Optimal Unemployment Insurance with Consumption Commitments -- Can Current UI Policy Be Justified?

Transcription

1 Opimal Unmplomn Insuranc wih Consumpion Commimns -- Can Currn Polic B Jusifid? Jia Luo Univrsi of California, Los Angls Fbruar 2006 [Absrac] Unmplomn insuranc programs ar imporan ingrdins of social wlfar policis in dvlopd counris. Ovr h pas wo dcads, hr has bn a branch of liraur ha dals wih h opimal dsign of insuranc plans. Howvr, no hor so far has bn abl o jusif h currn program in h U.S. Classical modls claim ha unmplomn bnfis should dcras graduall in ordr o induc h appropria incniv o b rmplod, whil h currn program is a fla rplacmn raio of approxima 66% for 6 monhs ha hn drops down o zro afrwards. This papr considrs an nvironmn wih consumpion commimns in which popl canno frl subsiu among diffrn goods wihin a singl priod. In his papr, h opimal unmplomn conrac is dsignd in a rpad principal-agn problm wih unobsrvabl job sarch ffor. Wih consumpion commimns, h opimal plan involvs a rlaivl fla dcrasing squnc of insuranc pamns ovr som duraion, which is hn followd b a larg drop o a vr low lvl of ransfr. Th rsuls fi currn polic wll, and hrfor giv an xplanaion o jusif h currn polic. Addiionall, h modl prdics ha if w chang from h currn unmplomn insuranc program o h opimal conrac, h govrnmn will onl sav.7% in unmplomn insuranc pamns, which shows ha currn polic is no as flawd as rsarchrs hav radiionall blivd. In fac, o achiv fficinc, unmplomn ransfrs should includ a jump, similar o wha w obsrv in pracic. Th diffrnc bwn h opimal conracs I obain in hor and h currn polic in h ral world is small and can b xplaind b adminisraion cos. JEL classificaion: E24; C6; H53 Kwords: Unmplomn Insuranc; Consumpion Commimns; Adjusmn Coss Dparmn of Economics, Univrsi of California, Los Angls, CA, jialuo@ucla.du

2 I. Inroducion Unmplomn insuranc () program is an imporan ingrdin of social wlfar policis in dvlopd counris and h var across counris and ovr im in rms of hir siz and duraion. In h U.S., unmplomn bnfis ak h form of approxima 66% rplacmn raio for 6 monhs which hn drop down o zro afrwards. Howvr, currn polic has bn widl criicizd bcaus of is prvrs ffcs on h incnivs for rmplomn. Ovr h pas wo dcads, hr has bn a branch of liraur ha dals wih h opimal dsign of insuranc plans. Thr is a big discrpanc bwn h opimal plans proposd horicall and wha w obsrv in pracic. For xampl, classical modls ha focus on unobsrvabl job sarch ffor argu ha unmplomn bnfis should dcras graduall in ordr o induc h appropria incniv o b rmplod. So far no hor has bn abl o jusif currn polic, whr w hav a fla insuranc pamn for a fixd duraion followd b a larg drop in bnfis. Th purpos of his papr is o giv an xplanaion for h xisnc of disconinui in h squnc of insuranc pamns. W wan o show ha h currn unmplomn polic isn as bad as rsarchrs hav radiionall xpcd; acuall h fficin unmplomn ransfrs should includ a larg drop as wha w obsrv in pracic. W modl h incniv problm causd b unmplomn insuranc in a rpad principal-agn problm whr h moral hazard coms from unobsrvabl job sarch ffor. Th innovaion of his papr is ha w dsign h opimal unmplomn insuranc conracs wihin h nvironmn in which popl canno subsiu among diffrn goods frl wihin a priod. W propos his sing bcaus in h ral world, man goods, such as housing and vhicls, involv commimns -- adjusmn coss hav o b paid o chang h consumpion of hs goods. Thr is a conrac bwn h principal (govrnmn) and h agn (unmplod workr) ha spcifis a squnc of pamns from h principal o h agn condiional 2

3 on h agn s prvious unmplomn hisor and h commimn goods h prviousl had. Th opimal conrac minimizs h xpcd discound coss, which includ boh ransfrs and h adjusmn coss if commimn goods ar changd, subjc o h consrain ha h unmplod ar nsurd a prspcifid wlfar lvl. W know ha opimal is o srik a balanc bwn h bnfi of consumpion smoohing and h cos of h insuranc. Th poin is whn w incorpora consumpion commimns ino h modl; h adjusmn among consumpion goods is no compll flxibl an longr. Thrfor h dmand for consumpion smoohing would chang. In his cas, w nd o rconsidr h opimal unmplomn insuranc plan. Th opimal plan I obaind has a rlaivl fla dcrasing insuranc pamn from h govrnmn o h unmplod during priods of unmplomn for som duraion hn followd b a significan drop o a vr low lvl of ransfrs. Th rsuls fi currn polic much br han hos of classical modls. Th logic bhind m rsuls is as follows: firs, in ordr o provid inr-mporal incnivs for job sarch, h conrac mus punish workrs for coninuing o b unmplod b rducing hir claims for fuur consumpion. This is h rason for dcrasing ransfrs during priods of unmplomn; scond, h dcrasing ra is much slowr han hos of classical modls. This is bcaus consumpion commimns chang prfrnc risk avrsion. To pu i concrl, l s us food and housing o rprsn adjusabl goods and commimn goods rspcivl. Suppos h agn suffrs an unmplomn shock. Bfor h dcids o pa h adjusmn cos and sll his hous, h has o cu back sharpl on food xpndiur in ordr o pa his morgag and mainain h housing commimn ha h prviousl mad. This will amplif his risk avrsion, bcaus h sharp concnrad rducion in food xpndiur cras a largr incras in marginal uili han would occur wihou commimns, whr housing xpndiur could also b cu. Thrfor, as long as commimn goods ar mainaind, h unmplod ar wors off han w hav xpcd. Thir dmands for consumpion smoohing ar largr, which rquirs a highr lvl of unmplomn bnfis; and las, h disconinui parn is du o h adjusmn of commimn goods. Onc h commimn goods ar downgradd, h agn has mor flxibili in adjusing his consumpion bundl. Th govrnmn no longr nds o 3

4 mainain such a high lvl of insuranc pamn; hrfor a drop in opimal unmplomn insuranc is xpcd. Th papr procds as follows. Scion II givs a brif liraur rviw and som facs abou consumpion commimns. In scion III I dscrib h modl and sa som horical rsuls. Numrical rsuls and snsiivi analsis ar providd in scion IV. Scion V consiss of h conclusion and xnsions. II. Liraur Rviw and Facs Liraur Rviw Th liraur on opimal is rlaivl nw. Much of rsarch on opimal unmplomn insuranc dsign has bn dvod o h sud of h ffc of bnfis on h duraion of unmplomn splls, mphasizing h moral hazard crad b insufficin monioring of job sarch ffor. Th analsis of Shavll and Wiss (979) focuss on h moral hazard associad wih h inabili of h providr o monior h job sarch ffor and rsrvaion wag of h unmplod. Th main insigh of hir papr is ha h bnfis should dclin monoonicall wih h duraion of unmplomn splls. Building upon his work, Hopnhan and Nicolini (997) discuss h dsign of an opimal program in which h insurr s powr o rward or punish includs h abili o ax or supplmn h agn s incom afr h is rmplod. Th k faur of hir modl is ha h liklihood of finding a nw job in an givn priod dpnds on h sarch ffor of h unmplod, which is priva informaion. Th show ha o moiva h unmplod o xr appropria job sarch ffor, h xpcd uili associad wih rmaining unmplod mus dclin ovr im. Ths conclusions sm complling, howvr h ar far from h program w hav in pracic. This papr is basd on Hopnhan and Nicolini (997). I incorpora commimn goods ino h opimal conrac dsign, in which h adjusmn of commimn goods is cosl, h rsul of which is a disconinui in unmplomn bnfis. Consumpion Commimns In mos xising modls, w assum ha individuals consum onl on good, ha is, consumpion can b rgardd as a composi good. Singl-good uili funcion is undr 4

5 h assumpion ha popl can subsiu frl among diffrn goods a all ims. In pracic, howvr, subsiuing among goods wihin a priod is cosl. Man housholds hav consumpion commimns ha ar cosl o adjus whn shocks such as job loss or illnss occur. Grubr (998) finds ha lss han 5% of h unmplod mov ou of hir homs during unmplomn and mos of hm sill hav morgag commimns. Housing is a lading xampl of a commimn good. Changing consumpion of housing picall nails moving xpnss and larg ransacion coss. Brokr fs ar around 5% of a hom s lising pric, and arl rminaion of a las ofn nails a las a on-monh pnal in rn. In addiion, laving a nighborhood and changing childrn s schools could hav dirc wlfar coss. Mor gnrall, an durabl good involvs som commimn, as mark rsal valus ar significanl lowr han acual valus. Th diffrnc bwn h pric of a nw car and a on-ar-old car suggss ha h loss from rslling a durabl good is a las 0% of is valu, prhaps du o phnomnon of lmons. Addiionall, a numbr of srvics rquir xplici conracs and pnalis for arl rminaion, such as: halh insuranc, halh clubs, cllular phons, and cabl lvision. Goods involving commimns compris a significan porion of mos housholds budgs. In mpirical sudis, Warran and Tagi (2003) show ha narl 70 prcn of h avrag houshold s budg is o som xn non-discrionar in h shor run. Raj Ch (2003) argus ha h pical houshold in h US allocas approximal half of is n-of-ax incom o hs cosl-o-adjus goods. III. Th Modl Th Environmn In his scion w characriz h opimal unmplomn insuranc conrac bwn h govrnmn and h unmplod workr wih consumpion commimns. Th prfrncs of h agn ar: 0 [ u( x, ) a ] = 0 E β (),2 W disrgard h disuili from working whn h agn is mplod. 5

6 whr x R+ and R+ ar adjusabl and commimn good rspcivl a im. a A is sarch ffor. β < is a discoun facor and E 0 is h xpcaion opraor a = 0. A is a closd inrval conaining zro. Th uili funcion is incrasing and sricl concav in boh consumpion goods: u >, u > 0, u < 0, u < Assumpion lim x u ( x, ) = lim u2( x, ) = 0 and lim 0 u( x, ) inf, u( x x x, = ) for all x. Assumpion2 Th marginal uili of adjusabl good is non-dcrasing in consumpion of commimn good: u x, ) 0.,2( Sarch ffor a nrs ngaivl in h uili funcion and i is priva informaion of h unmplod workr. Th liklihood of finding a nw job p dpnds on a p = p a ) (2) whr p ( ) is sricl incrasing, concav, wic diffrniabl and saisfis sandard ( Inada condiions o guaran an inrior soluion. In his modl, w assum h agn has no ohr sourc of incom xcp ransfrs from h govrnmn whn h ar unmplod. W also assum h agn has no savings and no accss o crdi. Undr hs assumpions, h govrnmn can dircl conrol h agn s consumpion sram. Hr w ar no going o considr h cas whr hr is hiddn walh or rad such as borrowing and lnding ha canno b obsrvd b h govrnmn. 3 W assum all jobs ar idnical, offring a prmann and consan wag. This assumpion is jus o xclud anohr kind of asmmric informaion whr h job offrs ar hrognous and canno b obsrvd. Lasl, w assum onc commimn goods ar downgradd, h can b rducd frl. This assumpion is for simplici. 4 Thr ar som rasons o jusif i: suppos h workr owns a hous whn h bcoms unmplod. Whn h downgrads from slf-ownd housing o aparmn rning, h 2 Th uili form hr is fairl sandard: concavi, addiiv im sparabili, and addiiv sparabili bwn consumpion and sarch ffor. 3 Following Hopnhan and Nicolini (997) argumns: hs assumpions ar qui pical in h rpad agnc liraur and provid an uppr bound on wha can b achivd hrough an opimal conrac. 4 I am going o adjus his assumpion in fuur work. 6

7 adjusmn cos canno b ignord. Bu afr ha, as an aparmn rnr, moving from a big aparmn o a small on inducs a much smallr adjusmn cos ha could horicall b nglcd. Anohr xampl is slling a car and changing o public ransporaion. Onc h commimn good (h car) is sold, h agn has almos compl flxibili o adjus h consumpion bundl. Th Conrac Squnial Problm A im = 0 h govrnmn offrs a conrac o h unmplod workr. Th conrac is v h x v an allocaion σ { τ, }, ), (,, ), (,, ), (,, ) = h = ( v h a v h, whr v0 is h iniial wlfar valu for h agn a im = 0. is h givn lvl of commimn good h agn prviousl consumd. h = h, h,.., h ) is h hisor up o h priod. H = { 0,} ( 0, whr 0 rprsns h sa of unmplomn and sands for mplod saus. τ v,, h ) is h ransfr from h govrnmn o individual a im ( 0, which can b allocad on h consumpion of adjusabl good x v,, h ) and ( 0 commimn good v,, h ). a v,, h ) is h rcommndd sarch ffor b h ( 0 ( 0 govrnmn. 5 Associad wih ach conrac, l C ( ) dno h xpcd discound valu of oal cos, which includs boh n ransfrs and adjusmn cos if commimn 0 σ 0 0 ) = 0 good is changd. L V ( ) = E β [ u( x ( σ ), ( σ )) a ( σ ] σ rprsn discound uili valu for h agn corrsponding o h conrac allocaion σ. Th opimal conrac is o choos h fficin allocaion σ o minimiz h cos of unmplomn insuranc subjc o h consrain ha h unmplod ar nsurd h iniial wlfar valu. 6 Th lr K is adjusmn cos. s.. C [ τ ( v,, h ) + ( ( v,, h ) K ] 0 ( v0, ) = Min E 0 β 0 0 ) { τ, x,, a } = 0 = 0 (3) 5 Rlvan onl if h workr is unmplod. 6 Assum h govrnmn discound h fuur h sam as h agn 7

8 Promis kping consrain v0 E Incniv consrain And [ u( x ( v0,, h ), ( v0,, h )) a ( v0,, h ] v0 0 ) = 0 β (4) h υ { a v,, h )} υ υ [ u( x ( v,, h ), ( v,, h )) a ] υ υ ( 0 arg max E = β υ υ 0 υ 0 υ (5) { aυ } υ = υ= τ v,, h ) = x ( v,, h ) ( v,, h ) (6) 7 ( In his problm, w assum h govrnmn has h abili o commi o h ransfr polic. Rcursiv Problm W wri h opimal conrac problm in a rcursiv form. Th xpcd discound uili o h agn is qual o v 0. Each priod, h dcision of whhr o chang h commimn goods is mad. If h commimn good is adjusd, govrnmn ransfrs τ o h unmplod which could b usd on adjusmn good x and commimn good. Th rcommndd sarch ffor a during his priod is spcifid in h unmplomn conrac. Th conrac also spcifis h nx priod promisd valu v o h agn. If h commimn good is mainaind, conrac includs ransfr τˆ, sarch ffor â and promisd valu v ˆ for h nx priod. Bcaus h commimn good is mainaind, a fixd amoun of ransfr τˆ gos o h commimn good, h lvl of which givn as. Th rs of ransfr τˆ is spn on adjusabl good xˆ. Th lr K is h adjusmn cos. Th opimal conrac can b dscribd as h minimizing problm followd: s.. C( v, ) = Min Promis kping consrains v Min { τ + ( } ) K + β ( p( a)) W ( v ) { τ, x,, a, v } { } Min ˆ τ + β ( p( aˆ)) C( vˆ ) { ˆ, τ xˆ, aˆ,ˆ v } ( p( a) v + ( p( a)) v' ) v u( x, ) a + β (8) (7) 7 Assum pric of commimn good is. 8

9 Incniv consrains and [ p( aˆ) v + ( p( aˆ)) vˆ ] v u( xˆ, ) aˆ + β (9) { u( x, ) a + [ p( a) v + ( p( a)) v' ]} { u( xˆ, ) aˆ + [ p( aˆ) v + ( p( aˆ))ˆ' ]} a arg max β (0) aˆ arg max β v () x + = τ (2) x ˆ = τˆ (3) + whr C v, ) is oal cos givn ha h promisd valu is v and h commimn good is (. p (a) is h probabili of finding a job during h nx priod. W (v ) is h discound cos in h nx priod whn h commimn good is adjusd during his priod: 8 W ( v ) = Min { τ + β ( p( a )) W ( v )} x,, τ a, v (4) s.. ) a + ( p( a ) v + ( p( a v ) v { u( x, ) a + ( p( a ) v + ( p( a )) v ) } u( x, β )) (5) a arg max a β (6) x + = τ (7) v is h discound uili valu for a workr who is mplod: 9,0 v u( x, ) = Max x, β (8) s.. x + = w, u x, ) p = u ( x, ) (9) ( 2 Sinc p (a) is sricl concav, h incniv consrains (0) and () can b wrin as: β p ( a)( v v' ) = (20) β p ( aˆ)( v vˆ' ) = (2) 8 Noic commimn good dosn nr h cos funcion bcaus h assumpion ha onc h commimn good is downgradd, h agn has h flxibili o adjus consumpion bundl afrwards. 9 v dosn dpnd on unmplomn hisor. 0 For simplici, w assum hr is no adjusmn cos for an upgrading in commimn good. W can rlax his assumpion and assum an adjusmn for upgrading, bu his will onl chang h quaniaiv valu of v, which will no chang h propris of h opimal conrac. As long as v > v, a is grar han zro. 9

10 Characrizaion of h Opimal Conrac Sinc h cos funcion is h minimum of wo minimizing problms, i is no smooh a h poin whn adjusmn is mad. Bu if w disrgard his kink and look a ohr pars of h cos funcion, w can sill us firs-ordr condiions o g som propris of h opimal conrac. Bfor h commimn good is adjusd, h firs-ordr condiions ar: C ( vˆ ) = u ( ˆ τ, βηp ( aˆ) ) ( p( aˆ)) (22) p ( aˆ)) C( vˆ ) ˆ + η p ( aˆ)( v vˆ' ) = 0 (23) Th nvlop condiion is: βηˆ p ( aˆ) C ( v, ) ( ˆ C v = =, ) + u ( ˆ τ, ) ( p( aˆ)) (24) whr ηˆ is h muliplir on h incniv-compaibili consrain. W also driv h firs-ordr condiions for h cas in which h commimn good is adjusd. Sinc h ar similar o quaion (22)-(24), w pu hm in appndix []. Proposiion : Th promisd uili valu in h nx priod is lss han h promisd valu in his priod, i.., v ˆ < v or v < v, if cos funcion is convx. Proof. Firs w considr h cas in which h commimn good has no bn adjusd. Sinc w hav p ( ) > 0, p ( ) < 0 and v > v, from (23) w g ˆ η > 0. From (24) w obain C v, ) > C ( vˆ, ). If h cos funcion C is convx, w g v > vˆ. Afr h ( commimn good is adjusd, w can follow h firs-ordr condiions in h appndix and h proof is similar. A h kink whn adjusmn is mad, sinc w choos h minimum of wo minimizing rsuls, if v ˆ < v, w hav alrad provd ha v > vˆ, so h conclusion is valid; if v ˆ > v, w hav v > vˆ > v. Q.E.D. Th inuiion of proposiion is ha in ordr o provid propr incniv for sking a job, h conrac mus punish workrs for coninud unmplomn b rducing hir claims for fuur consumpion. 0

11 Proposiion 2: Th rplacmn raio ( τ ) or ( τˆ ) is an incrasing funcion of currn w w sa variabl v. Proof. Bfor h commimn is changd, from nvlop condiion (24), w g C ( v, ˆ ) = u ( ˆ τ u( τ, ), τ ˆ,. Whn v, sinc C is convx, w hav C ( v, ), hrfor ). Bcaus w assum a prmann and consan wag whn h workr is mplod, w g ha h rplacmn raio dcrass. Afr h adjusmn, h proof is similar. Q.E.D. Lmma : Th uili lvl in his priod and h promisd valu in h nx priod mov in opposi dircions, givn hazard ra funcion u ( xˆ, ) ( ˆ < u x2, ) ; if v > v 2 hn u ( x, ) < u( x2, 2 ). Proof. S appndix [2]. p v hn ra ( a) =. If ˆ ˆ > v2 Proposiion 3: Promisd valu nx priod ( v ˆ or v ) is an incrasing funcion of currn sa v. Proof. S appndix [3]. Proposiion 2 and 3 ar inuiiv: h lss uili h agn is promisd oda, h lss h agn will g from h govrnmn for oda and his claims for fuur consumpion will b lss. Corollar : Transfr from h govrnmn o h agn is dcrasing during h unmplomn priod. Proof. Follows immdial b rpadl appling proposiion and 2. Corollar sas ha opimal unmplomn insuranc involvs a dcrasing squnc of insuranc pamn o h workr whil h rmains unmplod. No surprisingl, his dcrasing pamn squnc is h fficin rsul in a moral hazard problm, which is causd b unobsrvabl sarch ffor. Nx, w wan o show ha h unmplomn bnfi squnc dcrass mor mildl in our modl han would occur in classical

12 modls and i involvs a disconinui as w hav in a ral conom. In ordr o compar our modl wih classical modls, I also provid a bnchmark modl in appndix [3], in which all sings ar h sam as our modl xcp ha consumpion goods can b subsiud frl. Proposiion 4: Compard o h bnchmark, h promisd valu in h nx priod is smallr bfor h commimn good is adjusd v ˆ < v~, if u > 0. Proof. S appndix [5]. Proposiion 5: Bfor h commimn good is adjusd, h ransfr from h govrnmn o h unmplod workr is highr ( ˆ τ > ~ τ ), compard o h bnchmark. Proof. From Proposiion 3 w g v ˆ < v~. Lmma in appndix [2] shows if v ˆ < v~, hn u ( xˆ, ) ( ~, ~ > u x ). W wan o prov ha ˆ τ > ~ τ. Suppos no. B consrains x ˆ + = τˆ and ~ x + ~ = ~ τ, sinc is a fasibl choic whn oal ransfr ~ τ > ˆ τ, bu i is no chosn ( ~ ). So w should hav u ( ~ x, ~ ) > u( xˆ, ), which is a conradicion. Q.E.D. Proposiion 4 and 5 show ha wih consumpion commimns govrnmn will ransfr mor bu promis lss o h unmplod workr. Th inuiion is ha bfor h commimn good is adjusd, h ransfr canno b rducd much bcaus h agn has o spnd a fixd amoun on commimn goods. If govrnmn lowrs h pamns oo much, h unmplod workr would jus cu back sharpl on his consumpion of adjusabl goods. This would induc a larg incras in marginal uili and cra a larg wlfar cos, which couldn b opimal. Wih commimn goods, govrnmn has o mainain a rlaivl highr lvl of unmplomn bnfi o h unmplod han would occur whn all consumpion goods can b adjusd flxibl. Howvr, h highr pamn is a h xpns of fuur claim of consumpion. So h bs srag wih a commimn good is: bcaus of h adjusmn cos, h ransfr should b highr in ordr o kp h commimn good, bu claims for fuur consumpion should b lowr. Proposiion 6: Wih consumpion commimn, sarch ffor is highr compard o bnchmark ( a ˆ > a~ ). 2

13 Proof. From proposiion 4 w hav Q.E.D. v ˆ < v~, hn a ˆ > a~ follows immdial from (2). Th logic of proposiion 6 is as follows: Sinc h incniv for sking a job coms from h diffrnc bwn h uiliis whn mplod and unmplod, a lowr promisd valu in h nx priod mans a havir pnal for coninuing o b unmplod, which will induc a grar sarch ffor. From his propr, w undrsand ha wih consumpion commimns vn if h agn gs mor from govrnmn, h will no bcom lazir bcaus of hos highr pamns. Insad, h agn has o xr a largr ffor in sking a nw job. In his scion w hav discussd som propris of h opimal conrac. Now w procd o solv a paramrizd modl numricall and s h voluion of h opimal unmplomn insuranc conrac. IV. Quaniaiv Analsis Calibraion Now w giv funcion forms and assign paramr valus o h modl and r o g som numrical rsuls. Th uili funcion aks h sandard form γ γ ( x ) u ( x, ) = σ p a σ. W follow Hopnhan (997) o s h form of hazard funcion ra ( ) =, which is an xponnial disribuion wih paramr r. W s = 0. 5 σ, giving an inrmdia dgr of risk avrsion. This numbr ma sm small rlaiv o hos usd in h macro liraur, which ar gnrall abov on. Howvr, i should b akn ino accoun ha w ar using wkl daa for our calibraion xrcis, and h lasici of subsiuion on wkl consumpion is mos probabl largr han ha corrsponding o quarrl consumpion. 2 γ is h shar of oal xpndiur on adjusabl goods whil ( γ ) is consumpion shar of commimn goods. W ak γ as 0.5. Th lowr gamma is, h largr proporion of commimn goods in h consumpion bundl. W will do snsiivi analsis for diffrn valus of γ. Th valu of paramr 2 Argumns from Hopnhan (997). 3

14 r is s b following h sam procdur as Hopnhan (997) o mach a 0% hazard ra as in Mr (990). Th paramr r w g in his xrcis is Discoun facor β is s o b 0.999, rprsning a arl discoun facor of Adjusmn cos K is s o 4 monhs pa. Also, w will chck snsiivi for adjusmn cos in h nx subscion. Th wag ra for an mplod workr is normalizd o b 00. Numrical Rsuls In his subscion, w us h calibraion w g abov o compu numrical rsuls for wo rgims: opimal conrac wih and wihou consumpion commimns. Th lar corrsponds o h on in h classical modl. W wan o xplor h voluion of h opimal unmplomn insuranc conrac in h circumsanc whn consumpion commimns ar considrd. Th rsuls ar compud b sing h iniial promisd valu o b h sam as h on offrd o h unmplod agn undr h currn unmplomn polic a h bginning of h unmplomn priod.. Coss and Valu of h Currn Insuranc Polic Th currn insuranc program is rprsnd b a conrac ha provids 66% rplacmn raio o h unmplod for 26 wks and zro bnfis afrwards. W can solv h cos and valu of h currn insuranc program b using a backward inducion mhod from wk 26 o h bginning of h unmplomn priod. L V 26 dno h valu funcion for an unmplod workr a wk 26. L au V rprsn h auark valu funcion for an unmplod workr wih no ransfrs from h govrnmn. A wk 26, h problm for h unmplod workr is o choos an ffor lvl o maximiz h xpcd uili funcion: whr V x V 26 s.. 26 au = max + a a 26 au [ u( x, ) a + β ( p( a ) V + ( p( a )) V ] = T = max au au au au [ u(0) a + β ( p( a ) V + ( p( a V ] au )) T is a fixd amoun of ransfrs from h govrnmn

15 Whn w solv his problm, w obain h valu funcion in priod 26, which can b usd for solving h valu funcion and ffor lvl in priod 25. Mor gnrall, if w coninu his rcursion, w can solv for h cos and valu funcion up o h firs priod of unmplomn. Th gnral form of h rcursiv problm is as follows: V s.. x = max + a = 25, 24,... [ u( x, ) a + β ( p( a ) V + ( p( a )) V ] = T + 2. Opimal Conracs wih and wihou Consumpion Commimns W compar h rsuls of h wo rgims and show h diffrncs crad b h inroducion of commimn goods and adjusmn coss. Figur shows h squnc of promisd valus ovr h duraion of unmplomn in wo modls. 3 Th smooh lin rprsns h promisd uili valus o h agn ovr unmplomn splls. Th dcrasing rnd is du o h incniv problm: considring unobsrvabl job sarch ffor, h opimal conrac has o rduc h agn s claims for fuur consumpion as punishmn o induc appropria ffor lvl. Wih consumpion commimns, h promisd valus spcifid in h opimal conrac dcras fasr han hos in classical modl bfor h commimn goods ar adjusd. Th kink corrsponds o h momn whn adjusmn is mad. Figur : Promisd valu in h fuur Classical Modl w ih Consumpion Commimns promisd valu nx priod (+004) unmplomn spll (w ks) 3 This graph is obaind wih paramr γ = 0. 5 and K = 4.5 monhs pa. 5

16 Figur 2 provids h main rsuls of his papr. In his graph w plo h rplacmn raio in hr rgims: h classical modl, h modl wih consumpion commimn and h currn polic. Th smooh dcrasing lin illusras h rplacmn raio suggsd b classical modls. Th argu ha h opimal dsign of unmplomn insuranc involvs a coninuous dcrasing rplacmn raio hroughou h unmplomn spll. Th bold lin is h rplacmn raio of h currn unmplomn insuranc program, which is approximal a 66% rplacmn raio for 6 monhs dropping o zro afrwards. From h graph w can s ha hr is a significan diffrnc bwn h opimal conrac providd b classical modls and h currn program. This is probabl wh h currn unmplomn insuranc program has bn widl criicizd: firs, h fla unmplomn bnfi canno b fficin wih unobsrvabl sarch ffor; scond, h jump canno b xplaind as an opimal allocaion. 90% 80% Figur 2: Rplacmn Raio (gamma=0.5, Classical Modl cos=4.5 monhs wag) Currn Polic Opimal wih Consumpion Commimns 70% rplacmn raio 60% 50% 40% 30% 20% 0% 0% unmplomn spll (wks) In his papr w show ha h opimal conrac involvs a rlaivl fla dcrasing rplacmn raio for 25 wks followd b a larg drop. Th dod lin rprsns h opimal rplacmn raio w obain. Th inuiion is as follows: wih adjusmn cos i is opimal for h commimn good no b adjusd a h bginning of unmplomn priods. Th unmplod ar wors off han w hav xpcd bcaus h hav o spnd a fixd amoun of h govrnmn ransfr on commimn goods and consum adjusabl goods wih wha rmains. This will amplif prfrnc risk avrsion and dmand for consumpion smoohing, which rquirs h govrnmn o mainain a highr 6

17 lvl and flar dcrasing rplacmn raio for a priod of im. Howvr, as unmplomn duraion incrass, i will b opimal o pa adjusmn coss and amp o r-opimiz h consumpion bundl. Th logic is ha on h on hand h unmplomn bnfi should b rducd ovr im o induc incnivs; on h ohr hand mainaining commimn goods wih a dcrasing ransfr bcoms mor and mor cosl. 4 So a som poin i will b opimal o pa adjusmn coss afr comparing insuranc bnfis and coss. Whn i s im o chang commimn goods, a larg drop in pamn squnc occurs. Hr h drop is du o wo kinds of ffcs. On ffc is h r-opimizaion of h consumpion bundl: sinc popl can frl balanc bwn adjusabl goods and commimn goods, h govrnmn dosn hav o giv much o h individual. Th ohr, which is mor imporan, is rducd promisd valu. Figur shows ha h promisd valu dcrass fasr wih commimn good han wihou. Wih consumpion commimns, bcaus of h sharp incras of marginal uili, ransfr dosn dcras much bu promisd valu dos. This mans w don hav a mach bwn ransfr during his priod and promisd valu in h fuur as w do in classical modls. Whn i s im o chang h commimn goods, h promisd valu has bn rducd o a vr low lvl. From proposiion 2, in which w sa h ransfr is an incrasing funcion of currn promisd valu, w can xpc h opimal ransfr afr h adjusmn would b rducd o a vr low lvl. From h graph, w can s ha h opimal conrac in our modl fis h currn program wll, which involvs a rlaivl fla pamn squnc and an abrup disconinui. Furhrmor, w can also compu how clos h opimal conrac w obaind horicall is o h currn polic in pracic. Th xpcd coss of h opimal conrac in hor and h conrac undr currn polic can b obaind as follows: 4 Bcaus of h concavi of uili funcion. 7

18 C( V C 0 ( V, 0, ) = β K + = 26 ) = β K + 26 = = ( p( aˆ )) β 26 = ( p( a ˆ τ + )) β T = + = + ( p( a )) β τ whr C V, ) and C V, ) ( 0 ( 0 ar h oal coss of h opimal conrac in our modl and h currn unmplomn program rspcivl. is h im whn commimn goods ar adjusd in h opimal conrac. a ˆ, a and a ar opimal choics of sarch ffor for h unmplod workr. V 0 is h promisd wlfar valu ha h currn unmplomn ssm provids a h bginning of a priod of unmplomn. Ohr noaions ar consisn wih prvious sings. Tabl : Cos Savings Iniial wlfar lvl Cos Normalizd Cos Cos Saving Currn ssm % 0% Opimal conrac %.7% Tabl compars oal coss of h opimal conrac and h currn ssm givn h sam iniial wlfar lvl. From hs numbrs, w can s ha opimal conrac is mor fficin han currn program in ha i provids h sam wlfar bu incurs lss cos. Th cos savings com from incniv fficinc ha h currn program canno achiv wih a squnc of fla unmplomn bnfis. This is also wh h currn program has bn widl criicizd. Howvr, w find ha wih consumpion commimn, h cos savings achivd b h opimal conrac compard wih currn polic is vr modra, onl.7%, which mans our currn unmplomn program is vr clos o h opimal rsul w can achiv. Thrfor our rsuls giv an xplanaion o jusif currn polic. I shows ha currn polic isn as bad as rsarchrs hav radiionall blivd. Th diffrnc bwn h opimal conracs w g horicall and h on in pracic is small and could b xplaind b adminisraion cos. Figur 3 compars ffor lvl in h wo modls. Sarch ffor is highr in h modl wih commimns han wihou. This is du o h sharp dcrasing promisd valu, which inducs a highr incniv o find a job. Th rsul hr is inrsing: w don los fficinc in incnivs vn if a highr ransfr o h workr is mainaind. Popl 8

19 don bcom lazir bcaus of h ransfrs. On h conrar, h hav o xr mor ffor in sarching o kp h commimn good h prviousl mad. Figur 3: Sarching Effors Classical Modl w ih Consumpion Commimns sarching ffors unmplomn spll (w ks) Figur 4 plos h probabili of rmaining unmplod for boh modls. Th uppr smooh lin is h probabili of rmaining a unmplomn saus ovr im in h classical modl. Sinc during vr priod h workr spnds som amoun of im sarching for a job, h probabili of rmaining unmplod is dcrasing ovr im. Th lowr dod lin rprsns ha probabili in h modl wih consumpion commimn. Th lowr probabili w g is consisn wih h rsuls in figur 3. Th highr sarch ffor will rsul in a lowr probabili of saing unmplod. Figur 4: Prob of Rmaining Unmplod 00% 80% Classical Modl w ih Consumpion Commimns probabili 60% 40% 20% 0% unmplomn spll (w ks) W also plo h graphs for consumpion of boh goods (commimn, adjusabl) in h wo modls and pu hm in appndix [6]. 9

20 Snsiivi Analsis In his subscion w wan o chck if h propris of h opimal conrac w g in h modl ar snsiiv o changs in paramrs. Firs, w chck h snsiivi o chang in consumpion shar γ. Th highr γ mans lowr proporion of commimn goods in oal xpndiur. For diffrn γ from 0.3 o 0.7, w plo opimal rplacmn raio valus in figur 5. Figur 5 gamma=0.3, cos=3 monhs w ag gamma=0.4, cos=3 monhs w ag 00% 00% rplacmn raio 80% 60% 40% 20% rplacmn raio 80% 60% 40% 20% 0% unmplomn spll (w ks) 0% unmplomn spll (w ks) 00% gamma=0.5, cos=3 monhs w ag 00% gamma=0.6, cos=3 monhs w ag rplacmn raio 80% 60% 40% 20% rplacmn raio 80% 60% 40% 20% 0% % unmplomn spll (w ks) unmplomn spll (w ks) To s h ffcs du o chang in consumpion shar, w fix h adjusmn cos a 3 monhs pa. From h graphs w can s ha as γ incrass wo hings nd o b noicd. On is ha h ra of dcras of h rplacmn raio bcoms spr. Th inuiion is ha whn commimn goods accoun for a smallr proporion of oal xpndiur, h consumpion bundl can b adjusd mor flxibl; hrfor unmplomn bnfis can b rducd mor quickl. Th ohr hing w can find is ha h grar γ, h longr h priod susaind bfor adjusmn cos is paid. Th logic hr is ha a smallr proporion of commimn goods man popl hav mor flxibili 20

21 o balanc consumpion, so h urg o adjus commimn goods will b smallr. From hs graphs w g ha, ovrall, h im and magniud of h jump ar no snsiiv o chang in γ. Nx w do snsiivi analsis on adjusmn cos K. W choos h cos from monh s pa o 4 monhs pa, whil kping h consumpion shar a 50% prcn for boh goods. W plo h rsuls in figur 6. Ths graphs show ha h smallr h cos, h soonr commimn goods ar changd, which is vr inuiiv. Manwhil w sill s a larg drop whn adjusmn occurs. Snsiivi analsis shows ha our rsuls ar robus. Figur 6 gamma=0.5, cos=4 monhs w ag gamma=0.5, cos=3 monhs w ag 0 0 % 00% rplacmn raio 80% 60% 40% 20% rplacmn raio 80% 60% 40% 20% 0% % unmplomn spll (w ks) unmplomn spll (w ks) rplacmn raio 0 0 % 90% 80% 70% 60% 50% 40% 30% 20% 0 % 0% gamma=0.5, cos=2 monhs w ag unmplomn spll (w ks) rplacmn raio gamma=0.5, cos= monh w ag 00% 80% 60% 40% 20% 0% unmplomn spll (w ks) V. Conclusion and Exnsions Currn unmplomn insuranc polic has bn widl criicizd bcaus of h prvrs ffcs i has on h incnivs for rmplomn and h disconinui in is squnc of pamns. Classical modls argu ha h opimal unmplomn conrac 2

22 should involv a coninuous dcrasing rplacmn raio hroughou h unmplomn priod. In his papr w incorpora consumpion commimns ino h opimal unmplomn insuranc dsign. Commimn goods amplif prfrnc risk avrsion so ha dmand for consumpion smoohing bcoms largr. Th rsuls w g ar qui diffrn from hos of classical modls. Th opimal conrac w obaind involvs a rlaivl fla dcrasing rplacmn raio for 25 wks followd b a larg drop o a vr low lvl of ransfr. Th rsuls fi currn polic wll, which givs an xplanaion o jusif h currn unmplomn insuranc program in pracic. Furhrmor, w hav anohr inrsing finding: vn wih highr and mor mildl dcrasing unmplomn bnfis, h unmplod workrs don bcom lazir. On h conrar, hir incnivs o find a job ar highr if h wan o kp h commimn goods h prviousl had. Th implicaion hr is ha w don los fficinc in incnivs vn if a highr ransfr o h workr is mainaind. In his papr, w show ha currn polic isn as bad as rsarchrs hav radiionall blivd. Th cos savings if w chang from currn polic o h opimal conrac ar onl.7%, which mans ha h currn unmplomn program is vr clos o h opimal rsul w can achiv. Th diffrnc bwn h opimal conracs w g in hor and h currn polic in h ral world is small and could b xplaind b adminisraiv coss. Th firs xnsion of his papr is o add asss ino h modl and allows for saving bhavior. In our modl, w solv h social plannr s problm in which h adjusmn cos is akn as a social cos. Howvr in pracic i is h individual who maks h dcision o chang consumpion. And adjusmn cos usuall aks h form of ass dvaluaion. For xampl, h ransacion cos of slling a hous is a kind of dcras in h ass valu. Anohr xampl is slling a car; h cos is h diffrnc bwn h mark rsal valu and h acual valu. Sinc so far m modl dosn accoun for asss, such dvaluaion cos canno b rflcd in h budg consrain. So wha w do is o add asss ino h modl and dcnraliz h problm. Th scond xnsion is rlad o h firs on: whn govrnmn dosn hav h abili o conrol h agn s consumpion bhavior, w hav o considr h incniv compaibili o induc h individual o mak adjusmn a h righ im. Anohr xnsion dals wih h discound uili valu for an mplod workr. In our modl w assum h wag ra 22

23 dosn dpnd on unmplomn hisor bu mos liraur considrs h wag ra o b a funcion of h workr s prvious unmplomn xprinc. W ar going o rlax his assumpion in h fuur. In addiion, w ar going o do som wlfar analsis o compar h classical modl, our modl and h currn polic. 23

24 Rfrncs []Ch, Raj. Consumpion Commimns, Unmplomn Duraion and Local Risk Avrsion. NBER Working Papr 02, [2]Grubr, Jonahan. Unmplomn Insuranc, Consumpion Smoohing, and Priva Insuranc: Evidnc from h PSID and CEX. Rsarch in Emplomn Polic, 998 [3]Hopnhan, Hugo, and Nicolini, Juan Pablo. Opimal Unmplomn Insuranc. Journal of Poliical Econom 05, 997. [4]Mr, Bruc D. Unmplomn Insuranc and Unmplomn Splls. Economrica 58, Jul 990. [5]Phlan, Chrisophr, and Townsnd, Robr M. Consumpion Muli-priod, Informaion-Consraind Opima. Rviw of Economic Sudis 58, Ocobr 99. [6]Shavll, Svn, and Wiss, Laurnc. Th Opimal Pamn of Unmplomn Insuranc Bnfis ovr Tim. Journal of Poliical Economics 87, 979. [7]Spar, Sphn E., and Srivasava, Sanja. On Rpad Moral Hazard wih Discouning. Rviw of Economic Sudis 54, Ocobr 987. [8]Warrn, E. and Tagi, A.E. Th Two-Incom Trap: Wh Middl Class Mohrs and Fahrs ar Going Brok. Basic Books,

25 APPENDIX [] Afr h commimn good is adjusd, w can g h firs-ordr condiions: F. O. C. L = τ + β ( p( a)) W ( v ) + λ + ηβ Envlop condiion: [ p ( a)( v v' )] { v [ u( τ, ) a + β ( p( a) v + ( p( a)) v' )]} p ( a) p ( a) W ( v ) = λ + η = η (A) p( a) u ( τ, ) p( ) a β p ( a) W ( v ) ηβp ( a)( v v' ) = 0 (A2) u τ, ) = u ( τ, ) (A3) ( 2 p ( a) W ( v) = λ = = W ( v ) + η (A4) u ( τ, ) p( ) a [2] Proof of Lmma. Proof. If v ˆ ˆ > v 2 h u ( xˆ, ) ( ˆ < u x2, ) ; If v > v 2 h u ( x, ) < u( x2, 2 ), givn hazard ra funcion p ra ( a) =. Proof. From (9) u( xˆ, aˆ ) + β ( p( aˆ) v + ( p( aˆ)) vˆ ) = v u( xˆ, u( xˆ, u( xˆ, ) aˆ + β ( v ) aˆ + β ( v ) aˆ + β ( v ( p( aˆ))( v p ( aˆ) ( v r ) = v βr vˆ )) = v vˆ )) = v (A5) (A6) whr (A5) is go givn p β ˆ)( vˆ' (2). ra ( a) =, whil (A6) uss p ( a v ) = If v ˆ ˆ > v 2, from (2) w hav a ˆ ˆ < a2. From (A6), w g u ( xˆ, ) ( ˆ < u x2, ). Th scond par of lmma can b provd similarl. Q.E.D. [3] Proof of Proposiion 3 Proof. W wan o prov if v < v2, hn v ˆ ˆ < v 2. Suppos no. From Lmma, w g u ( xˆ, ) ( ˆ < u x2, ), hrfor x ˆ ˆ < x2 < (A7) u ( xˆ, ) u ( xˆ, ) From (23) w g η p ( aˆ) C( vˆ ) p ( aˆ)( v vˆ' ) ˆ = 2 25

26 From (2) βp ( aˆ)( v vˆ' ) = η From (22), w hav C ( vˆ C ) = u ( ˆ τ [ p ( aˆ) ] 2 C( vˆ p ( aˆ) ˆ = ) = u ( ˆ τ, [ p ( aˆ) ] ) βηp ( aˆ) ) ( p( aˆ)) 3 C( vˆ ) β ) p ( aˆ)( p( aˆ)) ( vˆ, = βp ( aˆ) C( vˆ u ( ˆ τ, ) ) (givn p ra ( a) = ) = C ( vˆ ) ( ˆ) ( ˆ + βp a C v, ) u ( ˆ τ, ) From (A0) w g (A8) C vˆ ) + β p ( aˆ ) C( vˆ ) < C ( vˆ, ) + βp ( aˆ ) C( vˆ, ) (A9) ( Sinc w suppos v ˆ ˆ > v 2, from (2) β p ( aˆ)( v vˆ' ) =, w can g p ( aˆ ) ( ˆ > p a2). Bcaus h cos funcion is incrasing and convx, w should hav C ( vˆ ) ( ˆ ) ( ˆ, ) ( ˆ, ) ( ˆ ) ( ˆ + β p a C v > C v2 + βp a2 C v2, ), which conradics wih (A9). Q.E.D. [4] Bnchmark Modl C v) = Min ~ τ + β ( p( a ~ )) C( ~ ) { } ( v ~ x, ~, ~ τ, a ~, ~ v (A5) s.. u( ~ x, ~ ) a~ + β ( p( a ~ ) v + ( p( a ~ )) v ~ ) v (A0) { ( ~ ~ ) ~ ( ~ u x, a + ( p a) v + ( p( a ~ )) ~ ) } a ~ arg max β v (A) ~ x + ~ = ~ τ (A2) whr v is dfind as: v u( x *, * ) = β * * * * s.. u ( x, ) = u ( x, ) and x * + * = w 2 [5] Proof of Proposiion 4. Proof. As in proof of Proposiion 3, w g C vˆ ) ( ˆ) ( ˆ = βp a C v, u ( ˆ τ, ) ( 26 ) (givn p ra ( a) = )

27 β p ( aˆ) C( vˆ ) (, ) ( ˆ = C v C v, ) = u ( ˆ τ, ) u ( ˆ τ, ) (A3) Similarl, w can g β p ( a ~ ) C( v ~ ) = C ( v) C ( v ~ ) = u ( ~ ~, ~ ) ( ~ ~, ~ τ u τ (A4) ) Suppos v ˆ = v~. Thn from Lmma, w hav u( ˆ τ, ) ( ~ ~, ~ = u τ ). Sinc h givn lvl of saisfis > ~, w hav ( ˆ τ ) ( ~ ~ < τ ). Sinc w suppos h bs soluion o a givn sa ( v, ) is ha v ~ = vˆ, w g v ~ = v ˆ. Again from Lmma, w hav u ˆ τ, ) = u( ~ τ ~, ~ ). Similarl, w g ( ( ˆ τ ) ( ~ ~ < τ ).Bcaus ~ < ~ and uili funcion is sricl concav, hn w g [ ~ ~ ) ( ˆ τ )] < [( ~ τ ~ ) ( ˆ τ )] ( τ. β p a~ ) C( v ~ ) βp ( aˆ) C( vˆ ( ) = u ( ~ ~ ~ ( ~ ~, ~ ( ˆ ( ˆ τ, ) u τ ) u τ, ) u τ, ) (A5) = u ( ~ ~, ~ ) ( ˆ ~, ~, ) ( ~ ) ( ˆ τ u τ u τ u τ, ) If w assum u > 0, hn (A5)<0, which implis C ( v~ ) C( vˆ ), whr v ~ = vˆ. Conradicion. Sinc β p aˆ) C( vˆ ) = C ( v, ) C ( vˆ, ) is oo small whn v ~ = vˆ, w should hav [6] ( v ~ > vˆ, givn h convxi of cos funcion. Q.E.D. 50 Consumpion of Adjusabl Good (gamma=0.5, cos=4 monhs wag) classical modl wih consumpion commimns consumpion unmplomn spll 27

28 consumpion Consumpion of Commimn Good (gamma=0.5, cos=4 monhs wag) classical modl wih consumpion commimns unmplomn spll