Integration of Unemployment Insurance with Pension Through Individual Savings Account

Transcription

1 Intgation of nmloymnt Inanc ith Pnion Thogh Inivial Saving Accont Joh Stiglitz Datmnt of Economic Stanfo nivity Phon: Fax: Jngyoll Yn Datmnt of Economic Eha nivity Sol Koa Phon: Fax: GENERAL FIELD: Lao Economic icoconomic ABSTRACT Thi a analyz a ocial inanc ytm that intgat nmloymnt inanc ith nion ogam thogh an inivial aving accont alloing ok to oo againt thi ft ag incom an th imov thi ach incntiv hil cing ik. Taking into accont th avantag of tax-fn inanc thi a intifi facto on hich th otimal g of nion-fn lf-inanc thogh an inivial aving accont n. W ho that hn nmloymnt ation i vy hot coma to th io of mloymnt o timnt th otimal ytm involv an xcliv lianc on nion-fn lf-inanc hich yil a ngligil ik n fo ok ithot attnating thi ach incntiv. W alo coni a ca of mltil ik - nmloymnt an iaility - in hich th intgation of oth nmloymnt an iaility inanc ith a nion ogam thogh a joint aving accont i ial nl th to ik a fctly colat to ach oth. Financial ot of th Wol Bank i gatflly acknolg.

2 . Intoction Th Eat Aia cii oght hom to mch of th vloing ol a lon that th Gat Dion oght hom to mo avanc conti vnty ya ago th imotanc of a afty nt. Bt a conti lik Koa go aot contcting thi afty nt thy a cognizant of th comlaint that hav n ai againt nmloymnt inanc ytm: thy attnat incntiv. To th a av incntiv ffct o a thy a toay gnally f to moal haza ffct in all inanc ogam. What oi citic i that th ik ction nfit might on th fac of it otigh y th av incntiv ffct. Fo mot inivial a tyical ll of nmloymnt i l than ix month an that ll ol maly hot oily mch hot in th anc of nmloymnt inanc. Ov a oking tim of ay foty fiv ya an inivial ith th ch ll ol lo ha 4% of hi liftim incom a ik hich maly th inivial col aily ao if h ha fficint aving o col oo againt ft aning. With th lk of aving fo timnt an motly icat to ocial city ogam th amont inivial hav to ff thmlv againt th incom hock i limit; an ll ocmnt limitation in caital makt mak it ifficlt fo inivial to oo mch againt ft aning. Comloy ol ag lic nion ogam hil thy hl olv on olm -th tnncy of inivial not to av nogh fo thi ol ag ca of th oclivity of lic ail-ot - xacat anoth. Thi natally la to th ggtion of an intgat nmloymnt an nion ogam. Sch intgation mak aticla n ith th ytm of inivial accont hich a incaingly foming th ai of vn lic nion ogam. In ch ogam nfit a lat to contition y iml fomla; in th imlt th i no itition. Sch ogam a lik fin contition nion ogam thogh om of th contition can to cha inanc.g. againt inflation o intt flctation hich i not availal on th makt. Bt it i ay to imo itition on to. Fo imlicity in thi a ill igno th ititiv comonnt. n th intgat nmloymnt-nion ytm thogh an inivial aving accont an inivial ho i nmloy can hav hi nmloymnt aymnt takn ot of hi inivial accont. Th th inivial otain th liiity to maintain hi tana of living; th comloy an nival nat of th contition ovi in ffct fct collatal i.. aly on in hi lif hi alanc col go ngativ. Th fact that nomally th ik i mall man that th inivial can a thi ik hn it i a ot ov hi nti lif; an inc th inivial a th ik th i no attnation of ach incntiv. If hov th lo fom nmloymnt i lag nogh it i otimal to hav om tax-fn nmloymnt inanc th inivial hol not a th cot vn ov hi liftim. In gnal inivial hol not ly xclivly on th aving-fn lf-inanc n th intgat ytm. Th ytm hol lmnt y a taxfn ogam. Taking thi into accont chaactiz in thi a th otimal nfit tct of th intgat ytm that i th otimal comination of th to ty of nfit tax-fn an aving-fn. Th lo th ik-avion o th

3 gat th laticity of ach ith ct to th inanc nfit th l lianc hol lac on th tax-fn inanc a oo to hat might vi a imlicit aving-fn lf-inanc. In an xtm ca if a ok i ik-ntal thn th hol no n fo tax-fn nmloymnt inanc an if th i no incntiv olm aociat ith nmloymnt inanc th i no n to ly on aving-fn lf-inanc. Not iingly th lag th ik hich in tn i lat to th lngth of th io of nmloymnt lativ to th oking io th gat th n fo tax-fn inanc. In th limit if th io of nmloymnt i vanihingly mall thn th inivial can a all th ik thogh aving-fn lfinanc ith no lfa lo. Th mo intting olm ai hov hn inivial fac mltil ik. W coni a imlifi ca h th a to ik on aly in lif nmloymnt an on lat in lif iaility h th con ik i lag nogh that if th inivial ay fo hi aly ot of nmloymnt ot of hi liftim aving h cannot affo to ay fo th lat. Th intgat liftim inanc ytm hich intgat oth nmloymnt inanc an iaility inanc ith nion ogam n a joint aving accont ha to notal act. Fit th aving-fn nmloymnt nfit can gnat av iincntiv to th oiility of a govnmnt ailot in th ca of a ngativ alanc. Th inivial ill kno that if h o not ach th nt aymnt fom th govnmnt hol h com ial ill gat. Th vn n th flly intgat ytm fn olly y aving h inivial alay ay fo th initial ot of nmloymnt mingly ot of thi inivial accont th i om attnation of incntiv. Thi olm ill nv ai if th to ik a fctly ngativly colat t it i mch mo likly to occ hn th to ik a highly colat. Scon th intgat ytm thogh joint aving accont nal th iaility inanc to on iffntly to iffnt alanc in th inivial aving accont an th i lfa-imoving. Thi nfit of intgation having a common ool fom hich to a on gt lag a th colation gt mall. Whn th to ik a fctly colat a ingl fn ill claly o a ll a to fn imly ca th i in ffct a ingl ik. W a al to ho that o long a th colation tn th to ik i not fct thn it alay ay to hav om g of intgation that i om of th nmloymnt an/o iaility nfit hol com ot of inivial aving accont. Not iingly th gat th g of colation th l lianc on inivial accont n th intgat ytm. In th nxt ction nt th aic mol fo th intgat ytm to chaactiz it otimal nfit tct an to ho ho it vai ith th lativ iz of nmloymnt ik an oth aamt. Sction 3 alo nt a iml mol fo an intgat liftim inanc ytm that intgat oth nmloymnt an iaility inanc ith nion aving thogh a joint aving accont an xamin ho it otimal tct chang ith th colation tn iffnt ik. Som concling mak a givn in Sction 4.. Th ol 3

4 Coni an infinitly-liv ok ho n io of oking an ti thaft. In io a ok com nmloy ith oaility an th lngth of nmloymnt ol n on hi ach ciion. In thi a am that a ok ith nmloymnt hock can choo ith no ach o ach hich la him ith to nmloy fo on io o to nmloy fo zo io i.. to mloy immiatly aft nmloymnt hock ctivly. Th ning on hi ach ciion a ok ith nmloymnt hock can ith nmloy in io o avoi maining nmloy. Th cot of ach i a anom vaial hich i itit ith itition fnction F. Th ach ciion y a ok i ma thogh hi choic of th thhol ach cot ch that h choo to ach o not to ach if o >. Th th oaility of ing nmloy in io ol -F. Th incom ot ytm fo th nmloy in th mol i th on that intgat nmloymnt inanc ith nion thogh a timnt aving accont RSA. Th nmloymnt nfit ovi y th intgat ytm com fom to oc: nmloymnt tax an th at an octiv aving of a ok that i manat to ma in hi timnt aving accont RSA. A citical oint of thi ytm i that a ok can oo againt hi ft timnt aving to financ at of hi nmloymnt nfit. Any ngativ alanc in on RSA a ca hich ill alt ith in th nxt ction ill ail ot y th govnmnt. In aition to th minimm lvl of aving manat y th govnmnt an mloy ok may mak volntay aving to hi RSA. It i am in thi ction that th timnt aving of a ok i gat than th manat lvl o that a ok may alay n ith a oitiv alanc in hi RSA at th tim of timnt. Thi ol aonal ca in thi ction th nmloymnt i th only ik a ok ha in hi liftim. A ok may alo contit to hi ivat aving accont fom hich h can itha at anytim. In io a ok may ant to av om mony in hi ivat aving accont to financ any aitional conmtion ing nmloymnt that i in xc of th nmloymnt nfit ovi y th govnmnt. 3 Not hov that a ok ol not n a ivat aving accont aft io ca any aving h accmlat aft io ol fo timnt in thi mol. 4 Lt an n th -nmloymnt aving at in io th otnmloymnt aving at fo tho ith nmloymnt xinc an th otnmloymnt aving at fo tho ithot nmloymnt xinc ctivly. W ill mak a col of imlifying amtion in th mol. Fit I tax i ai in io only fo nmloymnt hock. Scon th i no iconting ing th fit io of on ca hil oth incom an tility ing th timnt io a icont at th am at. Th xct tility of a ok ith ag V ; ol thn In th nxt ction a ok fac mltil ik incling nmloymnt an thfo h may n ith a ngativ alanc in hi RSA to a ction in aving ca y anoth nt hock. 3 It ill hon lat y Pooition hov that a ok ol not n a ivat aving accont vn in io. 4 Not that in thi mol th i no ik a ok fac in hi liftim oth than nmloymnt. 4

5 V ; ax n ax h I J ; v n ; t I J ; n ax ax n { n { } F t t hil i th icont at ing timnt io F an t. } Not that I. o J. inicat th ayoff that i xct at io fom not ing nmloy o fom ing nmloy in io ctivly. Haft fo imlicity ill th ag notation in th agmnt of tility fnction xct hn it i ncay. On thing that hol mntion h i that th ayoff J. m that an nmloy ok conm jt th amont of th ocially otimal conmtion lvl ovi y th govnmnt. Thi ill ov lat y Pooition. Th nl th govnmnt ovi a ok ith an nmloymnt nfit l than th ocially otimal amont of conmtion h ol not n to mak cationay aving in ivat aving accont in io againt oil nmloymnt in io. Chaactization of Otimal Bnfit Stct of Intgat Sytm In chaactizing th otimal nfit tct of th intgat ytm ill am fo th momnt that th govnmnt ovi a ok ith th fll amont of th otimal aving-fn nfit fom th RSA. Thi imli that nl i l than hat a ok ol lik to conm hil nmloy h ol t all hi -nmloymnt aving in hi RSA fo hi timnt conmtion. W ill chang thi amtion lat hov to ho ho mch govnmnt oviion of avingfn nfit i n hn a ok itha -nmloymnt aving fom hi ivat aving accont to financ hi nmloymnt conmtion. Lt tat ith chcking th choic of aving an th thhol ach cot n y an inivial ok. Fit th ciion on ot-nmloymnt aving at hich ill ma to maximiz th ayoff I. an J. yil th folloing lt:. t t 5

6 n o n { n } 3 { }. 3 Nxt th aving ciion in io ill ma a follo: t { n } { } 4 Not fom 4 that th aving in io i tmin a a at of timnt aving not a a cationay aving againt nmloymnt ik. In collcting comaativ tatic of th -nmloymnt aving ill o that i o mall lativ to an that. Thn a th folloing Lmma ho th ffct of om aamt on -nmloymnt aving a not ignificant. Lmma i 5 ii. iii a. Th oof of thi Lmma i in th Anix. Fit th tax-fn nfit ol c -nmloymnt aving ca of th tax n hil th aving-fn nfit ol not affct th aving. 6 Scon th -nmloymnt aving may alo n on oth aamt ch a an. A th ot-nmloymnt hock io 5 Thi lt nal to igno th inict ffct of any aamt chang on aving thogh it ffct on th ach ciion. 6 Th aving-fn nfit col affct th aving in io ca a ok might inca hi aving in on to th ction in RSA alanc. Hov th magnit of thi ffct ol vy mall ca -nmloymnt aving i jt a mall otion of liftim aving an ca th oaility of nmloymnt hock i in fact mall. 6

7 lngthn -nmloymnt aving ol ca fo it ol act a a titt fo ot-nmloymnt aving in contiting to timnt aving. Th long timnt io o th mall ol ly inca th -nmloymnt aving vn if it atio to th ot-nmloymnt oking io i kt contant a a. Th high oaility of nmloymnt ik ol c th aving to th high tax n. 7 A ok alo mak hi ciion on th thhol ach cot to maximiz V taking th tax t a givn. I J ;. 5 W can collct om comaativ tatic of th thhol cot of ach ffot. Lmma i { n }. iv v. { } Th oof a in th Anix. Fit th -nmloymnt aving ol affct ach ffot ciion ngativly ca a ok ith gat ivat aving ol not tak a io th ction in hi RSA alanc. 8 Th inivial ok ach ciion ill alo affct y th aamt of th intgat ytm hich i th oc of lfa cot aociat ith th nmloymnt inanc ytm. Th taxfn nfit ill avly affct ach ciion ca it inca th conmtion n nmloymnt y that amont. Th aving-fn nfit hov imov th ach incntiv of a ok lativ to th tax-fn nfit to th xtnt that it i chag to th timnt incom. That i >. 7 Th oaility of nmloymnt hock may hav th ooit ffct on ca it inca th aving fo th nmloymnt nfit. Thi ffct i mall in thi mol hov ca th aving in io i mall lativ to th hol timnt aving that can in th ca of nmloymnt. 8 Sinc th ffct of -nmloymnt aving on ach ffot i not ngligil th inict ffct of a aamt chang on ach ffot thogh it ffct on aving hol coni hich o in th mol. 7

8 Th ach ciion alo n on th ca tct aamtiz y. A long ot-nmloymnt oking io ol avly affct th ach incntiv of a ok y naling him to mitigat th n of th ction in RSA alanc thogh th ajtmnt of hi ot-nmloymnt aving. 9 No lt chaactiz th aamt of th otimal ytm. If iffntiat V ith ct to hav [ t t H ] 6 f. h H inicating ach laticity o th nitivn of mloymnt F. oaility to th inca in thhol ach cot. H am that H i contant fo all. Similaly if iffntiat th xct tility fnction V ith ct to taking into accont th aov inivial ciion hav [ { } }[ { n } H ] o 7 hich yil th folloing y 3 : Not fom 7 that th aving-fn nfit i tmin olly y conmtion moothing an that it i not affct y incntiv coniation. Thn can ov th folloing. Lmma 3 Th oltion fo 6 an 7 i ni if it xit. Althogh th oof can fon in th Anix th folloing Fig giv an infomal xlanation of th ni oltion. 9 Th ffct of th timnt ation on ach incntiv i amigo ca th long timnt ation ol inca oth I. an J. thogh mo ffctiv conmtion moothing. 8

9 A B C D Fig > W can fom 6 an 7 that th intio oltion ill atify th folloing conition: Xδ Y H 9 h " δ X n an Y W can fom 9 that th tax-fn nfit i affct y ik-avion δ an ach laticity H. Alo imli that fo a givn tax-fn nfit th aving- 9

10 fn nfit i tmin y th conmtion moothing aft ing nmloy. Not that Y inicat th al lacmnt atio fin a th atio of th nmloymnt nfit to th ag. Th vaial X nt th inca in aving hich a ok ol lik to hav aft xincing nmloymnt in io to cov fom th ction in RSA alanc. A X can ittn a X n n Y n X flct th iffnc tn th total amont of nmloymnt nfit an th amont of conmtion y a ok if h not nmloy. That i X nt th g of th incomltn of nmloymnt inanc. If X fo xaml it imli that a ok i flly in againt nmloymnt ik. In gnal hov X i gat than zo to th incntiv olm aociat ith th nmloymnt inanc ytm. In thi n X alo inicat th lfa cot minimiz y th otimal intgat ytm. Bfo analyzing ho th nfit tct of th otimal ytm chang ith om aamt mt mak an imotant oint. Th amont that an nmloy ok ol lik to itha fom hi accont fo hi conmtion givn tax-fn nfit i th am amont a th ocially otimal aving-fn nfit that th govnmnt ol ovi to him. That i Pooition fo any givn tax-fn nfit. Poof> Th ivatly otimal aving-fn nfit of an nmloy inivial ill th on that maximiz hi xct tility taking I tax t a givn: [ { } ] hich i th am conition 7 fo. Th ivat maginal nfit of th aving-fn nfit i in gnal not al to it ocial maginal nfit ca an inivial ok ol not tak into coniation th incntiv ffct of hi aving-fn nfit. Sinc th ivat maginal nfit i ootional to th ocial maginal nfit hov th to otimal choic an ol th am. In oth o oth an a tmin olly y th am conmtion moothing. Thi oint i imotant ca nl th govnmnt ovi

11 a ok ith l than th otimal aving-fn nfit a ok col hi aving in io jt fo hi timnt. Comaativ Static Aming that an intio oltion xit ill xamin om imotant facto that tmin th otimal nfit tct of th intgat ytm. Fit ill tat ith th tyical facto affcting th a tct n moal haza ik-avion δ an ach laticity H. Pooition i H H > δ > δ. Y Y ii >. H δ Th oof of Pooition i incl in th Anix. A th ach laticity inicat y H inca th otntial incntiv cot of nmloymnt inanc go alo lag hich tn to favo aving-fn nfit mo than tax-financ nfit. Sinc high ach laticity imli high fficincy cot aociat ith nmloymnt inanc hov it ol c th total amont of nmloymnt nfit. Th gat ik-avion of a ok on th oth han hich imli th gat n fo inanc againt nmloymnt ik i likly to favo tax-financ nfit mo than aving-financ nfit an an inca in th total nmloymnt nfit. An intio oltion may not xit in th mol hov. W can nt th folloing con oltion hich may infomativ. Coollay If δ If H Th aov lt a cla fom conition 9 an. Whn a ok i ik-ntal th i no n fo nmloymnt inanc. If th i not an incntiv olm aociat ith nmloymnt inanc aving-fn nfit i nncay. Coollay th imli that th ach incntiv i i th vy aon th intgat ytm i intoc. Tning ack to th intio oltion can intify a col of oth facto that can affct th otimal nfit tct of th intgat ytm. In oth o a ok ol not n hi ivat aving accont to mak any cationay aving againt oil nmloymnt in io.

12 Pooition 3 Y i >. Y ii o o o a δ o. Th oof i alo lgat to th Anix. Pooition 3 i imli that a high nmloymnt ik la to high xct incntiv cot lting in lo tax-financ nfit an a lo lvl of total nmloymnt nfit. A high nmloymnt ik hov inca aving-fn nmloymnt nfit to th ction in taxfn nfit an a ca in -nmloymnt aving. Pooition 3 ii ggt that ovi that a ok n ith oitiv RSA alanc at th tim of timnt th lationhi tn a ok tax-fn nfit an hi ag i nnt on hi lativ ik-avion. No ill mov on to mo imotant comaativ tatic. Th otimal nfit tct of th intgat ytm chang ith th inivial ca tct ch a th ation of mloymnt an timnt lativ to nmloymnt ation hich a inicat y an in th mol. In on to a ction in hi RSA alanc a ok ith nmloymnt xinc ol inca hi aving ov th otnmloymnt oking io to atially offt th chang in timnt incom. In oth o a ok ol otimally ajt to th ction in RSA alanc y cing conmtion not only ov th timnt io t alo ov th otnmloymnt oking io. Th th chang in th ca tct hich affct th ok attn of conmtion moothing ill affct th otimal nfit tct of intgat ytm. Sinc th lngth of mloymnt o timnt io ol alo affct conmtion io of a ok th aolt amont of nfit ol not lvant in intifying ffct of th ca tct on th otimal nfit comination. Lt γ i i i hich inicat th ha of tax-fn o of aving-fn nfit to th total nfit conm y a ok. Lt alo Tchnically th lativ ik-avion of a ok col affct om oth comaativ tatic thogh th chang of { XX} in X. Whnv th lativ ik-avion i gat than o al to o l than { XX} i caing o contant o incaing in X ctivly. In o to avoi th tchnical comlication ca y thi hov ill haft am that th lativ ik-avion of a ok i i.. that δ.

13 Y y n. Th vaial y inicat th lacmnt atio hich i fin a th atio of nmloymnt nfit to hat a ok col conm if not nmloy. Riting conition a follo: X Z Y Hγ 3 can collct th folloing lt. Pooition 4 So that δ. Thn γ γ y i > > γ γ y ii > a a a Th oof i in th Anix. Pooition 4 i ci th ffct of th otnmloymnt oking io on th nfit tct. A th ot-nmloymnt oking io gt long o a nmloymnt occ in th ali tag of on ca th otimal ytm ntail a high ha of aving-fn nfit to tax-fn nfit an a gat lacmnt at. Alo a Pooition 4 ii montat th otimal ytm ntail th am lt a th timnt io lngthn hil king it atio to ation of mloymnt contant a a. Th lt com fom th fact that a long ot-nmloymnt o timnt io can c th av ik ffct of avingfn nfit. Fit a ok ith nmloymnt xinc can a th n of a ction in RSA alanc y incaing hi aving ing th ot-nmloymnt io mitigating th lfa n y aing it ot mo ffctivly ov th long ot-nmloymnt io. Scon a ok can mitigat th lfa n of a ction in RSA alanc mo ffctivly y aing it ot ov th long timnt io. Th th intgat ytm ol ing to mo lfa gain a th otnmloymnt o timnt io lngthn lativ to nmloymnt ation. 3 Wlfa Pfomanc of Intgat Sytm: Poiility of Th Fit-Bt Stat In thi ction ill ic ho th intgat ytm col oily c lfa itotion aociat ith th inanc ytm a th ot-nmloymnt o timnt io gt long. W ill fit chaactiz th fit-t tat a a nchmak. Th fit-t tat can nt y to lmnt. On i th al maginal tility of incom i.. th al amont of conmtion fo vy tat in ach io hich i aliz thogh fll nmloymnt inanc. Th con lmnt i th fficint choic of th thhol ach ffot. 3

14 4 Th fit-t thhol ach cot o ill tmin a th lvl hich maximiz V givn th aamt val: z o o 4 h z o i th fit-t conmtion io. Sinc th xct total ag incom W ol W o h o o F th fit-t -io conmtion z o ol z o o o. 5 Th th fit-t ayoff fo a ok not y V o ill o o o o o o o o F z z F z F z V " 6 h z. W can it conition 5 fo th thhol lvl of ach cot a follo: " } " { } { } {.}. { X X X J I n n n 7

15 Th ayoff fo a ok thn n th otimal ytm not y V ol V t I. J. t n " { X F } F 8 W can no talih th folloing imotant ooition: Pooition 5 o o i A hich aoximat th fit-t tat h z. a ii A hil a o γ a o a o a th fit-t tat h z. a a hich aoximat Poof> i X a y hich imli that y 9 an th that y n. By an hav a go to infinity. Sinc X a hav fom 7 { I. J.} hich imli th fit-t ach ciion y 4. A n y an 4. Alo y 3 an 3 z an z. Th y 6 an 8 hav a V o V z n n z n z n ca a. 5

16 ii X a y hich imli that y 9 an Not that thi imli that γ. By an hav zo. Sinc X a hav fom 7 n a y. a a go to a { I. J.} a hich imli th fit-t ach ciion y 4. A n y a a an 4. Alo y 3 an 3 z an z. Th y a o th am aoning a aov V V a. Q. E. D. Pooition 5 highlight on of th imotant act of th intgat ytm. A th io of ot-nmloymnt o timnt gt long coma to th io of nmloymnt th intgat ytm mak th amont of lfa itotion aociat ith aving-fn nfit aitaily mall. Thi i ca th ytm mak it av ik ffct a mall a oil hil maintaining th i lvl of ach incntiv thogh th ction in RSA alanc. In th limiting ca thi ill la to th comlt lacmnt of tax-fn nfit y aving-fn nfit an ill la to th fit-t otcom. Whn th timnt io an ot-nmloymnt oking io inca ootionally th fit-t conmtion io i tmin y th atio a of th oking io to th timnt io. Sinc th conmtion cot al l than th ag th lacmnt atio fin a th conmtion n nmloymnt ov ag hol l than on in th fit-t tat. Not hov that all nmloymnt nfit a financ y RSA aving not y tax n th fit-t intgat ytm. 4 Th Rol of Govnmnt Poviion of Saving-fn Bnfit Th ol of th govnmnt in th intgat ytm i citical to th xtnt that it allo a ok to oo againt hi ft aving in o to financ hi nmloymnt nfit. Withot th govnmnt oviion of aving-fn nfit a ok ol hav to itha hi -nmloymnt aving to maintain th ivatly otimal conmtion lvl ing nmloymnt. H thn n to mak om cationay In fact a fll oviion of aving-fn nfit y th govnmnt may hav om olm. On of thm ol th cot of th manatoy aving n to cov th nfit. Thi cot i to th fact that a ok cannot itha hi aving ntil th tim of timnt. 6

17 aving in hi ivat aving accont to lmnt th tax-fn nfit fo hi conmtion ing nmloymnt. Thi ol an xact ca fo th I ytm. W ill fit ifly xamin th I ytm an coma it ith th intgat ytm. Lt th -nmloymnt aving a ok mak n th intgat ytm h h i off th otimal aving-fn nfit if nmloy. To foc on th ca hn th govnmnt oviion of aving-fn nfit i ncay ill o that fo any tax-fn nfit. 3 Thn a th folloing Pooition ho th conmtion lvl of an nmloy ok i lo n th I ytm than n th intgat ytm. Pooition 6 So that. Thn > ˆ ˆ h ˆ ˆ a tax-fn I nfit a ok cationay aving n th I ytm ctivly. Th oof of th Pooition 6 i lgat to th Anix. n th I ytm a ok ho timnt aving in io i not nogh to lac th otimal aving-fn nfit ol hav to mak aitional aving to a fo nmloymnt ik. Sinc thi cationay aving involv om fficincy cot hov it ol till in hot of th otimal aving-fn nfit fo th nmloy. Althogh th tax-fn nfit ill inca to fill th ga to om xtnt it ol not nogh to c th otimal conmtion fo th nmloy ca of it incntiv cot. Pooition 6 imli that th I ytm i infio to th intgat ytm in that th I ytm o not allo an nmloy ok to oo againt hi ft aving. To ho mch th govnmnt n to allo th nmloy ok to oo in o to c otimal conmtion lt o that th govnmnt ovi th inivial ith th ncay minimm aving-fn nfit. Not y Pooition that th ivatly otimal aving-fn nfit i al to th ocially otimal on. Taking thi into accont can nt th i govnmnt oviion of aving-fn nfit a hich ill call th i RSA-fn nfit. Onc th govnmnt off an nmloy ok thn h ill av in io an it togth ith to conm ing nmloymnt. Th th ill no chang in o n 3 If > th ol no ol fo th govnmnt to ovi aving-fn nfit. In thi ca th taitional I ytm ol achiv th am lt a th intgat ytm. 7

18 thi n gim. Th amont of i RSA-fn nfit may in fact nt th iz of lfa gain that th govnmnt ing to a conmtion-contain nmloy ok. By th i RSA-fn nfit ovi y th govnmnt ol Th folloing ooition montat ho th i RSA-fn nfit ol chang ith ach laticity ik-avion an th oaility of nmloymnt hock ith th oof ing lgat to th Anix. Pooition 7 H > δ >. Th i amont of RSA-fn nfit fo conmtion-contain nmloy ok inca n ctain cicmtanc. Fo xaml hn th conomy i jct to io incntiv itotion tax-fn nfit ca hil th n fo aving-fn nfit go. Alo an conomy ith high nmloymnt ik tn to hav lag i RSA-fn nfit ca th otimal ytm i a lo lvl of tax-fn nfit an a high lvl of aving-fn nfit. On th oth han th i RSA-fn nfit ol ca a ok com mo ik-av ca lag tax-fn nfit i ovi hil th otimal aving-fn nfit gt mall. No lt tn to th ffct of th ca tct th lngth of mloymnt an timnt io timing of nmloymnt on th i RSA-fn nfit. Lt γ hich inicat th ha of th i RSA-fn nfit to th total conmtion n nmloymnt. Lt alo Y n y hich inicat th ha of th i RSA-fn nfit to hat a ok ol hav conm if not nmloy. Thn can tat th folloing. Pooition 8 8

19 i γ y > > if δ. ii a a iii a hil king a. a Th oof can fon in th Anix. A th ot-nmloymnt oking io gt long -nmloymnt aving ca ca oth th nmloymnt aving an oth aving act a titt fo ach oth. Th ction in nmloymnt aving col ith th inca n fo aving-fn nfit a hon in Pooition 4 ol fth inca th ha of i RSA-fn nfit to total nfit. Thi lt imli that th lfa gain thogh th inttmoal conmtion moothing n th intgat ytm inca fo tho ho a nmloy ali in thi ca. In th limiting ca h th ot-nmloymnt io i vy long nmloymnt aving in io ill go to zo an th th ha of th i RSAfn nfit ol go to. By Pooition 5 thi i th ca h th fit-t tat i aoximat. Th it i thogh govnmnt oviion of th i RSAfn nfit that th fit-t fficincy can achiv hn th ot-nmloymnt oking io i vy long. Thi ca can contat ith th ca hn th timnt io an th ot-nmloymnt oking io a vy long coma to th nmloymnt io. In thi ca -nmloymnt aving i it lag o that govnmnt oviion of RSA-fn nfit may not a n a in th vio ca. Whn a i.. hn th timnt ation i long coma to th otnmloymnt oking io RSA-fn nfit may not n to achiv th fit-t fficincy. In oth o th ok ivat aving tak ca of any av ik ffct o that govnmnt intvntion may nncay. Th timing of nmloymnt in on ca i citical fo th lvanc of th intgat ytm. If a ok tn to fntly com nmloy lat in hi ca thn th i RSA-fn nfit ol com mall o nil. Thi i ca th ok -nmloymnt aving may lag nogh an ca th n fo aving-fn nfit itlf ol mall. Anoth facto that affct th lvanc of th intgat ytm i th oiility that a ok ill n ith a ngativ RSA alanc at th tim of timnt 4 hich affct th incntiv-ffctivn of th intgat ytm. W ill xamin thi i in th nxt ction. 4 S Fltin an Altman 998 fo a imlation ty on thi oiility. 9

20 3. Intgat Liftim Inanc Thogh Joint Saving Accont So fa hav coni th ca h th i no oiility of a ngativ alanc in on RSA an h th RSA only cov nmloymnt ik. Sinc in ality th amont of conmtion ing nmloymnt i a mall otion of on timnt aving th chanc of a ngativ RSA alanc a lim if RSA only cov nmloymnt ik. In thi ction ill coni a mlti-ik ca in hich a ok may hav mo than on hock nmloymnt an iaility fo xaml - in hi ca. W ill analyz th ytm that intgat oth nmloymnt an iaility inanc ith a nion ogam thogh a joint RSA hich ill call th intgat liftim inanc LI ytm. In a mlti-ik ca a ok may n ith a ngativ alanc in hi RSA fom th to hock in hi ca. In thi ction ill xamin ho th oiility of a ngativ alanc an conntly th govnmnt ailot of it affct th otimality of th intgat LI ytm an it nfit tct. Coni a ok ho liv fo th io. H ok fo th fit to io an ti in th lat io to conm hi timnt aving. Th ok i jct to to ik ing hi ca nmloymnt an iaility ik. So a ok ha nmloymnt hock ith oaility in io an iaility hock ith oaility in io. Onc a hock occ h col ith choo o not choo to xt om ffot to vnt nmloymnt. A ok n nmloymnt hock fo xaml can choo to ach to avoi ing nmloy in io hil a ok n iaility hock can choo to mak om xta ok ffot to ok in io. Th th oaility of nmloymnt in ach io n not only on th oaility o of hock t alo on th ffot ciion y a ok. A in th vio ction it i am that th cot of ach ffot o of ok ffot c i itit ith itition fnction F o ith G ctivly an that a ok choo th thhol ffot cot o c ctivly. Th oaility thn of maining nmloy ol F in io an F c in io. In aition to a ctain amont of initial aving a ok can accmlat aving at any io of mloymnt. Th aving a to accmlat in hi RSA. Inta of intocing a cific amont of manatoy aving am that th RSA i th only aving accont availal to a ok. W ill mainly foc on th cicmtanc h th initial aving alon i not fficint nogh to cov aving-fn nfit fo th nmloy. Whn a ok xinc th to hock an i hi only timnt aving thfo h may n ith a ngativ alanc in hi RSA. 5 Th ngativ alanc i cov y th tax vn an th it may c hi ach an ok incntiv. In oth o to th xtnt that th aving-fn nmloymnt nfit a ok ha civ i not chag to him hn h i nmloy again in io a ok ol lo incntiv fo ach an ok. A ok ay tax t in io an tax t o T in io to cov th tax-fn nfit an th ngativ alanc. In thi mol th tax fo th xct amont of non- 5 A ill n lat th otimal aving-fn nfit fo th nmloy gt mall a th to ik a mo oitivly colat to ach oth. Whn thy a tongly oitivly colat thfo th aving-fn nfit ol o mall that it may l than an thfo no ngativ alanc may occ. Thi ca ill mntion lat in thi ction.

21 chagal aving-fn nfit i ai in io ath than in io. 6 On th oth han th govnmnt ay a ctain amont of nfit to th nmloy o th ial. In io th govnmnt ovi th nmloy ith tax-fn nfit an aving-fn nfit. In io it off th ial iffnt comination of tax an aving-fn nfit ning on hth thy hav violy xinc nmloymnt. Th ial ith no nmloymnt hitoy a givn th tax-fn iaility nfit an th aving-fn nfit hil tho ith vio nmloymnt xinc ill off only th tax-fn nfit D inc thy o not hav a oitiv alanc in thi RSA. Lt I. an J. th xct ayoff of a ok fom ing mloy an nmloy in io ctivly. Th xct tility V of a ok can fin a follo: ; ; ; F D J I D V ax h } { } { ; } { } { ; C S C c c cg S T S D D J cg t t I ax ax an o i th oaility of iaility hock occing conitional on nmloymnt o no nmloymnt in io ctivly hil C G c G F an D T t t 9 6 Thi tax ytm imov a ok ach incntiv givn th oiility of ngativ alanc at th xn of fth iincntiv fo ok in io. Fo th analytical imlicity th tax ytm mol in thi ction lav all th av incntiv ffct aociat ith ngativ alanc to io hil maintaining th am ach incntiv in io. If a ok ay th aitional tax in io ach incntiv in io a ll a ok incntiv in io ill affct hich comlicat th mol. Not hov that th total amont of iincntiv ol main th am givn any tax ytm intoc in th mol.

22 High o lo i aociat ith high colation ρ tn nmloymnt an iaility hock. Th a to thing to not h. Fit th tax T in io incl th xct val of th non-chagal aving-fn nfit -. Scon it i am fo th momnt that > an that > hich i confim lat y Lmma 5 a inivial aving an nfit tct a otimally chon. Thi amtion imli that in th hyothtical ingl-ik ca hn th oaility of nmloymnt o iaility hock i zo th i no oiility of ngativ alanc. 7 Lt fit xamin an inivial ok choic fo aving. H choo th th lvl of aving an S a follo. : t : t : S T S S Som comaativ tatic of inivial aving ciion a nt in Lmma 4 in th Anix. A ok alo tmin th thhol lvl of hi ach an ok ffot c an C in th folloing ay: ; ; D J I 3 an } { t c 4 } { D S T S C 4 ing th nvlo thom can collct om comaativ tatic of inivial ach an ok ciion. Lt fit chck ho a ok ach ciion i affct y th ytm. D. t 5 S 5 7 Othi th ol no incntiv ffct of aving-fn nfit.

23 > 6 Givn that iffnt t of otimal iaility nfit a off fo ok ith an ithot nmloymnt hitoy th aamt D of th nfit tct fo th ial ol not affct th ach ciion of a ok. 8 By 5 an 5 can ay that hich jtifi th aving-fn nfit fo th nmloy in th intgat LI ytm. Th comaativ tatic of inivial ok ciion ith ct to th aamt of th nfit tct fo th ial i c c an t 7 7 C D C S T D S 8 A col of oint v to mntion. Fit a ha n th ca th avingfn nfit i mo incntiv-ffctiv than th tax-fn nfit i... c c Sconly an mo imotantly a 8 ho th aving-fn nmloymnt nfit avly affct inivial ok incntiv n iaility hock. Thi i ca a ok ith nmloymnt hitoy i not chag fo th aving-fn I nfit if h com nmloy again in io an th tax fo th non-chagal 8 A 5 ho ach ffot i not affct y th oiility of a ngativ alanc. Thi i to th tax ytm in th mol in hich th cot of non-chagal aving-fn nfit i ai in io thogh tax a mch a ath than in io. Th th imact of th cot n of th ngativ alanc ill avly fall on th ok incntiv of a ok in io hich ill xlain lat. 3

24 I nfit i ai y tho ho a mloy in io. Bca of thi olm aociat ith th govnmnt ailot of a ngativ alanc om iincntiv on th at of a ok ill alay nt vn in a flly intgat LI ytm that i oly fn y th aving. In chaactizing th otimal nfit tct of th intgat LI ytm an xamining th otimality of th ytm in th mlti-ik ca can am that th intgat ytm i otimal in th ca of hyothtical ingl-ik. In oth o hn th oaility of nmloymnt hock o th oaility of iaility hock i zo th otimal nfit tct ntail a oitiv amont of aving-fn nfit. Thi amtion ill nal to fig ot ho th mltil ik an th oiility of a ngativ alanc ol chang th nfit tct an th otimality of th intgat ytm. Otimal Bnfit Stct of Intgat Liftim Inanc Sytm On imotant fat of th intgat LI ytm i that it off iffnt comination of tax- an aving-fn nfit fo th ial ith iffnt hitoi of mloymnt an nfit aymnt. To th xtnt that mloymnt hitoy affct RSA alanc th iaility nfit hol conitional on mloymnt hitoy. Fit ill analyz th ca fo ok ho hav n nmloy in io. Th otimal tax-fn nfit D hol atify Q S T D δ 9 Q g c h δ an H hich i am to contant fo all c nt th G c ok ik avion an th laticity of ok ffot ctivly an S T Q D H. Taking into accont th aving ciion can it 9 a follo: T D Q δ 9 Q Not fom 9 that a al th tax-fn nfit D i incaing an caing in th ik-avion an th laticity of ok ffot ctivly. Sinc th i no oitiv RSA alanc th otimal aving-fn iaility nfit D i zo: D 9 Lt tn to th otimal nfit tct fo th ial ho hav not n nmloy in io. Diffntiating th xct tility fnction V ith ct to hav 4

25 Q t δ 3 Q 3 h Q H. Taking inivial aving ciion an into coniation can nt th conition fo a follo: t Q δ 3 Q 3 Not that th otimal nfit tct fo th ial ithot nmloymnt hitoy in io i th am a that in th hyothtical ingl-ik ca hn th oaility of nmloymnt hock i zo ca thi RSA alanc a. Sinc th otimal nfit tct i am to ntail a oitiv aving-fn nfit in th hyothtical ingl-ik ca th aving-fn iaility nfit fo th ok n th intgat LI ytm ill alo oitiv i.. >. 3 No lt tn to th nfit tct fo th nmloy n th intgat LI ytm. On imotant facto that tmin th nfit tct fo th nmloy i th iincntiv ffct aociat ith th oiility of a ngativ alanc. A cogniz in 8 th oiility of ngativ RSA alanc an th nt govnmnt ailot of it can aggavat a ok incntiv. Not that thi iincntiv olm affct th nfit tct fo th nmloy not that fo th ial. Diffntiating th xct tility fnction V ith ct to hav P [{ t } δ ] 3 P H h P t. Similaly if iffntiat V ith ct to taking into accont th inivial ciion 5 5 an 8 hav Q [{ S T } δ ] 33 P 5

26 D H h Q S T. Th lat tm of 33 inicat th iincntiv ffct aociat ith th ngativ RSA alanc hich c th avingfn nfit. Not that th iincntiv ffct i oitivly lat to o to th colation tn th to ik. Sinc th otimal aving-fn nfit i am to oitiv in th ingl-ik ca hn - hav > hn an - > y Pooition 9 nt lo. Finally can ov that a S an a otimally chon th ill not a ngativ RSA alanc hn only on hock occ i.. Lmma 5 > S >. Th oof i lgat to th Anix. Colation tn Rik Otimality an Bnfit Stct in th Intgat LI Sytm On of th mot imotant facto that tmin th nfit tct of th intgat LI ytm i th colation tn th to ik. So a fo that th otimal aving-fn nmloymnt nfit i gat than o that a ngativ RSA alanc i oil. A th folloing Pooition ho a high colation la to a mall aving-fn nfit an a gat tax-fn nfit. Pooition 9 i > ρ ρ ii > ρ ρ Pooition 9 montat that th otimal aving-fn nfit fo th nmloy o fo th ial n th intgat LI ytm ca a th colation tn th ik inca. Not that th aving-fn iaility nfit i not affct y th iincntiv ca y th oiility of th govnmnt ailot of a ngativ alanc. High colation affct thogh th lo oaility of iaility hock fo tho ithot nmloymnt hitoy cing. Th lt on aving-fn nfit in ii fo th nmloy hov lt fom th iincntiv ffct aociat ith 6

27 th oiility of a ngativ alanc. A a n fo th oiility of th govnmnt ailot of a ngativ RSA alanc hich inca in th colation tn th to ik ol c a ok incntiv n iaility hock. 9 Th a aitional facto that a onil fo th ffct of th colation on th aving-fn nfit fo th nmloy. A th oaility of iaility hock fo tho ith vio nmloymnt hitoy inca fo xaml th xct ayoff of ing nmloy ol ca cing th aving-fn nfit. Sinc th amont of aving-fn nfit in th intgat LI ytm ca in th colation ρ on may on if th intgation of nmloymnt an/o iaility inanc ith a nion ogam ol till ncay hn th to ik a highly colat to ach oth. Th a to oint to coni on thi i. If th aving-fn nfit fo th nmloy ca lo th initial aving a a lt of tong colation th i no oiility of a ngativ alanc an th no iincntiv olm. If th colation i vy high hov th nmloy ol hav a high chanc of ning ith th timnt aving ith no aitional RSA alanc. Thi ol la to a lo o vn zo aving-fn nfit fo th nmloy hn i lo. Th th intgation of nmloymnt inanc alon ith nion may not otimal in a mlti-ik ca h th ik a highly colat. Th folloing agmnt hov c th otimality of th intgation of nmloymnt an iaility inanc ith nion thogh a joint RSA. A citical oint to mhaiz i that th intgat LI ytm allo th iaility nfit tct to on otimally to iffnt nmloymnt hitoi of a ok. To th xtnt that iffnt nmloymnt hitoi in io la to iffnt alanc in th joint RSA thy can alo affct th otimal nfit tct fo th ial. In oth o a can fom an 3 D > D Sinc a DI ytm that i intgat ith nion t not ith an I ytm off th am nfit fo th ial gal of thi vio nmloymnt hitoy it i infio to th intgat LI ytm. Th oc of th lfa imovmnt y th intgat LI ytm com fom it aility to ha th RSA alanc tn nmloymnt an iaility nfit. Fo xaml fo tho ho hav not n nmloy an th hav not n ai nmloymnt nfit th ytm allo th aving to fo th iaility nfit. In oth o th aving v a a common ool of aving to ha ithin th ytm. Th ytm alo allo th aitional aving hich tho ok mak in io to fo iaility nfit. With th aving mm in a joint RSA th ytm off a lativly lag amont of aving-fn nfit fo th ial ho hav not n violy nmloy. 9 Not alo that ning on ho th tax fo th non-chagal I nfit i collct th oiility of th ngativ alanc col alo affct ach incntiv a ll a ok incntiv. To avoi nncay comlxiti in thi mol ill not xamin xlicitly th mlti-ik ca ith no oiility of ngativ alanc. 7

28 Fo tho ho a nmloy in io on th oth han th RSA alanc ca fo th to aon; om of th alanc may ithan to financ a at of th nmloymnt nfit t an nmloy ok ill not al to mak aitional aving in hi RSA. If th aving-fn nmloymnt nfit i gat than th initial aving a cifi aov th lting ngativ RSA alanc ol la to a zo aving-fn nfit D fo th ial. Evn if i l than th lativly mall amont of RSA alanc ol la to a mall o zo aving-fn iaility nfit hich ol claly iffnt fom. Th agmnt hav talih folloing Pooition on th otimality of th intgat LI ytm: Pooition nl ρ that i nl th to ik a fctly colat to ach oth it i otimal to intgat nmloymnt an iaility inanc ith a nion ogam thogh a joint RSA. Pooition ag that th intgat LI ytm i otimal it th iincntiv ca y th oiility of a ngativ alanc o it th oiility of a lo RSA alanc lting fom th occnc of to conctiv hock. Whn th to ik a fctly oitivly colat to ach oth hov only th tax-fn nfit D might ovi to th ial if th initial aving i mall a illtat in th vio ction. In thi xtm ca thfo th ol no lfa gain xct fom th intgat LI ytm coma ith th nintgat ytm. 4. Conclion Th fail of makt to ovi aat ocial inanc ha long n cogniz. Thi comin ith th fact that ocial nom o not allo inivial in thi ol ag to ff fom infficint incom vn hn thi mifotn ai ca thy hav chon to av fficintly ovi a ational fo a lic comloy nion ogam. Thi a ha vlo a fth avantag to th lic manatoy ogam; it allo fo th collatalization of ft ag incom in a ay hich i not aily oil othi th alloing inivial to in ffct lf in. Thi a ha a to lat i. Th oiility of aving-fn lfinanc o not liminat th iaility of om tax-fn inanc xct n xtm cicmtanc. W hav intifi th facto on hich th otimal g of aving-fn lf-inanc n. O analyi i conitnt ith th ggtion in th intoction of a havy lianc on aving-fn lf-inanc. Fo th o of imlicity xlicit analyi of th otimal nfit tct in thi ca i omitt in thi a. In thi ct th Povint Fn in Singao an alayia might of th ial fom to th xtnt that it cov a nm of contingnci fo a ok y aving h ha accmlat in hi accont. 8

29 Whn th a mltil ik incling th ik of mltil ot of nmloymnt again om lianc on aving-fn lf-inanc i in gnal ial nl th ik a fctly colat. Althogh th mlti-ik ca may involv om av iincntiv to th govnmnt ailot in th vnt of a ngativ alanc th intgat ytm can alay gnat lfa gain fom alloing a common ool of nion aving to ha. Th gnal incil natally la to th ggtion of a flly intgat liftim inanc ytm thogh a joint accont imila to th Povint Fn of Singao an alayia 3 h not only nmloymnt an iaility ik t alo halth ik a intgat ith nion. 3 Fo tail infomation on th ytm Ah

30 REFERENCES Ah.G. 994 Social Scity in alayia an Singao actic i an iction Intitt of Statgic an Intnational Sti alayia 994 Bjokln An 993 A Comaion tn Actal Ditition of Annal an Liftim Incom: Sn 95-9 Rvi of Incom an Walth Fltin atin an Danil Altman 998 nmloymnt Inanc Saving Accont NBER Woking Pa 686 Folt S. an Tofimov G. 999 Social Inanc a on Ponal Saving Accont: A Thotical Analayi. Woking Pa Th Sih Rach Intitt of Ta Stockholm Folt S. An Evalation of Social Inanc Saving Accont mimo Ozag.J. Ozag P.R. Sno D.J. an Stiglitz J.E. 999 Th Imact of Inivial Accont: Picmal v. Comhniv Aoach nt At th Annal Bank Confnc on Dvlomnt Economic Th Wol Bank. OECD 996 Emloymnt Otlook OECD Pai 3

31 3 APPENDIX Poof of ooition:. Pooition Not that δ H an that δ H. i H y 9 hich la to > H y. Alo hav > δ y 9 an δ y. ii Fom hav H H H Y hil H H ca H. Sinc H H Y. Sinc δ δ δ an δ hav > δ δ δ δ δ Y.. Pooition 3 i y 9 ca > hich la to th con lt of y. Sinc hav th thi lt y. ii 9 can ittn a } { H Y X X δ δ h } { o Y a δ o. Th th fit to lt com fom th aov conition an an fom that hich la to th lat lt. 3. Pooition 4 i Sinc > Y B X A hav th fit to lt. Th fit lt imli that X ca a inca y an n inca in y 3 hich la to th lat to lt. ii Fit ill ho that } { a X. Riting 5 a

32 I. J. n δ. X Th { } o not contain th tm / hich imli that X { } a a ca. Nxt tting ZH k contant in 3 an iffntiating oth i ith ct to Y an hil king a a hav a a a k a taking into accont th conition that. ing thi inality hav a X X X a > { a } > a k a { a a } a a ca y. Thi ov th fit lt hich imli th con on. a By 9 >. Alo >. Fom th finition of x thn hav a a th thi lt ca > y Lmma. Th lat lt com fom th thi a on an th finition of y. 4. Pooition 6 Not that an atify t n Ω > ca >. Th >. Lt th cationay aving of a ok ho i not off any RSA-fn nfit y th govnmnt. Thn it atifi 3

33 Ω n t. Not that Ω ca > an y th aov conition fo. Th >. Sinc an hav fom Fig th i lt. 4. Pooition 7 Th lt com fom Pooition -3 an th fact that. H σ 5. Pooition 8 γ i By Pooition 4 an y Lmma th fit lt otain y x th finition of γ. That y Pooition 6 an Lmma la to th con lt. ii Sinc a go to infinity Pooition 5 gt th i lt y th finition of γ an y. iii Sinc a go to zo Pooition 5 gt th i a lt y th finition of γ an y. 6. Pooition 9 i Fit can ov th folloing lmma. Lmma A Q > Poof> Q [ " ] > ca δ an ca y Lmma 3. Q. E. D. 33

34 Q By 3 hav o > fo givn ca > y Lmma 4. ρ Sinc > hav > o y 3. ρ ii To fig ot th ffct of th colation tn th to ik on th nfit tct of th intgat ytm ov th folloing tchnical lt. Lmma B a So i chon otimally fo givn y 3. Thn P > So that D i chon otimally y 3. Thn Q >. Poof> a Fom 3 hav t P 33 t Thn can that -P i caing in not that δ an that RHS of th aov conition 33 i incaing in not that >. Sinc > y Lmma 4 it i cla that oth -P an RHS of 33 a incaing in fo any givn. Th gt th i lt. Fom 3 hav Q. 34 S T It i cla that Q i incaing in D hil RHS of 34 i caing in D Not that S T i caing in D. Sinc S T i caing in can that oth Q an RHS of 34 a incaing in ing δ. Th gt th i lt. Q. E. D. Not that 34

35 > > ρ > ρ hich imli that > fo givn y 3 an Lmma B. Thi an Lmma B ρ la to th lt that y 33. ρ Poof of Lmma. Lmma X n { }. Th conition 4 can ittn a o t n { δ X } 4 t { } { }. 4 i Diffntiating 4 ith ct to ing an th amtion that yil th lt. ii If iffntiat 4 ith ct to an ing an aming that an a o mall that an that hav th i lt. iii Diffntiation of 4 ith ct to δ an la to th fit lt. Diffntiation of 4 ith ct to king contant an yil th t of th lt.. Lmma i Diffntiation of 5 ith ct to an th nvlo thom fo n an yil th i lt. 35

36 36 ii }] { [ ] [ n y Lmma an an y. Alo } { } } { { iii Diffntiating 5 ith ct to an ing Lmma hav " } { X n 4. Lmma 3 Sinc t y an Lmma ii hav y iffntiating 6. Diffntiating 7 ith ct to an hav. Th to conition la to th ni oltion y Fig. 3. Lmma 4 S > Poof> 4 > y 9 an t y 9 an

37 37 D S y 9 an t y 9 an 4. Lmma 5 Fom 3 hav that >. Alo fom hav T S. Sinc T y 33 hav