for the femle popultions of Denmrk, the US, Jpn nd Frnce in 25. Contents 1. Introduction 2 2. Min results The effect on life disprity cused by verting deths Approching the threshold ge numericlly Illustrtions Constnt hzrds Numericl findings Proofs Proof of Theorem Proof of Theorem Appendix: Clculting the perturbed life disprity 14 References 15 1

3 1. Introduction Following Keyfitz s ide tht everybody dies premturely since every deth deprives the person involved of the reminder of his expecttion of life Key77, p.61-68, the mesure e for the verge life expectncy lost due to deth, hs widely been studied in the literture. It first ppered in Mit78 nd ws developed further by Vupel Vu86 nd recently in VCR3, ZV8 nd SAZ + 9. Zhng nd Vupel ZV9 initited new direction of nlysis, by studying the impct on e of concentrted decrese in mortlity t ge. Life disprity is mesured by life expectncy lost due to deth e = where e() is the remining life expectncy t ge e() = 1 l() e()d()d (1.1) d() is the life tble distribution of deths l(x)dx (1.2) d() = l()µ() (1.3) l() is the probbility of survivl to ge ( ) l() = exp µ(x)dx (1.4) nd µ() is the ge-specific hzrd of deth. Note tht, by equtions (1.4) nd (1.3), d 1 l() = l()µ() = d() (1.5) d Letting H() be the cumultive hzrd function H() = µ(x)dx (1.6) in ZV9, perturbtion is considered tht lters H() by the step function with single negtive step of size s t ge to get the new cumultive hzrd function H,s (x) = H(x) s 1, ) (x) (1.7) It cn be derived, s done in the Appendix, tht the corresponding new vlue of life disprity stisfies e,s = e + exp(s) 1l()e() H() 1 + e () e() 2

4 where e () is life expectncy lost due to deth mong people surviving to ge Now e () = 1 l() e(x)d(x)dx (1.8) s e,s = exp(s)l()e() H() 1 + e () e() nd s e,s = l()e() H() 1 + e () s= e() (1.9) In ZV9, the function k, where k() = 1 l() = e() s e,s s= H() 1 + e () e() = e () + e()h() 1 (1.1) is nlysed nd shown to hve positive first derivtive t every root, from which it follows tht k hs t most one root. It is shown tht k() < implies the existence of unique root (then, of course, positive), k() < for < nd k() > for >, nd tht k() implies k() > for >. Since, by eqution (1.1), k() nd s e,s hve the sme sign, nd since s= k() = e e(), this is equivlent to sying Theorem 1.1 There re the following three cses: e (i) < 1 e() > : < s e,s s= < = s e,s s= = > s e,s s= > e (ii) = 1 e() = : = s e,s s= = > s e,s s= > e (iii) > 1 : e() s e,s s= > 3

5 The unique root of k() nd clled threshold ge. s e,s s=, if it exists, tht is, if e 1, is e() In this pper we provide refinement to the bove result by chrcterizing the ges t which verting deths hs n extreml impct (or, in other words, effect) on life disprity. In ddition, procedure is given for pproching the threshold ge numericlly. The results re illustrted using dt for the femle popultions of Denmrk, the US, Frnce nd Jpn in 25. The pper is orgnised s follows. The min results re presented in Section 2. An explicit exmple s well s some numericl results using dt from the Humn Mortlity Dtbse re given in Section 3. The proofs of the theorems re collected in Section 4. A formul for the perturbed life disprity is derived in the Appendix. 2. Min results The function s e,s (cf. (1.9)) mesures the effect on life disprity of verting deths t ge. It turns out tht looking t this function directly llows s= us to mke interesting sttements bout its behviour, beyond those bout the sign obtined by looking t k (cf. (1.1)) nd summrised in Theorem The effect on life disprity cused by verting deths The min result of this pper is the following Theorem 2.1 The function ϕ, defined by ϕ() = s e,s (2.1) s= hs the following properties. (i) Let ã, the ge of cumultive hzrd unity, be defined vi H(ã) = 1 Then ϕ is strictly incresing on, ã nd strictly decresing nd strictly positive on ã, ), hving globl mximum of ϕ(ã) = l(ã)e (ã) t = ã nd locl minimum of ϕ() = e () e() t =. More precisely, d ϕ() = l()1 H() d 4

6 (ii) If, for some ε > nd A, we hve µ() ε A then lim ϕ() = (iii) Let be defined by H( ) = 2 Then ϕ is strictly concve on, nd strictly convex on, ). More precisely, d 2 ϕ() = d()h() 2 d2 Regrding the existence of ã nd, note tht, by eqution (1.6), H is strictly incresing with H() =, nd tht lim H() = is generlly ssumed. Theorem 2.1(i) implies tht ϕ hs t most one root, which exists if nd only if ϕ() e () e() e 1, so tht Theorem 2.1(i) e() implies Theorem 1.1. Refining Theorem 1.1, our Theorem 2.1 highlights some helpful monotonicity properties nd drws ttention to ã nd, the ges of extreml effect on life disprity cused by verting deths. Consider ges >, if exists, nd consider ny ge otherwise. By Theorem 1.1, life disprity increses by verting deths t ny such ge. More precisely, by Theorem 2.1(i), the increse in life disprity by verting deths is the higher the closer the ge is to ã, where the increse is highest. Consider ges <, if exists. By Theorem 1.1, life disprity decreses by verting deths t ny such ge. More precisely, by Theorem 2.1(i), the decrese in life disprity by verting deths is the higher the closer the ge is to, where the decrese is highest Approching the threshold ge numericlly According to Theorem 2.1(iii), since ã < clerly holds, ϕ is strictly concve on, ã. This is the property we cn mke use of for pproximting, provided such n pproximtion mkes sense, tht is, ϕ() <. First, we cn construct strictly incresing sequence tht pproches from below. Second, we cn construct strictly decresing sequence tht pproches from bove. Theorem 2.2 Assume e e() < 1 nd let be the threshold ge, defined by ϕ( ) =. 5

7 (i) The sequence (x n ), defined by x n+1 = x n + e(x n ) e (x n ) 1 H(x n ), x = strictly increses nd rpidly (qudrticlly) converges to. In prticulr, x 1 = e() e () is first non-trivil pproximtion of from below. (ii) The sequence (y n ), defined by y n+1 = y n e() e () l(y n )e(y n )H(y n ) 1 + l(y n )e (y n ) + e() e (), y = ã strictly decreses nd (linerly) converges to. In prticulr, y 1 = ã e() e () l(ã)e (ã) + e() e () is first non-trivil pproximtion of from bove. Concluding for x 1 nd y 1 of Theorem 2.2, we hve (x 1, y 1 ), ã. This, of course, does not locte precisely, but constitutes new non-trivil bounds on. First, we cn sy tht life disprity decreses by verting deths t ny ge x 1 = e() e () (with the decrese being the higher the closer the ge is to ). Second, we cn sy tht life disprity increses by verting deths t ny ge y 1 = ã increse being the higher the closer the ge is to ã). e() e () l(ã)e (ã)+e() e () (with the 3. Illustrtions 3.1. Constnt hzrds Constnt hzrds re unrelistic for humn popultions, where, roughly speking, between ges 3 nd 9 the hzrd is exponentil. However, hypotheticl popultions with constnt hzrds of deth cn be used to illustrte Theorem 2.1. The effect on life disprity cused by verting deths t ge should be strictly incresing from ge to ã nd strictly decresing (nd strictly positive) from ge ã onwrds. Indeed, suppose By eqution (1.6), H() = µ() = C µ(x)dx = C (3.1) 6

8 By equtions (1.4) nd (1.6), l() = exp( C) (3.2) By eqution (1.3), d() = l()µ() = C exp( C) By eqution (1.2), e() = 1 l() l(x)dx = exp(c) exp( Cx)dx 1 = exp(c) C exp( Cx) 1 = exp(c) C exp( C) = 1 C (3.3) By equtions (1.8) nd (1.5), e () = 1 e(x)d(x)dx l() 1 = Cl() 1 l(x) = 1 C (3.4) Finlly, by equtions (4.2), (4.1), (3.2), (3.3), (3.1) nd (3.4), ϕ() = l()e()ψ() = l()e() H() 1 + e () e() = exp( C) 1 C 1 + C = exp( C) Clerly, =. Since ã is defined by 1 = H(ã) = Cã, we hve ã = 1. It is C strightforwrd to see tht ϕ is indeed strictly incresing from ge to ã nd strictly decresing (nd strictly positive) from ge ã onwrds. 1 C 1 C 3.2. Numericl findings To illustrte the theoreticl results, we hve computed severl relevnt quntities for four life tbles from the Humn Mortlity Dtbse 21 Dt1. In the figures below the function ϕ, representing the effect on life disprity cused by verting deths (see Theorem 2.1), is displyed by solid line. The pproximtions x 1 nd y 1 of the threshold ge (see Theorem 2.2) re obtined by the intersections of the dshed lines nd the ge xis. 7

9 Figure 1: The effect on life disprity cused by verting deths for Dnish femles in 25 For femles in Denmrk in 25, the ge of cumultive hzrd unity ã is bout 87 yers, nd x 1, nd y 1 re pproximtely 7, 78 nd 85 yers, respectively, s shown in Figure 1. For US-femles in 25, ã is round 87 yers, nd x 1, nd y 1 re pproximtely 69, 79 nd 85 yers, respectively, s shown in Figure 2. For femles in Frnce in 25, ã is bout 9 yers, nd x 1, nd y 1 re pproximtely 74, 83 nd 88 yers, respectively, s shown in Figure 3. Finlly, for Jpnese femles in 25, ã is round 91 yers, nd x 1, nd y 1 re pproximtely 76, 84 nd 89 yers, respectively, s shown in Figure 4. Note tht the four obtined threshold ges concur with Figure 2 of ZV9. Note further tht our Figure 2 concurs with Figure 1 of ZV9. It is evident tht in ll four cses the pproximtions x 1 nd y 1 of lie within ten yers of, nd the rithmetic men of x 1 nd y 1 lies within two yers of. In fct, ã lso lies within ten yers of, nd the rithmetic men of x 1 nd ã even lies within one yer of. Whether the rithmetic men of x 1 nd ã is, in generl, better pproximtion of thn the rithmetic men of x 1 nd y 1, would be n interesting question for future reserch. If so, it would emphsise the importnce of ã, the ge of cumultive hzrd unity, nd, due to the simple definitions of x 1 nd ã, improve our understnding of. It should be mentioned tht in ll four cses, the extreml effects on life disprity cused by verting deths re fr from being equl. Indeed, for Denmrk, ϕ() = x nd ϕ(87) 1.44 with ϕ()/ϕ(87) 48.7, for the US, ϕ() = x nd ϕ(87) 1.56 with ϕ()/ϕ(87) 44.1, for Frnce, ϕ() = x nd ϕ(9) 1.29 with ϕ()/ϕ(9) 57.4, nd 8

10 Figure 2: The effect on life disprity cused by verting deths for US-femles in 25 Figure 3: The effect on life disprity cused by verting deths for French femles in 25 9

11 for Jpn, ϕ() = x nd ϕ(91) 1.34 with ϕ()/ϕ(91) 56.6, so tht, in bsolute vlue, the effect on life disprity by verting deths t ge zero is roughly 5 times lrger thn the effect on life disprity by verting deths t ge ã. Studying this rtio for other life tbles nd possible implictions might be nother interesting direction of future reserch. Figure 4: The effect on life disprity cused by verting deths for Jpnese femles in Proofs 4.1. Proof of Theorem 2.1 For proving Theorem 2.1, we need the following result. Lemm 4.1 The function ψ, defined by ψ() = H() 1 + e () e() (4.1) is strictly incresing. More precisely, d d ψ() = e () e() > 2 1

12 Proof. By equtions (1.6), (1.2), (1.8) nd (1.3), d d ψ() = d d H() + d d = d d = µ() + e () e() µ(x)dx + d d d d l()e () l()e() e(x)d(x)dx l()e() l()e () d d l()e() 2 = µ() + e()d() l()e() l()e () l() l()e() 2 = µ() d() l() + e () e() 2 = e () e() 2 > showing tht ψ is strictly incresing function. Proof of prt (i) By equtions (1.9), (2.1) nd (4.1), l(x)dx ϕ() = l()e()ψ() (4.2) First, by Lemm 4.1 nd eqution (4.1), we hve ψ() > ψ(ã) = e (ã) e(ã) >, for ll > ã. Consequently, by equtions (4.2) nd (1.2), ϕ is strictly positive on ã, ). Then, by equtions (4.2), (1.2), (4.1) nd Lemm 4.1, we hve d d ϕ() = d d l()e()ψ() { } d = d l()e() ψ() + l()e() d d ψ() d = l(x)dx H() 1 + e () + l()e() e () d e() e() 2 = l() H() 1 + e () + l() e () e() e() = l()1 H() (4.3) So the first derivtive of ϕ is strictly positive on, ã) nd strictly negtive on (ã, ). Thus, ϕ is strictly incresing on, ã nd strictly decresing on ã, ). Consequently, ϕ hs globl mximum t = ã, where its vlue, by equtions (4.2) nd (4.1), is ϕ(ã) = l(ã)e (ã) nd locl minimum t =, where its vlue, by equtions (4.2) nd (4.1), is ϕ() = e () e() (4.4) 11

13 Proof of prt (ii) By equtions (4.2), (4.1), (1.2), (1.8), (1.4) nd (1.6), ϕ() = l()e()ψ() = l()e() H() 1 + e () e() = H() = H() l(x)dx exp H(x)dx l(x)dx + l(x)dx + e(x)d(x)dx e(x)d(x)dx Clerly, lim l(x)dx = nd lim e(x)d(x)dx =, so tht it remins to show lim H() exp H(x)dx =. Let c = ε. Since µ() ε for ll A, we hve 2 nd H(x) cx H() c = H() c It follows from eqution (4.5) tht A exp H(x)dx = x µ(y) cdy x A (4.5) µ(x)dx c c 2cA A (4.6) exp{ H(x) cx cx}dx exp{ H() c} exp( cx)dx = 1 exp{ H() c} A (4.7) c One obtins further with equtions (4.7) nd (4.6) tht H() exp H(x)dx 1 H() exp{ H() c} c 1 H() c exp{ H() c} + c exp (c 2cA) A 1 By eqution (4.6), lim H() c exp{ H() c} =, nd clerly c lim exp (c 2cA) =. Hence lim H() exp H(x)dx =, which finishes the proof. Proof of prt (iii) By equtions (4.3), (1.5), (1.6) nd (1.3), d d d d ϕ() = d {l()1 H()} d 12

14 d = d l() 1 H() + l() d 1 H() d = d() 1 H() + l() µ() = d()h() 1 l()µ() = d()h() 2 So the second derivtive of ϕ is strictly negtive on, ) nd strictly positive on (, ). Thus, ϕ is strictly concve on, nd strictly convex on, ) Proof of Theorem 2.2 Note tht < ã nd, by Theorem 2.1(iii), ϕ is strictly concve on, ã. By Newton s method, the sequence (x n ), defined by x n+1 = x n ϕ(x n) d d ϕ(x n), x = strictly increses nd rpidly (qudrticlly) converges to. In our prticulr cse, by equtions (4.2) nd (4.1) nd Theorem 2.1(i), x n+1 = x n l(x n)e(x n )ψ(x n ) l(x n )1 H(x n ) e(x n ) H(x n ) 1 + e (x n) e(x n) = x n 1 H(x n ) = x n + e(x n ) e (x n ) 1 H(x n ) By the Regul Flsi method, the sequence (y n ), defined by y n+1 = y n ϕ() ϕ(y n ) ϕ(), y = ã strictly decreses nd (linerly) converges to. equtions (4.4), (4.2) nd (4.1), In our prticulr cse, by y n+1 = y n = y n = y n e () e() l(y n )e(y n )ψ(y n ) e () e() e() e () l(y n )e(y n ) H(y n ) 1 + e (y n) e(y n) + e() e () e() e () l(y n )e(y n )H(y n ) 1 + l(y n )e (y n ) + e() e () 13

15 5. Appendix: Clculting the perturbed life disprity Suppose tht the cumultive hzrd function H gets replced by H,s, defined in eqution (1.7). According to eqution (1.6), µ then gets replced by µ,s, where µ,s = µ s δ According to equtions (1.4) nd (1.6), l then gets replced by l,s, where l,s = l exp{s 1, ) } (5.1) According to eqution (1.5), it cn be shown tht d then gets replced by d,s, where d,s = {1 + exp(s) 1 1, ) } d + 1 exp(s)l() δ (5.2) Indeed, for x <, we hve x d,s(t)dt = x d(t)dt = 1 l(x) = 1 l,s(x). For x, we hve x d,s(t)dt = 1 exp(s)l() + d(t)dt + x exp(s)d(t)dt, where the right hnd side simplifies to 1 exp(s)l()+1 l()+exp(s)l() l(x) = 1 exp(s)l(x) = 1 l,s (x). According to eqution (1.2), e then gets replced by e,s, where By eqution (5.1), we obtin e,s (x) = For x <, we thus hve e,s (x) = 1 l,s (x) 1 (l exp{s 1, ) })(x) x x l,s (y)dy (l exp{s 1, ) })(y)dy (5.3) e,s (x) = 1 (l exp{s 1, ) })(y)dy l(x) x = 1 l(y)dy + exp(s)l(y)dy l(x) x = 1 l(y)dy l(y)dy + exp(s) l(x) x = 1 l(x)e(x) l()e() + exp(s)l()e() l(x) l(y)dy = e(x) + exp(s) 1 l() e() (5.4) l(x) Note tht if x < is very close to, then e,s (x) is pproximtely exp(s)e(). For x, by eqution (5.3), we hve e,s (x) = 1 exp(s)l(x) x exp(s)l(y)dy = 1 l(x) 14 x l(y)dy = e(x) (5.5)

16 Hence, by combining equtions (5.4) nd (5.5), e,s (x) = e(x) + exp(s) 1l()e() { 1,) l } (x) (5.6) Finlly, ccording to eqution (1.1), e then gets replced by e,s, where e,s = By equtions (5.2) nd (5.6), we obtin e,s (x)d,s (x)dx e,s = = e(x){1 + exp(s) 11, ) (x)}d(x)dx + e(x)1 exp(s)l()δ (x)dx + exp(s) 1l()e() 1,)(x) {1 + exp(s) 11, ) (x)}d(x)dx + l(x) exp(s) 1l()e() 1,)(x) 1 exp(s)l()δ (x)dx l(x) ( ) e(x)d(x)dx + exp(s) 1 e(x)d(x)dx + e()1 exp(s)l() + exp(s) 1l()e() d(x) l(x) dx + Furthermore, by using equtions (1.1), (1.3), (1.6) nd (1.8), we obtin e,s = e + exp(s) 1l()e () + 1 exp(s)l()e() + Acknowledgments exp(s) 1l()e()H() = e + exp(s) 1l()e() H() 1 + e () e() The uthor gretly thnks Jmes Vupel, Trifon Missov nd Zhen Zhng for helpful comments. References Dt1 Humn Mortlity Dtbse. electronic resource. University of Cliforni, Berkeley (USA) nd Mx Plnck Institute for Demogrphic Reserch (Germny). Avilble t or (dt downloded on Jnury 7, 21)

17 Key77 Nthn Keyfitz. Applied Mthemticl Demogrphy. New York: John Wiley, Mit78 S. Mitr. A short note on the Teuber prdox. Demogrphy, 15(4): , SAZ + 9 Vldimir M. Shkolnikov, Evgueni M. Andreev, Zhen Zhng, Jmes Oeppen, nd Jmes W. Vupel. Losses of expected lifetime in the US nd other delevoped countries: methods nd empiricl nlyses. MPIDR WORKING PAPER WP DECEMBER 29, 29. Vu86 Jmes W. Vupel. How chnge in ge-specific mortlity ffects life expectncy. Popultion Studies, 4(1): , VCR3 Jmes W. Vupel nd Vldimir Cnuds-Romo. Decomposing chnges in life expectncy: A bouquet of formuls in honor of Nthn Keyfitz s 9th birthdy. Demogrphy, 4(2):21 216, 23. ZV8 ZV9 Zhen Zhng nd Jmes W. Vupel. The Threshold between Compression nd Expnsion of Mortlity. Pper presented t Popultion Assocition of Americ 28 Annul Meeting, New Orlens, April 17-19, 28. Zhen Zhng nd Jmes W. Vupel. The ge seprting erly deths from lte deths. Demogrphic Reserch, 2(29):721 73,