A sufficient statistic approach to disability insurance with application to the United States Social Security Disability Insurance program

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1 Insurne Mrkets n Compnies: Anlyses n Aturil Computtions, Volume, Issue, 202 Henry R. Hytt (USA) A suffiient sttisti pproh to isility insurne with pplition to the Unite Sttes Soil Seurity Disility Insurne progrm Astrt The Unite Sttes Soil Seurity Disility Insurne (SSDI) progrm s enefit level is perioilly moifie y poliymkers. However, few eonomists hve ompre its enefit level ginst welfre riterion. This pper tkes moel of optiml isility insurne enefits, presente in Dimon n Sheshinski (995), n imposes smll numer of wek ssumptions tht llow suffiient sttisti pproh to ssess the optiml SSDI enefit level. Tking empiril results from stuies on the inentive effets of soil insurne n the onsequenes of isility, lirtion implies tht it is iffiult to onlue tht the existing replement rte for SSDI prtiipnts is inonsistent with the optimum inite y the Dimon-Sheshinski moel, given the lrge rnge of estimtes for the optimlity onition s key prmeters: risk version, SSDI prtiiption responsiveness to the enefit level, n the employment effets of pyroll txtion. Keywors: isility, txtion, enefits, soil seurity, isility insurne, soil insurne. Introution In orer to insure workers ginst ny long-term employment-limiting isility ours prior to retirement eligiility, the Unite Sttes government mntes tht ll employees prtiipte in Soil Seurity Disility Insurne (SSDI) progrm. This progrm ministers roughly $80 illion ollrs in enefits nnully to nerly 0 million enefiiries. Empiril stuies of SSDI enrollment, suh s Boun (989) n Kreier (999), inite tht progrm ministrtion involves ommon form of morl hzr: higher enefit levels inue greter enefit utiliztion. While lrge numer of stuies hve onsiere the prolem of the optiml soil insurne in the presene of morl hzr, most stuies with strightforwr poliy onlusions hve fouse on Unemployment Insurne (UI). Reltively little work hs resse the optiml SSDI enefit level. In this stuy, I propose n implement metho for pplying the well-known optimlity onition propose y Bily (978) to SSDI. Strting with the moel of optiml SSDI evelope y Dimon n Sheshinski (995 herein revite DS), I erive formul for the optiml isility enefit tht is funtion of reltively smll set of estimle prmeters. The formul is then pplie to ssess whether the SSDI enefit is optiml or not. The enefit rte is the only spet of the SSDI progrm uner onsiertion in this pper. I o not onsier other very importnt poliy questions, suh the optiml sreening rule or the witing perio for SSDI, espite the importne of these questions. I lso ignore the metho of estlishing n ernings history for enefit lultion, n justments for epenents. The eqully importnt question of the optiml egree of progressivity in SSDI txtion n enefit istriution is lso eyon the sope of this nlysis, s is the question Henry R. Hytt, of the optiml ssessment of SSDI pyroll txes etween firms n employees. I emonstrte tht the first orer onitions of the DS moel imply reltionship etween the optiml ifferene in mrginl utility etween workers n progrm prtiipnts, n the morl hzr inue y rising revenue n proviing enefits. This optimlity onition n e lirte fter mking two restritions tht o not pper in the DS moel: tht the utility of onsumption is onstnt etween workers n enefit limnts, n tht the higherorer terms of the utility funtion re smll. I then present wht the optimlity onition implies, given the existing empiril estimtes on () the effet of pyroll txes on lor supply, (2) the effet of the enefit level on SSDI prtiiption, n () iniviul risk version. Clirtion of the optimlity riterion permits inferene on whether the rop in onsumption tht ours when someone prtiiptes in SSDI is optiml, given smll set of prmeters. However, review of the literture yiels wie rnge of estimtes for eh of the three key prmeters liste ove. We n nevertheless rw onlusions out the sets of prmeter vlues tht imply tht enefits re t their optiml level: if iniviul risk version is low (oeffiient of reltive risk version roun -2) n SSDI prtiiption is not very responsive to the enefit level ( prtiiption elstiity of 0.5 or lower), the rop is roughly optiml. However, if rtes of risk version re higher (roun 4-5), then the rop in onsumption is only optiml when the SSDI prtiiption elstiity is higher, roun. Despite its iffiulty in rwing n ext onlusion out whether SSDI enefits re t their optimum, this stuy shoul serve s useful referene point for empiril reserh, euse it very lerly speifies wht empiril estimtes re most neessry for welfre lultions of the SSDI enefit level. This pper is similr in priniple to the suffiient sttisti literture tlogue y Chetty (2009), ut one importnt methoologil ifferene ples it

2 Insurne Mrkets n Compnies: Anlyses n Aturil Computtions, Volume, Issue, 202 outsie of the frmework of these ppers. The ppers he tlogues propose setting in whih soil plnner is onsiering the level of one istint poliy instrument, n solving involves implementtion of the envelope onition for gent mximiztion. In this pper, I onsier the DS moel in whih the soil plnner hs two poliy instruments, nmely, the level of enefits for SSDI prtiipnts n the level of enefits for those who neither work nor prtiipte in SSDI. This mens tht I onstrut formul tht inlues elstiities onstrute from prtil erivtives. The pper proees s follows. I first isuss the SSDI progrm n review the eonomi literture ssoite. I then present the DS moel, erive n impliit optimlity onition, n isuss the simplifying ssumptions neessry to lulte the optiml enefit. I then isuss the prmeters utilize y the moel n provie n ssessment of the onitions uner whih the urrent enefit regime is n optiml poliy. I then isuss the implitions of this lirtion exerise n ttempt to provie some iretion for future empiril reserh.. Disility insurne.. rogrm etils. SSDI is feerlly mnte inome support progrm for iniviuls whom the government fins unle to engge in sustntil ginful tivity ut re not eligile for retirement enefits euse of their ge. In orer to prtiipte, pplints must ern no more thn $860 per month uring perio of five months, fter whih the pplint is sreene y otor. If the otor, following government-mnte set of riteri (whih inlues eution n ouptionl history), etermines tht the pplint is inple of engging in sustntil ginful tivity ue to illness, he is mitte into the DI progrm. Roughly fifty perent of pplints re rejete nnully, n they must wit one full yer efore re-pplying. Those who re epte reeive enefit, whih is inresing in pre-isility inome t elining rte. As reporte in Autor n Duggn (2006), limnts who, efore pplying to isility, erne inome t the 0 th perentile of the ntionl wge istriution reeive pproximtely 60% of their pre-progrm ernings, n those erning inome t the 90 th perentile reeive 25% of their pre-progrm ernings. The Soil Seurity Aministrtion ministers SSDI n mntory retirement progrm Ol Age n Survivors Insurne (OASI), long with other soil insurne progrms. SSDI n OASI re finne y pyroll tx of 2.6% of the first roughly one hunre thousn ollrs of the nnul ernings of eh employe person, hlf of whih is pi y employers, the other hlf is pi y employees. Soil Seurity Disility Insurne is itself finne y pyroll tx of.8% (OASI is finne with the remining 0.8%). The SSDI pyroll tx is reset infrequently n oes not hnge from yer to yer se on the progrm s revenues n expenitures. The two most reent hnges in the pyroll tx rte, in 994 n 2000, hve oinie with opposite hnges in the OASI pyroll tx, so tht the pyroll tx use to finne these two progrms hs remine t 2.6% of pyroll sine revious eonomi nlyses. Severl stuies of SSDI hve emonstrte tht progrm prtiiption is to the enefit level. Some stuies, suh s rsons (980) n Boun (989), employ rosssetionl estimtes, while other stuies, suh s Hlpern n Husmn (986) n Kreier (999), ientify the effets of the SSDI enefit level using struturl moels. These ppers o not ssess the welfre onsequenes of their prtiiption estimtes. Other stuies, inluing Dimon n Sheskinski (995) n Lozhmeur (2006), speify struturl moels in orer to rw qulittive onlusions out the struture of optiml isility insurne progrms. The gol of these stuies is to speify the optiml poliy of welfre-mximizing soil plnner. However, these stuies o not formlly inorporte estimtes of empiril stuies into their optimlity onitions. From the SSDI literture, the nlysis in Boun et l. (200) is the losest to this stuy. The uthors tie together empiril nlysis of SSDI on prtiiption n the welfre onsequenes of inresing SSDI enefits y %. The uthors mke lrge numer of ssumptions in orer to onut this welfre lultion, suh s ssuming tht the eweight uren of the itionl pyroll txes neessry to finne this inrese in the enefit level is zero, tht there is no utility gin from those who re-optimize their lor supply eision fter the enefit inrese, tht there is n elstiity of - etween the SSDI enefit level n the ernings of SSDI limnts spouses, n tht the representtive iniviul hs utility funtion tht exhiits onstnt reltive risk version of prtiulr form. The pproh I tke iffers from the nlysis one in Boun et l. (200) in tht it fouses on estimting moel using smll set of suffiient sttistis, n explores the optiml level of isility insurne rther thn welfre lultion. I lso resses roer question: whether, given the rnge of empiril estimtes for its key prmeters, the null hypothesis tht the urrent enefit level is optiml n e rejete. This nlysis gretly simplifies the si welfre prolem n highlights wht the key prmeters for welfre nlysis re. 2. The moel This setion presents the DS moel n erives testle optimlity onition. This moel onsiers n eonomy in whih the popultion is ivie etween those who work, those who o not work n 27

3 Insurne Mrkets n Compnies: Anlyses n Aturil Computtions, Volume, Issue, 202 reeive SSDI, n those who o not work n reeive nother enefit, whih n e thought of s welfre enefit. The government s prolem is to set the SSDI enefit n welfre enefit levels. This moel iffers from UI moels tht explore simpler optimlity onitions. Bily (978) n Chetty (2006) onsier moels where gents n e in one of two sttes: either employe or prtiipting in UI: the uthors o not onsier unemployment prt from prtiiption in single progrm. Kroft (2008) moels enefit tke-up in frmework in whih gents either work, prtiipte in UI, or re unemploye without progrm prtiiption. Those who re unemploye without prtiipting in UI reeive no enefits. In ll of these moels, the government only hs one poliy mehnism, the UI enefit level. 2.. Bsi form. Following DS, onsier n eonomy in whih gents proue one unit of single onsumption goo when working n nothing otherwise. These gents vry in isutility of working, whih is istriute oring to F(). The government hs n imperfet sreening mehnism tht it uses to seprte iniviuls with high isutility of working from those with low vlue. Speifilly, eh iniviul hs proility p 0, of eing mitte into isility ompenstion progrm, where p is ontinuous funtion n wekly inresing in. The government sets the onsumption of iniviuls who work, those who re sreene s hving high isutility of work, n those who o not work ut o not signl high isutility of work. Eh iniviul reeives onsumption utility u(.) n v(.), while working n not working, respetively, oth of whih re inresing in onsumption. Then there re unique vlues of n efine y u v n u v, where n iniviul is inifferent etween not working n either prtiipting in the isility progrm or not working t ll without isility ompenstion respetively. Then ll iniviuls with vlue of less thn will lwys work n ll iniviuls with greter thn will never work, regrless of whether the iniviul is on isility. Assume in the following tht there is n interior solution to the optiml enefit prolem, i.e. tht, 0 <. In this pper, I efine the goverment s optimiztion prolem s how to set onsumption levels, n. Before eginning the forml mthemtil tretment, it is helpful to onsier, roly, wht hppens when the government justs the level of onsumption for group. Consier, for exmple, the effet of n inrese in the isility enefit,. This hnge is shown in Figure. Fig.. An inrese in the enefit level in the Dimon- Sheshinski moel The isutility vlue t whih the utility of employment is equl to the utility of isle onsumption is = u( ) v( ). In orer to finne this inrese in enefits ereses n, onsequently, u( ) ereses. Likewise the inrese in enefits uses the v( ) inrese of. These two effets use to erese to : in other wors, when isility enefits re higher, less isle iniviuls will hoose to prtiipte in the progrm. The ifferene etween the onsumption vlue of employment, u( ), n the outsie option v( ) grows s well, lowering the isutility threshol t whih gents hoose to never work regrless of prtiiption in the progrm = u( ) v( ) from to. In summry, n inrese in enefits will rise the numer of iniviuls on isility, rise the numer who pursue no ginful tivity ut re not on Disility Insurne, n lower the numer who work. Following DS, the plnner prolem is: mx u F p v p u F p v p v F sujet to the uget onstrint 0 F p p F p p F 0. (2) () Note tht this pper only onsiers the enefit level s speifie y the DS moel. DS onsiere the se in whih the government tightens or relxes the sreening rule. Aing severity optimiztion prolem woul hnge the optiml enefit levels through hnging the elstiity prmeters n popultion frtions t n optiml sreening rule. 28

4 Insurne Mrkets n Compnies: Anlyses n Aturil Computtions, Volume, Issue, 202 The first orer onitions with respet to, n re, respetively, u F u p F p f p f u, 0 () v p F p f v (4) n v pf p f v. (5) These first orer onitions ppere in the originl pper y Dimon n Sheshinski (995). It is possile to go further n write own n ext formul for the optiml enefit level s follows. For onveniene, let the frtion of iniviuls who work e lle 0 F p n the frtion of iniviuls who prtiipte in the isility ompenstion progrm 2 p F, n the frtion of iniviuls who neither prtiipte in the SSDI progrm nor work p F. Let F, e the elstiity of the frtion of the lor fore tht works with respet to the pyroll tx rte,, the elstiity of the frtion of the lor fore tht works with respet to its own onsumption, p 2, e the elstiity of the frtion of the lor fore on SSDI with respet to the enefit rte, n e the elstiity of the frtion of the popultion reeiving the seon enefit with respet to the level of tht enefit. Note tht these elstiities re expresse in terms of the prtil erivtive of the frtion of the popultion either working or on isility with respet to the enefit level or pyroll tx rte, respetively, holing the other ifferentitor onstnt. For exmple, 2 v p f v. u p f u p f u 2, It is useful to note the following prtil erivtives of the popultion shres with respet to the onsumption prmeters n : 2 p f v p F, (7) p 2 f u p F, (8) (9) pf u pf n f u p f u p f u F p F. 0 (0), p, is the elstiity of lor fore prtiiption with respet to the pyroll tx rte, holing the two enefit levels onstnt Derivtion of n optimlity onition. I now erive n impliit optimlity onition. Rerrnge these first orer onitions n ivie them to get: Now, we n rerrnge terms to show: 2 (6) 2 v, u () p f v. The government s lne uget onstrint implies tht 2, (2) Comining the rerrnge first-orer onitions with the lne uget onstrint n rerrnging terms, the proportionte ifferene etween mrginl utilities etween onsumption while working n onsumption while reeiving isility enefit n e written s follows: 2 k 2 p2,, v u u 2 2,,, () 29

5 Insurne Mrkets n Compnies: Anlyses n Aturil Computtions, Volume, Issue, 202 where k is the rtio of the SSDI onsumption to the lternte enefit onsumption. Thus, the moel llows the proportionte ifferene in mrginl utilities to e expresse s funtion of few prmeters: the frtion of the popultion tht is employe, the frtion prtiipting in the isility progrm, the frtion of the popultion tht neither prtiiptes in SSDI nor works, the elstiity of SSDI prtiiption with respet to the enefit level, n the elstiity of lor fore prtiiption with respet to the tx rte. This optimlity onition hs numer of intuitive properties. The proportionte ifferene in mrginl utility t the soil optimum is eresing in the egree of prtiiption morl hzr, n is inresing in the extent to whih pyroll txes isourge work, so long s the frtion of the popultion tht works is suffiiently lrge. This is euse higher elstiity mkes the seon enefit less le to e use to inrese welfre, therey eing seon soure of inresing the optiml onsumption of those who eie to work. In other wors, when the lterntive enefit is more ostly to provie, the pyroll tx is lower. 2.. Fitting the optimlity onition into suffiient sttisti frmework. Eqution (2) is implie iretly y the DS moel. However, it is possile to mke two further ssumptions tht permit the lirtion of this moel. These re the itive seprility of onsumption n leisure n tht the thir-orer onitions of the utility funtion re smll. An itively seprle utility funtion mens tht ifferent levels of utility of leisure hnge the optiml enefit y hnging the vlues of where gents re inifferent etween work n its lterntives, n so it enters into the ove formul through the elstiities of progrm prtiiption n lor fore prtiiption with respet to the level of isility enefits. This is the reltionship tht hs een lirte y Bily (978), Gruer (997), Chetty (2006) n Kroft (2008) for unemployment insurne. Noting tht, when the thir-orer n higher terms of the utility funtion re smll, u u, (4) les to,, k 2 p2, 2, 2 2, where is the rte of reltive risk version.. rmeteriztion (5) In this setion, I lirte eqution (5) using prmeters foun in the roer literture on txtion n soil insurne. The vlues of the prmeters selete re shown in Tle, n their soures re expline in the susetions tht follow. Tle. rmeter vlues n soures for Dimon-Sheshinski lirtion rmeter Symol Vlue Soure Frtion working 9% Frtion reeiving isility enefit Frtion reeiving welfre enefit Elstiity of SSDI prtiiption to the SSDI enefit rte Inome elstiity of lor fore prtiiption Mrginl effet on SSDI prtiiption Mrginl effet on welfre prtiiption Rtio of SSDI enefits to welfre enefits Drop in onsumption etween work n SSDI prtiiption 2 6% % p 2, 0-0.2, Estimte from the following soures: US Bureu of Lor Sttistis (202): 2 million people ge in the lor fore. US Soil Seurity Aministrtion (2006): 8 million SSDI limnts in 2005 US Census Bureu (202):. million TANF househols in 2005 Boun n Burkhuser (999): with respet to pplition rte: with respet to nonprtiiption rte: Upper oun pinne own y 0-0.2,,, Upper oun pinne own y 0-0.2,,, k 2 20%-60% Risk version -5 Gruer n Krueger (990): Lor fore prtiiption elstiity in the rnge of Gruer (997): ~0 Anerson n Meyer (998): ~ 0 Soil Seurity Aministrtion (2006): Monthly DI enefit for 2005 is ~$760 US Census Bureu (202): In 2005, TANF-prtiipting househols reeive $5/month Autor n Duggn (2006): Replement rtes re in the rnge of 25%-60% Meyer n Mok (2007): Foo expenitures rop 20%, housing expenitures rop 25% Brsky et l. (997): ~4 Cohen n Einv (2007): Chetty (2006): -2 0

6 Insurne Mrkets n Compnies: Anlyses n Aturil Computtions, Volume, Issue, opultion shres. Clirtion of the DS moel involves settling on levels of the popultion tht re in the ifferent groups uner onsiertion: those who work, those who re on SSDI, n those who reeive welfre enefit. However, it is not ovious how to efine the popultion tht reeives welfre enefit. It oul e efine to e the group of rejete pplints who re not working, s the pool of iniviuls in the ge rnge tht mkes them eligile for SSDI ut o not work. These iniviuls my reeive inome from ny numer of soures, inluing from fmily memers. Government progrms tht trget this popultion inlue Supplementl Nutrition Assistne rogrm enefits (foo stmps), Meii, Temporry Ai to Neey Fmilies (TANF), n numer of other trnsfer enefits. In the lirtion tht follows, I onsier ult TANF reipients to e the group of iniviuls tht neither works nor reeives SSDI. In wht follows I will onsier the following popultion for I strt with the 2 million people in the lor fore ge 25-64, erive from the US Bureu of Lor Sttistis (202). In the lirtion tht Aoring to the BLS Intertive CS Tles, 67% of ll ults ge 6-64 re enrolle in the lor mrket, n Autor n Duggn (2006) report tht 4% of ll ults ge re on isility. There re. million reipients of Temporry Assistne for Neey Fmilies (TANF), less thn hlf the numer of SSDI enefiiries. Tking 2% s n upper oun on the numer of ults ge who re reeiving TANF, some rough strting figures for lirting the Dimon n Sheshinski moel oul e tht = 9% n 2 = 6%, so onsequently the frtion tht neither works nor re enrolle in SSDI is = %. Of ourse, one woul get ifferent popultion shres y onsiering the frtion of the popultion tht onsists of rejete SSDI pplints. Furthermore, there re roughly ten million ults who reeive foo stmps, so onsiertion of this s the lterntive enefit woul, of ourse, yiel ifferent nswer..2. The elstiity of SSDI prtiiption with respet to the enefit level. A survey y Boun n Burkhuser (999) summrizes most of the stuies of the responsiveness of SSDI prtiiption with respet to the SSDI enefit level. Two tles provie rnges of elstiity estimtes for two spets of prtiiption: the pplition rte n non-employment. The elstiity of the pplition rte with respet to the SSDI enefit level rnges from 0.2 to. the lrger of whih ome from the struturl estimtes of Hlpern n Husmn (978) n Kreier (999). The elstiity of the non-employment rte with respet to the enefit level rnges from 0.2-, with the highest estimte oming from rsons (980)... The rtio of SSDI enefits to welfre enefits. There re severl welfre enefits tht one oul onsier in lirting the DS moel. However, it my e useful to keep stnr outsie enefit in min. In prtie, limnts reeive some hunres of ollrs when prtiipting in vrious welfre progrms. However, SSDI reipients n reeive muh more, on the orer of two thousn ollrs per month. Given this sustntil heterogeneity, I will ssume tht the urrent strting point is tht SSDI reipients reeive roughly twie s muh in nnul inome s those who re not working ut prtiipting in nother welfre progrm. Aitionl preision of this prmeter woul of ourse e useful, ut even proportiontely lrge perturtion will not ffet the optiml rop in onsumption..4. yroll txes. The DS moel speifies tht workers rete output, some of whih the workers keep, n some of whih is use to finne the two soil insurne progrms for the non-employe. There is no istintion etween tht whih is pi y firms n tht whih is pi y workers euse in the DS moel, there re no firms. A ursory summry of the soil insurne literture inites tht this ssumption oes not mtter, n tht the elstiity of lor fore prtiiption with respet to the pyroll tx rte is quite smll regrless of the finning mehnism. In stuy of the employment effets of workers ompenstion enefits, whih re fully finne y employers, Gruer n Krueger (990) report tht employment is not prtiulrly responsive to premium rtes, the osts of whih re primrily psse on to workers in the form of lower wges. They fin lor fore prtiiption rte elstiity of Anerson n Meyer (998), onsiering unemployment insurne, whih is finne y employees through pyroll eutions, fin tht the employment rtes re essentilly not responsive to pyroll txes Leimn n Sez (2006) fin tht lor supply ptterns roun OASI/SSDI pyroll tx reforms re not onsistent with lrge responses to these txes. Gruer (997) estimtes the effets of 98 Chilen pyroll tx reform on wges n lor fore prtiiption for mnufturing. In the stuy tht onsiere the lrgest reform in soil insurne ost levels, the pyroll tx rte for this group went from 0% to 8.5%. The effet ws erese in wges n no isernle hnge in lor fore prtiiption. Overll, the onlusion of this smll literture is tht, inepenent of the finne struture of the enefits, pyroll txes hve very smll impt on employment. While most empiril stuies suggest tht lor fore prtiiption is not responsive to the pyroll tx rte, I will ssess the effets of ssuming ifferent inome elstiity of lor fore prtiiption prmeters s suggeste y Gruer n Krueger

7 Insurne Mrkets n Compnies: Anlyses n Aturil Computtions, Volume, Issue, 202 (990), who, while not rejeting tht the effet of lor fore prtiiption is equl to zero, fin n intervl of roughly 0-0.2, n lso Leimn n Sez (2006), who highlight lor fore prtiiption elstiies of 0.2 n 0.5 in their simultions on the OASDI progrm..5. yroll tx implitions for welfre, SSDI prtiiption. The optimlity onition ove inlues terms tht represent the mrginl effet of onsumption while working on the popultion shre prmeters for SSDI n welfre reipients, respetively. In the lirtion tht I implement elow, I rely on upper ouns for these mrginl effets, n the logi y whih they re erive is s follows. It must e the se tht the mrginl effet of onsumption while working equls the mrginl effet on the non-employe sttes: 2. We n ssume tht pyroll txes t lest wekly enourge iniviuls to prtiipte in SSDI n welfre, so 2 0 n 0. One we hve selete prmeter to represent the elstiity of lor fore prtiiption with respet to fter-tx onsumption, reltionship must hol for 2, 2, 0. n likewise for must hol for,, then the following, 0. Given tht the rtio of inome fter txes in support of SSDI n welfre is roughly 98% n the shre is 9%, we n sfely use s n upper oun tht.,.6. The rop in onsumption when prtiipting in SSDI. The simplest metho of lulting the rop in onsumption through progrm prtiiption is to use the replement rte. roportionte rops onsistent with the urrent level of SSDI replement n e estimte through the tul replement rte, whih spn pproximtely 25% to 60%, s reporte in Autor n Duggn (2006). There is little reserh to te tht ttempts to solve for the hnge in onsumption tht results from hnge in the soil seurity isility insurne enefit level. One pper tht omes lose to ressing this is presente y Brue Meyer n Wlle Mok (2007), who exmine the hnge in onsumption in the ten yers following the onset of isility in the nel Stuy of Inome Dynmis. They fin tht the most severely (in the pper s terminology hroni-severe those with persistent isility tht prelues mny of the Ativities of Dily Living from the SID helth supplement) isle experiene rop in nnul inome of pproximtely $2,000, from n verge ernings prior to isility of $4,000. These iniviuls reeive, on verge, $0,000 per yer in government trnsfers. The expenitures on foo rop, for this group, y 20%, n on housing y 25%. The uthors o not provie n estimte of wht the rop in onsumption woul e in the sene of isility insurne, ut they o report tht the hnges in the ernings of other memers of the househol re estimte to e smll n re sttistilly insignifint, suggesting tht the trnsfers generte ollr-for-ollr inreses in househol inome..7. Risk version. Brsky et l. (997), using survey t from the Helth n Retirement Stuy, fin tht iniviul risk preferenes re onsistent with rte of reltive risk version tht exhiits sustntil heterogeneity, ut is on verge roughly 4. Cohen n Einv (2007), who fin in tset of insurne eutile hoies rte of reltive risk version somewhere etween 0.5 n roun 50, epening on whether the mein or men risk version level is onsiere, respetively. Chetty (2006), exploring the reltionship etween lor supply estimtes n expete utility theory, ples n upper oun on risk version of roughly two. This short survey of estimtes the rte of reltive risk version suggests tht there is rther more unertinty out the level of risk version, n thus the vlue of insurne generlly, thn the unertinty regring the morl hzr ost of proviing SSDI enefits. In wht follows, I onsier risk version levels from to Clirtion In this setion, I omine the empiril estimtes isusse in the previous setion with the optimlity onition previously erive. The funmentl question is whether the urrent SSDI enefit regime n e si to e unresonle, given wht we know out the vrious prmeters involve: speifilly, the popultion tht works, tht prtiiptes in SSDI, n tht prtiiptes in the other enefit, n the elstiity of eh shre with respet to its primry eterminnt. roportionte rops onsistent with the urrent level of SSDI replement n e estimte through the tul replement rte, whih spn pproximtely 25% to 60%, s reporte in Autor n Duggn (2006). Alterntively, it is possile to estimte the rop in onsumption through tul onsumption t. Using the nel 2

8 Insurne Mrkets n Compnies: Anlyses n Aturil Computtions, Volume, Issue, 202 n the proportionte rop in onsumption in selete tegories for those who suffer severe injury, is reporte y Meyer n Mok (2006) to e pproximtely 20%-25%. Of ourse, this my unerestimte the relize rop in onsumption experiene y the isle euse the Meyer n Mok (2006) estimtes re not limite to SSDI prtiipnts. We egin y prmeterizing estimte in eqution (5) * , 2, p2, 2, (6) n I propose sets of vlues of the prmeter vetor, p,, 2, to ssess whether the onsumption rop implie y eqution (5) is onsistent with either the SSDI sheule replement rte or the Meyer n Mok (2007) estimte onsumption rop. Tles 2 to 5 present the results of this nlysis, showing the optiml proportionte rop in onsumption for ifferent vlues of risk version, SSDI prtiiption elstiity, lor supply responsiveness to pyroll txes, n welfre prtiiption responsiveness to enefit levels. Tle 2. Optiml onsumption rop with zero pyroll tx response : Coeffiient of reltive risk version p % % 7% 5% 4% %* 27%** 8% 4% % % 4%* 29%** 22% 7%. 9% 60%* 40%* 0%** 24%** Note: tle results show the onition for the optiml proportionte rop in onsumption for SSDI progrm prtiipnts when lor supply n welfre prtiiption re not responsive to pyroll txes n not very responsive to welfre enefit level, with n elstiity of 0,25. * inites tht enefits re roly onsistent with the rnge of SSDI replement rtes. ** inites enefits re roly onsistent with Meyer n Mok s estimtes of the eline in onsumption fter isility. Tle 2 presents the optiml proportionte onsumption rops when lor fore prtiiption oes not pyroll txes 0,. When risk version is reltively low, from out -2, moerte rnge of morl hzr, of pproximtely is onsistent with urrent onsumption rops. However, t higher levels of risk version, of -5, the enefit rte for SSDI is muh too low unless the higher levels of SSDI prtiiption responsiveness re orret. Tle presents the optiml rop in onsumption when lor supply is t the high en of responsive to pyroll txes, tht is, 0. 2,, whih is t the high en of the Gruer n Krueger (990) estimtes n supply elstiity highlighte y Limn n Sez (2006). In this se, the optiml onsumption rop is unerstnly greter thn when rising revenue is ostless in itself. In this se, the urrent level of replement rtes is resonle for risk version levels 2-5. However, if the Meyer n Mok s (2006) estimtes re goo estimtes of the onsumption rop for SSDI prtiipnts, then enefit levels re somewht too high. Tle. Optiml onsumption rop with lrge pyroll tx response : Coeffiient of reltive risk version p %* 4%* 2%** 7% 4% 0.5 % 56%* 8%* 28%** 2%** % 79% 5% 9%* 2%*. 20% 0% 68% 5%* 4%* Note: The results show the onition for the optiml proportionte rop in onsumption for SSDI progrm prtiipnts when lor supply hs n elstiity with respet to pyroll txes of -0.2 n welfre prtiiption is not very responsive to welfre enefit level, with n elstiity of * inites tht enefits re roly onsistent with the rnge of SSDI replement rtes. ** inites enefits re roly onsistent with Meyer n Mok s estimtes of the eline in onsumption fter isility. Conlusion I hve emonstrte tht, given the rnge of estimtes for vrious inputs in simplifie version of the Dimon n Sheshinski (995), the urrent rop in onsumption ssoite with SSDI prtiiption is not unresonle. However, if lor supply is very responsive to the SSDI pyroll tx rte, n if the relize onsumption rop is tully s smll s 20-25%, then SSDI enefits re more generous thn the DS moel suggests is optiml. This pper s ility to ssess the SSDI progrm s enefit replement rte is se on numer of simplifying ssumptions tht re tken from the DS moel, respeting tht the intent ws not to estlish n lirte n optimlity onition. The moel provies useful referene point when onsiering the mening of morl hzr estimtes. Future work shoul ttempt to relism to the moeling of isility insurne, while remining lose to the integrtion of empiril estimtes. One helpful venue my e to onsier the wie rnge of enefits n the speifi progrm requirements pursunt thereto, inste of onsiering single seon enefit for SSDI pplints who o not meet the sreening rule s requirements. Furthermore, the DS moel, long with most optiml soil insurne moels,

9 Insurne Mrkets n Compnies: Anlyses n Aturil Computtions, Volume, Issue, 202 onsier gents who o not iffer in their utility of onsumption. It my e useful for future isility insurne moels to onsier the reltionship etween isutility of lor n the mrginl utility of onsumption. Furthermore, this pper shows the iffiulty tht the very wie rnge of estimtes for responsiveness to progrm prtiiption n risk version present when ttempting to unertke welfre nlysis of the SSDI progrm. When the low en of estimtes of responsiveness is 0., n the high en of estimtes is roun, it is iffiult to mke ny istintion etween the optimlity of, for exmple, onethir or two-thirs replement rte. Hopefully, onsiertion of the welfre ens of empiril estimtes will motivte further empiril work whih istinguishes etween the high n low estimtes of prtiiption responsiveness. Referenes. Anerson,. n B. Meyer. Unemployment Insurne Tkeup Rtes n the After-Tx Vlue of Benefits // The Qurterly Journl of Eonomis, 998. Vol. 2, No.. pp Autor, D. n M. Duggn. The Growth in the Soil Seurity Disility Rolls: A Fisl Crisis Unfoling // Journl of Eonomi erspetives, Vol. 20, No.. pp Bily, M. Some Aspets of Optiml Unemployment Insurne // Journl of uli Eonomis, 978. Vol. 0, No.. pp Brsky, R., F.T. Juster, M. Kimll n M. Shpiro. referene rmeters n Behviorl Heterogeneity: An Experimentl Approh in the Helth n Retirement Stuy // The Qurterly Journl of Eonomis, 997. Vol. 2, No. 2 pp Boun, J. The Helth n Ernings of Rejete Disility Insurne Applints // The Amerin Eonomi Review, 989. Vol. 79, No. pp Boun, J. n R. Burkhuser. Eonomi Anlysis of Trnsfer rogrms Trgete on eople with Disilities // Hnook of Lor Eonomis, O. Ashenfelter n D. Cr, es., 999. Vol. C. pp Boun, J., J. Cullen, A. Nihols n L. Shmit. The Welfre Implitions of Inresing Disility Generosity // Journl of uli Eonomis, Vol. 88, No. 2. pp Chetty, R. Suffiient Sttistis for Welfre Anlysis // Annul Review of Eonomis, Vol.. pp Chetty, R. A Generl Formul for the Optiml Level of Soil Insurne // Journl of uli Eonomis, Vol. 90, No. 0-. pp Chetty, R. A New Metho of Estimting Risk Aversion // The Amerin Eonomi Review, Vol. 96, No. 5. pp Cohen, A. n L. Einv. Estimting Risk referenes from Deutile Choie // The Amerin Eonomi Review, Vol. 97, No.. pp Dimon,. n E. Sheshinski. Eonomi Aspets of Optiml Disility Benefits // Journl of uli Eonomis, 995. Vol. 57, No.. pp Gruer, J. Disility Insurne Benefits n Lor Supply // Journl of olitil Eonomy, Vol. 08, No. 6. pp Gruer, J. The Iniene of yroll Txtion: Eviene from Chile // Journl of Lor Eonomis, 997. Vol. 5, No. S. pp. S72-S0. 5. Gruer, J. n A. Krueger. The Iniene of Mnte Employer-rovie Insurne: Lessons from Workers Compenstion Insurne // NBER Working per #279, Hlpern, J. n J. Husmn. Choie Uner Unertinty: A Moel of Applitions for the Soil Seurity Disility Insurne rogrm // Journl of uli Eonomis, 986. Vol., No 2. pp Kreier, B. Soil Seurity Disility Insurne: Applitions, Awrs, n Lifetime Inome Flows // Journl of Lor Eonomis, 999. Vol. 7, No. 4. pp Kroft, K. Tke-up, Soil Multipliers, n Optiml Soil Insurne // Journl of uli Eonomis, Vol. 92, No. -4. pp Liemn, J. n E. Sez. Ernings Responses to Inreses in yroll Txes // Mimeogrph, Lozhmeur, J. Disility Insurne n Optiml Inome Txtion // Interntionl Tx n uli Finne, Vol., No. 6. pp Meyer, B. n W. Mok. Disility, Ernings, Inome n Consumption // Mimeogrph, rsons, D. The Deline in Mle Lor Fore rtiiption // Journl of olitil Eonomy, 980. Vol. 88, No.. pp Summers, L. Some Simple Eonomis of Mnte Benefits // The Amerin Eonomi Review, 989. Vol. 79, No. 2. pp US Bureu of Lor Sttistis. Lor Fore Sttistis from the Current opultion Survey. Aville t tles.htm (lst esse: June 8, 202), US Census Bureu. Annul n Monthly Soil Welfre rogrms, rtiiption Rtes n Amounts. Aville t (lst esse: June 8, 202), US Soil Seurity Aministrtion. OASDI Monthly Sttistis, Aville t os/sttomps/osi_monthly/2005/inex.html (lst esse: June 8, 202),