Health Insurance and Household Savings

Transcription

1 Health Insurance and Household Savngs Mnchung Hsu Job Market Paper Last Updated: November, 2006 Abstract Recent emprcal studes have documented a puzzlng pattern of household savngs n the U.S.: households covered by prvate health nsurance save more than comparable households wthout coverage, even when applyng econometrc methods to control for other varables. However, ths emprcal fndng s not consstent wth the standard model of precautonary savng, whch predcts that the ntroducton of health nsurance wll reduce precautonary savngs. Ths paper suggests that nsttutonal factors can account for ths nconsstency. To perform a comprehensve analyss, I develop a standard dynamc general equlbrum model ncorporatng two U.S. nsttutons, a means-tested socal welfare system and an employment-based health nsurance system. I show that, although health nsurance has a negatve effect on precautonary savng, t also has a postve savng effect due to the exstence of these two nsttutons. The observable net savng effect of health nsurance s consequently ambguous. Ths fndng brngs nto queston the approprateness of the conventonal emprcal approach to testng the precautonary savng hypothess through the effect of health nsurance. Ths s because the regressons capture the net savng effect of health nsurance rather than exclusvely the effect on precautonary savng. Ths paper also provdes some possbltes for mprovng the emprcal tests. Department of Economcs, UCLA. Emal: [email protected]. I would lke to thank my advsor, Gary Hansen and commttee members, Lee Ohanan and Hanno Lustg for many helpful comments and advce. I also benefted from conversatons wth many people ncludng Andrew Atkeson, Wllam Zame, Roger Farmer, Chrstan Hellwg and members of Prodec. All errors, of course, are mne. Revsed versons of ths paper can be downloaded from 1

2 1 Introducton Starr-McCluer (1996) was the frst to report a puzzlng phenomenon of household savngs n the U.S: households covered by prvate health nsurance save more than comparable households wthout coverage. In the data she reported, we also observe that the dfference n asset holdngs between nsured and unnsured households decreases wth ncreased ncome level. Even when applyng econometrc methods to control for other household characterstcs, ncludng selectvty, Starr-McClure stll found a sgnfcant and postve effect of health nsurance coverage on household asset holdngs. 1 Ths s a puzzle from the vantage pont of the standard model of precautonary savng. In the standard model, rsk averse agents wll save to guard aganst future uncertanty. 2 Because the ntroducton of health nsurance wll moderate medcal expendture uncertanty, t wll consequently reduce household s precautonary savngs. 3 Although some papers fnd emprcal results smlar to Starr-McClure, the other papers argue that selectvty can not be well controlled for and so causes the puzzlng fndng. 4 However, no evdence for ths selectvty argument and no further analyss of ths puzzlng fndng are provded n the exstng lterature. Ths paper suggests that two U.S. nsttutons can account for ths puzzlng phenomenon: a szable asset-based means-tested socal welfare system and an employmentbased health nsurance system. I show that, n a standard model ncorporatng these two nsttutons, although health nsurance has a negatve effect on precautonary savng, t also has a postve savng effect due to the presence of the two nsttutons. The observable net effect on savngs s consequently ambguous. Moreover, the quanttatve results show that f an economy has a szable means-tested socal welfare system and an employment-based health nsurance system as does the U.S., the smulated household nsurance-savng pattern wll be very close to that observed n the U.S. data. Ths fndng not only explans the puzzlng phenomenon, but also brngs nto queston the approprateness of the conventonal emprcal approach to testng the precautonary sav- 1 Selectvty means that people have heterogeneous rsk averson, and more rsk averse people may tend to save more and also buy health nsurance. 2 Ths s also known as the precautonary savng hypothess. For example, Ayagar (1994) presents qualtatve and quanttatve analyses of the mpact of unnsured dosyncratc rsk on aggregate savng. Because health care has been a large porton of ndvdual expendture, Kotlkoff (1989) and Hubbard et al. (1995) suggest that the uncertanty of medcal expendture has sgnfcant effects on savng behavors. 3 Many emprcal studes adopt ths result to buld regresson models for emprcal tests. For example, Starr-McClure(1996), Guargla and Ross (2004), Gruber and Yelowtz (1999), and Chuo et al (2003). 4 Guargla and Ross (2004), for example, use U.K. data and fnd results smlar to Starr-McClure. However others, for example, Gruber and Yelowtz (1999), and Chuo et al (2003) queston the approach that Starr-McClure used. 2

3 ng hypothess through the effect of health nsurance. Ths s because the regressons capture the net savng effect of health nsurance rather than exclusvely the effect on precautonary savng. To better understand the puzzlng pattern of household savngs, ths paper frst examnes the argument for selectvty, and fnds that a smple assumpton of heterogenety of rsk averson (.e. selectvty) s not suffcent to entrely explan the nsurance-savng features observed n U.S. data. In partcular, t wll lead to a result contrary to the savng pattern across ncome levels. 5 Instead of the focus on selectvty, I suggest that the two U.S. nsttutons can account for the puzzlng savng pattern. For an easer understandng of the effects of the two nsttutons, we can start from the assumptons that guarantee the valdty of the standard precautonary savng model. To have the conventonal result that health nsurance coverage decreases savngs, we at least requre: frst, that no other factors affectng savng decsons vary wth the nsurance coverage; and second, that the nsurance coverage s permanent, whch means there s no rsk of losng nsurance coverage. The two nsttutons volate these two assumptons and therefore can lead to results contrary to the standard model. The exstence of an asset-based means-tested socal welfare system volates the frst assumpton of the standard model. Prevous studes have dscussed the effects of ths knd of socal welfare programs on savngs. The socal welfare system provdes fnancal support for households n bad tmes and ths reduces precautonary savngs. In addton, the asset and means tests create an ncentve for households to reduce savngs n order to qualfy for socal welfare benefts. 6 Ths paper further explores the nteracton between health nsurance coverage and the socal welfare system and examnes how ths nteracton affects household savngs. As n a standard model, health nsurance stll has a negatve effect on precautonary savngs. However, because health nsurance coverage wll decrease the qualfcaton probablty and the expected benefts of the socal welfare programs, t wll reduce the negatve savng effect caused by the socal welfare system, and lead to a postve effect on savngs. Therefore, the net effect of health nsurance s consequently ambguous. I show that, f the means-tested socal welfare system s szable (n a reasonable range), the net savng effect of health nsurance wll be postve. The employment-based health nsurance system n the U.S. also volates the second assumpton of standard model. Prevous studes have dscussed the features of 5 See Appendx A1 for detaled dscusson. 6 Hubbard et al. (1995) provde a theoretcal analyss and suggest that ths knd of means-tested socal welfare programs can depress savngs and answer the queston of why many Amercan households hold very few or even no assets. Some recent emprcal studes have confrmed ther analyss, for example, Gruber and Yelowtz (1999) and Maynard and Qu (2005). 3

4 employment-based health nsurance (see McGarry, 2002), but few explore ts effects on savng. One feature of ths health nsurance system s that when people work full tme n good postons (usually n bgger companes), they and ther famles usually have health nsurance coverage provded by employers; however, when people lose ther jobs, change to part-tme jobs or move to worse postons (usually n smaller companes) and do not qualfy for publc health nsurance programs, they usually become unnsured. 7 Because health nsurance coverage s hghly contngent on work status, nsurance coverage s also uncertan, and currently nsured households wll save for future bad tmes wthout nsurance. Ths channel further strengthens the postve savng effect of health nsurance. 8 In addton, because both the means-tested socal welfare system and health nsurance have smaller effects on hgh-ncome-level households, ther savng behavor wll not change as much as the low-ncome-level households when recevng health nsurance coverage. Therefore, the dfference n asset holdngs between nsured and unnsured households n hgh-ncome-level groups wll be smaller than the dfference n low-ncome-level groups. Ths explans the nsurance-savng pattern across ncome levels observed n the U.S. data. To perform a comprehensve analyss of ths ssue, ths paper develops a standard dynamc general equlbrum model wth nfntely-lved rsk averse households who have dentcal preferences and face ncome and medcal expendture uncertanty. The two U.S. nsttutons are ncorporated n the model. Because the asset markets are ncomplete (.e. households are borrowng constraned), households have ncentve to accumulate assets aganst the uncertanty. 9 Wthout the addton of the two nsttutons, 7 The man reason for ths feature s that for workers (and the unemployed) wthout any offer of employment-based health nsurance who are not qualfed for publc nsurance programs, the only opton to access health nsurance s the ndvdual (non-group) health nsurance, whch s much more expensve than the employment-based (group) health nsurance. The hgh premum of ndvdual health nsurance s caused by many factors, such as hgher admnstraton costs, hgher advertsement costs and adverse selecton problems n the ndvdual health nsurance market (see dscusson n McGarry, 2002 and Swartz, 2003). In the U.S. data, we observe that over 90% of the prvate health nsurance s employment-based and that most of those wthout employment-based nsurance coverage and not qualfed for publc health nsurance programs are unnsured. See McGarry (2002) for more detaled nformaton. 8 In addton, because the currently-nsured households usually have hgher ncome, ths stuaton further lowers nsured households expected benefts from the means-tested socal welfare system, and then further ncreases ther savngs; on the other hand, the currently unnsured usually have lower ncome, and ths rases ther expected benefts from the socal welfare system, and then further reduces ther savngs. 9 If the asset markets are complete, households can become full nsured by tradng assets (.e. borrowng and lendng). Consequently, precautonary savng and nsurance wll be meanngless. Ths stuaton s not consstent wth the real world. 4

5 ths model s smple and standard: households save n a precautonary manner, and the ntroducton of health nsurance wll reduce savngs. I show clearly that n the presence of the two nsttutons, the net savng effect of health nsurance s flpped. 10 Moreover, when the model s calbrated to match the U.S. economy, the smulated household nsurance-savng pattern wll be very close to that observed n the U.S. data. The above fndngs call nto queston the approprateness of the conventonal emprcal approaches based on the standard model for testng the precautonary savng hypothess. Based on the standard precautonary savng model, emprcal studes regress savngs/asset holdngs on health nsurance coverage and also control for other household characterstcs to estmate the savng equaton. I apply ths regresson approach n the model economy. The regresson results show the same sgnfcant and postve coeffcent of health nsurance coverage as exstng emprcal studes. However, the postve coeffcent does not necessarly mply the nonexstence of precautonary savng, because ths regresson captures the net savng effect of health nsurance rather than exclusvely the effect on precautonary savng. In fact, households do have precautonary savng motves n the model economy. Ths paper suggests that f we can observe an economy experencng regme changes n ts socal welfare system and health nsurance system, we wll be able to separately dentfy these combned effects. However, f these data are unavalable, developng an adequate approach to examnng the effects on precautonary savng stll requres more research. The rest of ths paper s organzed as follows. I wll descrbe the puzzlng emprcal fndngs n the next secton. In Secton 3, I wll ntroduce the model. Secton 4 wll provde ntuton to show the mechansm that can lead to the puzzlng household nsurance-savng patterns. In Secton 5, I wll present specfcatons of the model wth standard settngs, calbraton of the parameters and measurement of the shocks. In addton, the numercal method wll also be ntroduced n ths secton. Secton 6 wll show the numercal results and provde quanttatve analyss. In Secton 7, I wll dscuss the mplcaton of ths model for emprcal studes. Secton 8 wll present the concluson and possble future research. 10 Unlke n a lfe cycle model, there are many factors whch together affect savng decsons, for example, lfetme ncome dstrbuton, retrement, death, bequest, and precauton aganst future uncertanty; n ths model, the only reason for savng s precauton aganst uncertanty. The goal of ths paper s to answer the queston If people do have precautonary motves and other varables are well controlled, why s ther savng behavor under nsurance coverage so dfferent from the expectaton of a standard model? Ths smple model shows the reason clearly, and prevents complex analyss across many dfferent factors that affect savng, and thereby avods the dffculty of dstngushng the effects of precautonary savng from the effects of other factors. 5

6 Table 1: Rato of medan asset holdngs of the nsured to the unnsured Rato of medan asset holdngs (a n /a un ) Measure 1 Measure 2 Measure 3 Income (Lqud assets) (Fnancal assets) (Net worth) Below $15K $15-30K $30-50K $50-100K Above $100K Source: Starr-McCluer (1996); orgnal source: SCF (1989), weghted under-65 sample. Lqud assets, fnancal assets and net worth are three measures of assets. The rato (a n /a un ) > 1 mples that the nsured hold more assets than the unnsured. The medan asset holdngs of unnsured households s zero n the bottom ncome group n the frst two columns. 2 Some Puzzlng Features of Health Insurance Coverage and Household Asset Holdngs n the U.S. Starr-McCluer (1996) frst studes the mpact of prvate health nsurance on household savngs n the U.S. workng age populaton, and uses t to test the precautonary savng hypothess. She fnds that households covered by prvate health nsurance mantan much hgher assets than those comparable households wthout coverage, contrary to what the precautonary savng hypothess predcts. Table 1 shows the ratos of asset holdngs between nsured and unnsured households computed from the descrptve statstcs reported n Starr-McCluer s paper. Three measures of assets are consdered: lqud assets, fnancal assets and net worth. 11 We can see a strong postve correlaton between health nsurance coverage and asset holdngs, regardless of measures of assets used. In addton, we can also observe that the rato, regardless of measures used, decreases wth ncreased ncome level whch means the dfference between hgher ncome nsured and unnsured households s smaller. 12 Because descrptve statstcs are not suffcent to show the true savng effect of health nsurance, Starr-McCluer also apples econometrc methods, both ordnary least 11 Please see Starr-McCluer (1996) page 288 for the defntons of these three measures. 12 The ncome groups n the table are classfed by current ncome. So the ncome groups n the table mght not precsely represent permanent ncome levels. Even so, the nformaton provded by ths table stll clearly shows the pattern of asset holdngs across ncome levels. 6

7 Table 2: Regresson of log asset holdngs - selected varables Measure 1 Measure 2 Measure 3 (Lqud assets) (Fnancal assets) (Net worth) HI coverage 2.66* 2.97* 1.72* (0.18) (0.16) (0.19) Perm t Income 1.53* 1.71* 1.69* (0.11) (0.11) (0.15) Health problems -0.34* -0.28* -0.41* (0.10) (0.10) (0.12) Source: Starr-McCluer (1996), maxmum lkelhood selecton model. Only reportng some selected varables n page 290 of the paper. *Sgnfcant at the 5-percent level. squares (OLS) and maxmum lkelhood selecton model, to control for other varables, ncludng selectvty. 13 The results stll show that health nsurance has a sgnfcantly postve effect on savngs. Table 2 shows part of the emprcal results n her paper. We can see that the coeffcents of health nsurance coverage are sgnfcant postve wth all three measures of assets. Guargla and Ross (2004) also study the effects of prvate medcal nsurance on household savngs but use the Brtsh Household Panel Survey data. They fnd results smlar to Starr-McCluer s fndngs: health nsurance coverage ncreases the probablty of savng. 14 In addton, because they have panel data, they can mprove the approach used n Starr-McCluer (1996) to control for selectvty. 15 Based on these emprcal fndngs, they suggest that Brtsh households do not have precautonary savngs for health rsk, and that selectvty s not the reason for these puzzlng emprcal results. However, some papers argue that selectvty can not be controlled for well when usng prvate health nsurance data, and they provde other evdence to support the precautonary savng hypothess. For example, Gruber and Yelowtz (1999), Chou et al.(2003, 2004) and Maynard and Qu (2005) all argue that people who are hghly rsk averse are more lkely to both purchase prvate health nsurance and accumulate assets, and ths se- 13 In order to control for selectvty, she uses a maxmum lkelhood selecton model n whch she smultaneously estmates the nsurance coverage and the savng equatons. 14 Because the survey only asks households f they save, a yes-or-no queston, they do not have actual values of the households savngs. 15 They use an Instrumental Varables (IV) estmaton technque, and a Full Model Maxmum Lkelhood approach for ths purpose. However, n all ther specfcatons, they stll fnd a postve assocaton between health nsurance coverage and savng. 7

8 lecton problem s dffcult to control for. They also provde evdence from other types of health nsurance (for example, Medcad and unversal health nsurance) to support the precautonary savng hypothess. However, there are no further analyses of the puzzlng savng patterns, and there s also no evdence for the argument about the role of selectvty provded n the current lterature. The man reason for ths phenomenon s stll an open queston. 3 The Model The heterogenety n rsk averson (.e. selectvty), as suggested by some papers, can explan why nsured households hold more assets than the unnsured, but can not explan the declnng asset-holdng rato by ncome level shown n Table 1, unless we make more assumptons about household preference and rsk averson. I provde a detaled explanaton and a numercal exercse to llustrate ths n Appendx 1. I suggest that two nsttutons can account for the puzzlng savng patterns: an assetbased means-tested socal welfare system and an employment-based health nsurance system. To provde a comprehensve analyss, n ths secton I develop a standard dynamc general equlbrum model wth unnsured dosyncratc uncertanty that ncorporates these two nsttutons. 16 The model economy s populated by a large number of nfntely-lved rsk-averse households who have dentcal preferences and rsk averson. Households have dfferent labor effcences/sklls, whch are gven exogenously and determne permanent ncome levels. All households face dosyncratc labor endowment shocks (equvalently, ncome shocks) and medcal expendture shocks. Snce the asset markets are ncomplete, households wll save n a precautonary manner to guard aganst the uncertanty. In addton to savng, households can have health nsurance aganst medcal expendture shocks. For smplcty and to capture the employment-based health nsurance system, I smplfy the feature of accessng health nsurance n group (employment-based) and non-group (ndvdual) markets as a probablty contngent on work status (labor 16 For better understandng of the whole structure of the economy, I use a general equlbrum approach. Ths means that the producton sde s also ncorporated nto the model economy. In ths economy, households provde labor and rent captal to frms (.e. hold assets) and the frms produce products whch can be consumed or nvested. Therefore, households savng decsons wll affect captal supply and then affect the equlbrum nterest rate and wage rate. The equlbrum nterest rate and wage rate are determned by captal demand (frms) and by captal supply (households). In addton, as dscussed n Ayagar (1994), n a partal equlbrum analyss, by choosng an nterest rate close enough to the rate of tme preference, one can generate arbtrary large precautonary savng. Ths s another reason for usng a general equlbrum analyss n ths paper. 8

9 endowment status, n the model): the probablty of beng nsured s hgher when a household has a better labor endowment status, whch represents a better job poston. 17 In the model economy, there exsts an asset-based means-tested socal welfare system that enables households to mantan a mnmum consumpton level (denoted by C) n all states. I employ a smple rule for the operaton of the socal welfare system: f a household s dsposable ncome and assets (denoted by H) are lower than C, the household qualfes and wll receve the socal-welfare benefts (transfer payment guaranteeng that households at least can have the amount of money C to spend). As ntroduced n Hubbard et al. (1995), ths smple socal welfare system s used to characterze the socal programs wth means tests and asset restrctons n the U.S., such as Ad to Famles wth Dependent Chldren (AFDC), Medcad, Supplemental Securty Income (SSI), and food stamps. 3.1 Household Problem In ths economy, every household faces dosyncratc labor-endowment and medcalexpendture shocks (denoted by l t and x t respectvely). Households can have health nsurance aganst the medcal expendture shocks but nsurance coverage s also uncertan and assumed to be contngent on the labor-endowment state. For smplcty, I assume the medcal-expendture and the labor endowment shocks are dstrbuted ndependently across households, and both of them are..d. over tme. 18 Households have dfferent values of labor effcency, λ, whch determnes household s permanent ncome level n 17 As mentoned n the Secton 1, the prce dfference n the group and non-group health nsurance markets leads to the dfference n accessng health nsurance. Therefore, for smplcty, the decson process of accessng health nsurance s smplfed to a probablty contngent on work status n the model. However, addng the decson process nto the model wll not change the analyss and results of ths paper. Snce every household has the same preference and rsk averson, f we allow households to choose whether they purchase health nsurance or not, the decson wll strongly depend on whether they are offered employmentbased health nsurance. If they are offered t, most of them wll accept t because of the reasonable premum of employment-based health nsurance; f they are not offered t, many of them wll not purchase health nsurance because the ndvdual nsurance plan s much more expensve. Hence, the dstrbuton of health nsurance coverage wth decson process wll be consstent wth that n ths smplfed model. Snce the focus of ths paper s on nsurance s effects on savng rather than analyzng nsurance purchase decsons or nsurance market structure, addng the decson process wll make the model more complcated wthout addng any accuracy to the results. 18 Labor endowment volatlty s used as a proxy for ncome volatlty. Illness mght be one of the reasons for ncome volatlty, but only very serous llness can really change work status and ncome level. As long as ncome and medcal expendture shocks are not perfectly related, the smplfed settng of ndependence does not change the analyss n ths paper. 9

10 the model. 19 Then household s effectve labor supply s λ l,t. I also assume there s only one knd of health nsurance polcy wth consurance rate (or copayment rate) ᾱ, 0 < ᾱ < 1, and premum q. If households are covered by health nsurance, they only need to pay ᾱx t for health care, and pay a premum q for the nsurance coverage. In addton, to characterze the employment-based health nsurance system, the probablty of recevng nsurance coverage, Pr(α = ᾱ), s set contngent on labor-endowment state l (thus an ncreasng functon of labor endowment). In the economy, household chooses consumpton level (c,t ) and asset holdngs (a,t+1 ) at each perod to maxmze ts expected dscounted lfetme utlty: ] E 0 [ t=0β t U(c,t ) (1a) subject to c,t + a,t+1 = (1 τ t )w,t l,t +(1+r t )a,t α,t x,t q,t + TR,t ; (1b) (α,t, q,t ) {(ᾱ, q), (1, 0)}, (1c) where α,t = ᾱ and q,t = q, f nsured; α,t = 1 and q,t = 0, f not nsured; the probablty of beng nsured Pr(α,t = ᾱ) s an ncreasng functon of labor status l,t ; c,t 0; a,t b; (1d) T R,t = max{0, C H,t } (1e) where H,t = (1 τ t )w,t l,t +(1+r)a,t α,t x,t q,t. Equaton (1a) ndcates that, at each perod t, consumpton c,t s chosen to maxmze the expected lfetme utlty wth the one-perod utlty functon U(c). U(c) s assumed wth standard propertes to characterze a rsk averse household, such as strctly ncreasng, strctly concave, contnuously dfferentable (.e. U (c) > 0, U (c) < 0), lm c 0 U (c) = (for nteror soluton), and U (c) > 0 (for precautonary savng) Ths settng creates several ncome groups for easer comparson of the model wth the data. 20 In a partal-equlbrum savng problem, t has been known snce Leland (1968) and Sandmo (1970) that precautonary savng n response to rsk s assocated wth a postve thrd dervatve of the utlty functon, but n a general equlbrum envronment, the postve-thrd-dervatve assumpton s not necessary (see the dscusson n Ljungqvst and Sargent, 2000). However, I stll use the stronger thrd-dervatve assumpton for precautonary savng because the goal of ths paper s to show that, even when households do have precautonary motves, we are stll able to observe a postve correlaton between health nsurance coverage and savngs. 10

11 E t s the expectaton operator condtonal on nformaton at tme t and β s the utlty dscount factor. Equaton (1b) ndcates the perod budget constrant. The left hand sde (LHS) of (1b) ndcates that household can spend the money on consumpton (c,t ) and asset holdngs (a,t+1 ). The rght hand sde (RHS) of (1b) s household s dsposable ncome and assets at perod t. Gven wage rate w,t and rate of return of assets r t, the household receves after-tax (tax rate τ t ) labor ncome (1 τ t )w,t l,t and prevous assets plus asset return (1 + r t )a,t (a,t s the assets held from last perod). The household may also receve TR,t, whch s the transfer payment from the socal welfare program at tme t. Addtonally, t s assumed that medcal expendture x,t s requred only to offset the damage brought on by poor health or llness; no utlty s delvered from medcal expenses. Equaton (1d) mples that households are lqudty constraned; they only can borrow by a lmted amount, b, and therefore can not fully nsure ther consumpton through asset tradng. For smplcty, b s set at zero for the followng analyss that means households can not hold negatve assets (.e borrow). Ths settng makes households precautonary savng motve even stronger. Equaton (1e) descrbes the transfer functon of the socal welfare system. I consder a smple transfer rule proposed by Hubbard et al. (1995). H,t s household s dsposable ncome plus assets (net after medcal expendture). C s the mnmum level of consumpton supported by government or socety. Transfer TR,t wll be made f H,t s smaller than C, and the transfer amount wll be equal to C mnus H,t Frm, Government and Insurance Companes On the producton sde, I assume there s a sngle frm wth a producton technology whch dsplays constant returns to scale (CRS). 22 The frm hres labor and rents captal from households. It s assumed that only effectve labor (λ,t l,t for each houshold ) s productve, and there s no uncertanty n producton. Let Y t denote the total output at perod t, and K t, N t denote the aggregate captal and aggregate effectve labor, respectvely. The total output producton functon s: Y t = F(K t,n t ), (2) where K t = a,t and N t = λ l,t. Let γ t denote the rental rate of captal and w,t denote the wage rate for labor wth effcency λ. Hence, the producton cost s (γ t K t + w l ). 21 The transfer payments s not requred to spend only on consumpton. 22 Ths mples that the frm makes zero proft n the equlbrum. The economy would behave the same as f there were many compettve frms. 11

12 Ths CRS producton functon has standard propertes: F K > 0, F KK < 0; F N > 0, F NN < 0; F KN > 0, and F(0, N)= F(K, 0)=0. From the frst order necessary condtons, we can see that the rental rate of captal (γ t ) equals margnal product of captal and the wage rate (w,t ) for each equals labor effcency tmes margnal product of effectve labor: γ t = F K (K t,n t ); (3) w,t = λ F N (K t,n t ). (4) Equaton (4) mples nequalty n labor ncome as well as n lfe-tme ncome. The government operates the socal welfare system n ths economy. It taxes household s labor ncome to fnance the socal welfare system. Snce labor supply s nelastc n the model economy, ths tax does not affect labor supply and wll less dstort household s savng decsons. 23 The government s requred to have a balanced budget represented by τ t w,t l,t = TR,t. Snce the structure of the health nsurance market s not the focus of ths paper, we assume nsurance companes all provde the same health nsurance wth consurance rate ᾱ and charge a far premum ( q) that equals the expected cost, (1 ᾱ)e(x). 3.3 Recursve Formulaton For convenence of analyss, we can rewrte the above household s problem n a recursve form and have the followng Bellman equaton: s.t. { V(a,l,x,α,q ) = max U(c )+βe [ V(a,l,x,α,q ) ]} (5a) a c + a = (1 τ)w l +(1+r)a α x q, f H > C; (5b) c + a = C, f H C; (5c) (α, q ) {(ᾱ, q), (1, 0)}, (5d) c 0; a 0; (5e) where α = ᾱ and q = q, f nsured; α = 1 and q = 0, f not nsured; H = (1 τ)w l +(1+r)a α x q. I also translate the constrants (1b) to (1e) nto (5b) to (5e). 23 If government taxes captal ncome, t wll dstort the asset prces drectly and affect the decsons of asset holdng and consumpton. 12

13 (5b) represents the budget constrant when the household does not qualfy for socal welfare benefts, and (5c) represents the budget constrant when the household does qualfy. Because asset holdng s stll allowed when household s qualfed for socal welfare benefts, the functon of the socal welfare system s provdng a lower bound for household s dsposable wealth (.e. H ) nstead of regulatng asset holdng/consumpton decsons. Hence, gven C, τ, w and r, the household s problem s standard, but a corner soluton for a mght exst (.e. zero asset holdngs, a = 0).24 If a has an nteror soluton, we can have the Euler equaton: U c =βev a { =β p E(V a H > C)+(1 p )E(V a H C)+ p [ a E(V a H > C) E(V a H C)] }. (6) where p = Pr(H > C H ), and (1 p ) = Pr(H C H ). So (1 p ) s the probablty of household beng qualfed for the socal welfare program n the next perod, condtonal on today s state. EV a n the equaton s the margnal expected value of asset holdngs, and I decompose the expectaton nto two parts: H > C and H C wth probabltes p and (1 p ) respectvely. Then EV a can be expressed as n equaton (6). By the envelope condton, 25 we have ] EV a =p (1+r )E [U c H > C + p { [ E U(c a ) H > C ] E [ U(c ) H C ]}. (7) Equaton (7) shows that the margnal expected return of asset holdngs and ndcates that the benefts of asset holdngs come from two parts: an ncrease n consumpton tomorrow (c ) n good states (defned as when H > C); secondly, an ncrease n the probablty of stayng n good states (.e. an ncrease n p ). 24 Standard means that households can make ther own decsons wthout any regulaton. The socal welfare system only changes the dsposable resources. 25 Takng a partal dervatve of the value functon V(a,l,x,α,q ) wth respect to a, we have: V a = (1+r)U c, f H > C; V a = 0, f H C. 13

14 Usng equaton (7), we can rewrte the Euler equaton (6) as follows: { U c =β p (1+r )E(U c H > C)+ p { [ E U(c a ) H > C ] E [ U(c ) H C ]}}. (8) Ths Euler equaton characterzes the household s asset holdng decson. The LHS of equaton (8) represents the margnal cost of holdng assets due to sacrfce of today s consumpton, and the RHS s the dscounted expected margnal return of holdng assets. Therefore, when the soluton s nteror, gven C, τ, w and r, the optmal asset holdng decson rule, a = A(a,l,x,α,q w,r,c), must satsfy the Euler equaton (8) and then solve the Bellman equaton (5a). However, there exst some possble crcumstances n whch even when a s zero, the margnal return of assets (the RHS n equaton (8)) s stll smaller than the margnal cost (the LHS). In these crcumstances, the optmal decson s not holdng any asset, a = Recursve Compettve Equlbrum The statonary recursve compettve equlbrum for the model s defned as follows. If a statonary recursve compettve equlbrum exsts, there wll exst values of aggregate captal stock K and aggregate labor employed L, a decson rule of asset holdng a for each household, wage rate w for each household, rental rate of captal γ, rate of return of assets r, tax rate τ and a dstrbuton of asset holdngs {a} n the economy (denoted by ψ(a)), such that: a) gven w for each and γ, K and L solve the frm s problem; b) gven w and r, the decson rule of a solves household s problem; c) ψ(a) s statonary; d) government budget s balanced: τ w l = TR ; e) break-even condton for nsurance companes s satsfed: q = (1 ᾱ)e(x); f) markets clear: l = L (whch mples λ l = N) and a = K; g) the allocaton s feasble: Y = C + X δk (n steady state), where C = c s the aggregate consumpton, and X = x s the total medcal expendture In equlbrum, we have Y = C+ X + K (1 δ)k; => Y δk = C+ X +(K K); (f1) 14

15 4 The Factors that Affect Household Savng Behavor Snce there s no smple closed-form soluton for the above stochastc dynamc programmng problem, I wll solve the model numercally and perform a quanttatve analyss. Before that, ths secton wll provde ntuton of the effects of the socal welfare system and health nsurance on household asset holdngs. Ths wll also show the mechansm that can lead to the puzzlng nsurance-savng phenomenon. 4.1 Effects of the Socal Welfare System The asset-based means-tested socal welfare system has a negatve effect on savng, and the magntude of ths effect decreases wth ncreased ncome level. 1) The asset-based means-tested socal welfare system decreases savngs. There are two channels through whch the socal welfare system can decrease household asset holdngs. The frst channel s the reducton of ncome and expendture uncertanty. Because of the socal welfare system, the household dsposable resources are guaranteed to never fall below C. Hence, the uncertanty s reduced and households can mantan less precautonary savngs. The second channel comes from the asset-based means test. Ths provdes an ncentve for households to reduce savngs and ncrease consumpton n order to receve the socal welfare benefts. Meanwhle, the expected return of asset holdngs s also reduced by the means test of the socal welfare system. Ths s because f households qualfy for the socal welfare programs, ther dsposable resources H wll be subsdzed to the same level (C), whch mples prevous savngs do not matter for qualfed households. Therefore, as long as there exsts some probablty of qualfcaton, there wll be a negatve effect on household savngs. Appendx A2 provdes a more detaled analyss based on the Euler equaton that we get from the model. where K s equal to total asset holdngs a. In steady state, we have Y = C+X δk because K = K. In addton, the RHS of (f1) s the total consumer expendture and nvestment, whch s equal to total ncome w l + r a (= w l + rk). Because of the CRS producton functon, Y = F K K + F N N = γk + w l. Thus, we can rewrte the LHS of (f1) as γk + w l δk. Then combne the RHS and the LHS, we have γk + w l δk = w l + rk; => γ δ = r. (f2) Ths means that the rate of return of assets equals the rental rate of captal mnus the deprecaton rate. In the general equlbrum model, the captal demand (from the frm) and the captal supply (.e. the asset demand from households) determne the equlbrum r. Snce there are no aggregate shocks on ether demand sde or supply sde, there wll exst an unque steady-state equlbrum r. 15

16 2) The negatve effect on savngs decreases wth ncreased permanent ncome level. Households wth hgher permanent ncome level (represented by a greater λ for ther labor effcency n the model) have stronger ablty to resst uncertanty, and therefore have a hgher probablty of stayng n good states (.e. a larger p ). On the other hand, they have a lower probablty of qualfyng for the socal welfare programs (.e. a smaller 1 p ). Therefore, they are less affected by the socal welfare system. In addton, hgher-ncome-level households have hgher expected utlty n good states. The socal programs can not greatly reduce ther utlty gap between good states and bad states (relatve to lower-ncome-level households), and so these households have greater ncentve to save for stayng n good states. Both factors lead to the concluson that the socal welfare system has a smaller (negatve) savng effect on hgher-ncomelevel households. 27 Another reason s that, n the general equlbrum envronment, when households all drop ther asset holdngs (because of the effects of the socal welfare system), the rate of return of assets r s drven up. Because of ths ncrease n r, households wll then ncrease some asset holdngs n response. If the utlty functon has the property of convex margnal utlty, such as the wdely used constant relatve rsk averson (CRRA) utlty, hgher-ncome households wll ncrease more asset holdngs (on the other hand, they wll drop more current consumpton) than those wth lower ncome. 28 Ths also offsets some of the drect negatve savng effects of the socal welfare system, and so the net negatve effects are smaller on hgher-ncome households. 4.2 Effects of Health Insurance on Savng under the Socal Welfare System Here I analyze the effects of health nsurance on savng n an economy wth the meanstested socal welfare system dscussed above. The nteracton between health nsurance and the socal welfare system provdes one mechansm responsble for the puzzlng nsurance-savng pattern observed n the U.S. 29 1) Health nsurance reduces the negatve effects of the socal welfare system on savngs. As analyzed above, the socal welfare system reduces the expected return of assets 27 Compared wth low-ncome-level households, hgh-ncome-level households have a smaller probablty of qualfcaton (1 p ), and have a larger utlty gap (E[U(c ) H > C] E[U(c ) H C]). Both factors lead to a hgher margnal expected return of assets (the RHS of the Euler equaton) for hgh-ncome-level households, wth other thngs beng equal. 28 That propery s U (c) > Another mechansm s the employment-based (work-contngent) health nsurance system. See further dscusson of ths later. 16

17 and then depresses savngs (nteror soluton) and also ncreases the probablty of zero asset holdngs (corner soluton). However, the ntroducton of health nsurance reduces the negatve savng effects of the socal welfare system n the followng two ways. Frst, health nsurance reduces the fluctuaton of medcal expendture shocks, and then reduces the probablty of qualfyng for the socal welfare benefts (concurrently, t ncreases the probablty of beng n good states). 30 Ths leads to an ncrease n the expected margnal return of asset holdngs, and so households have a hgher ncentve to hold assets, other thngs equal. 31 In addton, health nsurance also ncreases expected utlty n good states, and has no effect n bad states when qualfyng for socal benefts. Ths ncreases the expected utlty gap between good states and bad states, and ncreases the expected return of asset holdngs. 32 Therefore, households covered by health nsurance wll have a hgher ncentve to save. 33 A detaled dscusson of these effects s provded n Appendx A3. In general, the ntroducton of health nsurance lets households rely less on the socal welfare system and thereby re-creates an ncentve to save for ther future. 2) Health nsurance has opposng effects on savng n ths economy. In addton to the orgnal effect of nsurance reducng ncome and expendture fluctuatons, and consequently reducng precautonary savngs, health nsurance stll has effects that ncrease savngs n an economy wth the means-tested socal welfare system, as dscussed above. 34 3) The net effect of health nsurance depends on the sze of the means-tested socal 30 Ths s because households only need to pay the copayment of the medcal expendture when they encounter a medcal shock. Hence, ths leads to an ncrease n p and a decrease n 1 p. 31 An ncrease n p wll ncrease p (1+r )E(U c H > C), whch s the frst part of the RHS n the Euler equaton (8). Thus the expected return of assets becomes hgher. 32 The expected utlty n good states, E[U c H > C], s ncreased because of a a reducton of medcal expendture uncertanty. Health nsurance has no effect on the utlty n bad states (defned as when H C,.e. when households qualfy for the socal welfare programs). Once a household qualfes, the medcal expendture wll be covered and the household wll be subsdzed untl dsposable ncome s equal to C. Hence there s no uncertanty n the bad states n ths economy. So the utlty gap {E [ U(c ) H > C] E [ U(c ) H C] }, n the second part of the RHS n the Euler equaton (8), becomes larger. 33 However, ths postve savng effect may be largely offset. The postve effect comes from the second part of the RHS n the Euler equaton (8): ( p / a ){ E [ U(c ) H > C] E [ U(c ) H C]}. Health nsurance ncreases the utlty gap, but also decreases the mportance of savng that s reflected by the reducton of the multpler ( p / a ). See Appendx A3 for the detal. 34 The reducton of precautonary savngs s because E[U c H > C], whch s n the frst part of the RHS n equaton (8), becomes smaller. Wth the convexty of the margnal utlty functon (.e. U (c) > 0), when the fluctuaton of consumpton becomes smaller, the expected margnal utlty drops. 17

18 welfare system. Snce the postve savng effect of health nsurance s due to the reducton of the negatve effects of the socal welfare system, the sze of the socal welfare system affects how large the postve effect wll be, and consequently determnes the net effect of health nsurance. In the model economy, the means test crteron C represents the sze of the socal welfare system. If C s set lower than the mnmum of household dsposable ncome, denoted by H, the socal welfare system wll not have any effect on households. Therefore, the only effect of health nsurance s a reducton n savngs as n a standard precautonary savng model wthout the socal welfare system. When C s set hgher than H, the socal welfare system begns to affect household savngs. Consequently, an ntroducton of health nsurance wll reduce the (negatve) savng effects from the socal welfare system and thereby have postve effects on savngs. In a reasonable range, a hgher C mples a larger reducton of the socal-welfaresystem effect that health nsurance can make, and so leads to a greater postve savng effect that health nsurance wll have. 35 I provdes a detaled dscusson n Appendx A3, and wll show n the quanttatve analyss that when C s set hgh enough, the postve savng effect wll surpass the negatve savng effect. In addton, as mentoned above, the negatve savng effect of the socal welfare system decreases wth ncreased ncome level, and therefore the postve savng effect of health nsurance also decreases wth ncreased ncome level. 36 Ths leads to a smaller asset holdng raton of the nsured over the unnsured (a n /a un ) n hgher ncome groups, as we observe n the data (see Table 1). 4.3 The Role of the Employment-Based (Work-Contngent) Health Insurance System Above we have dscussed household asset-holdng behavor wth the ntroducton of health nsurance, wthout any specfcaton of health nsurance elgblty. As mentoned n the ntroducton of the model settng, ths paper assumes that health nsurance s workcontngent, n order to capture the current employment-based health nsurance system for the workng-age populaton n the U.S. Ths work-contngent health nsurance sys- 35 However, f C s set very hgh (the extreme case s C equal to average ncome), nsured households wll stll have a very hgh probablty of qualfyng for the socal welfare benefts. Then the postve savng effect of health nsurance wll become smaller and the net effect mght be negatve agan. See detaled dscusson n Appendx A3. 36 Ths s because the postve savng effects of health nsurance come from the reducton of the negatve savng effects of the socal welfare system. 18

19 tem renforces the ncentve of savng when households are covered by health nsurance n an economy wth the means-tested socal welfare system. Because health nsurance coverage s contngent on work status, when households are n good labor states (representng full-tme and good job postons), they have a hgh probablty of becomng nsured; when households are n bad labor states (representng part-tme or bad postons), they have a very low probablty of becomng nsured. Ths mples that nsured households on average have hgher labor ncome. On the other hand, when a household becomes unnsured, t usually also suffers from lower labor ncome. Ths provdes the ncentve for nsured households to save for future possblty of losng nsurance coverage and current ncome. Moreover, as dscussed above, health nsurance can reduce the negatve savng effect from the socal welfare system and so ncrease the expected return of asset holdngs. The work-contngent health nsurance system renforces ths ncrease. Because the dfference n nsurance status s now assocated wth a dfference n ncome, nsurance coverage together wth hgher ncome further reduces the mportance of the socal welfare system and further ncreases the savng ncentve. Therefore, n addton to the socal welfare system, the employment-based (work-contngent) health nsurance system provdes another mechansm that leads to the puzzlng postve nsurance-savng correlaton. A more detaled analyss of ths mechansm s provded n Appendx A4. 5 Parameterzaton and Numercal Soluton of the Model Accordng to the above analyss, we expect that the model wll show: (1) the net savng effect of health nsurance wll be postve f C s hgh enough; (2) the work-contngent health nsurance system wll strengthen the postve savng effects of health nsurance and so wll enlarge the rato of asset holdngs (a n /a un ); and (3) the postve effect s smaller on households wth hgher permanent ncome levels. Hence, the rato of asset holdngs (a n /a un ) wll decreases wth ncreased ncome level, as observed n the data. In the followng two sectons, I wll provde a quanttatve analyss to verfy ths expectaton, and wll compare the results wth U.S. data and exstng emprcal fndngs. In order to do ths, I begn by the specfcaton and parameterzaton of the model, and brefly explan the numercal method used to solve for ths stochastc dynamc general equlbrum model. 19

20 5.1 Model Specfcaton and Calbraton The model perod s set to be one year, and the utlty dscount factor (β) s chosen to be In addton, a standard and wdely-used CRRA (constant relatve rsk averson) utlty functon s specfed for households: U(c) = (c 1 µ )/(1 µ). (9) In the model economy, every household has the same preference. 37 The relatve rsk averson coeffcent (µ) n the utlty functon s set to be 3, whch s consstent wth many emprcal studes (see Hubbard et al., 1995). Other possble values of µ wll also be used to test the robustness of the results. The producton functon s taken to be a standard Cobb-Douglas type: F(K,N) = AK θ N 1 θ. (10) Lke the economy that Ayagar (1994) defned to analyze unnsured rsks and precautonary savng, I also focus on a detrended economy wthout producton uncertanty. So, as n Ayagar s economy, A s set at unty. The captal share (θ ) s taken to be 0.36, and as suggested by Stokey and Rebelo (1995), the deprecaton rate of captal (δ) s set at The above specfcatons and parameter values are chosen n order to be consstent wth aggregate features of the postwar U.S. economy and are commonly employed n aggregatve models of growth and busness cycles Labor endowment shock and labor effcency In the model, the dosyncratc labor endowment shock (l t ) s used to capture the ncome fluctuatons, and the labor effcency λ determnes permanent ncome level. Prevous studes suggest that a frst order autoregressve (AR(1)) process can well approxmate the pattern of logarthm of labor endowment shocks (or equvalently, ncome shocks). 38 The process s set as: log(l t+1 ) = ρ l log(l t )+ε lt, (11) where ρ l s the seral correlaton coeffcent on labor endowment shocks and ε lt s whte nose. The specfcaton and parameterzaton for ths AR(1) process are based on the results reported n Hubbard et al. (1995). Ther estmaton of ncome shock, ncludng the 37 Therefore, our results do not depend on heterogenety of preference. 38 See, for example, Ayagar (1994) and Hubbard et al. (1995) 20

21 unemployment nsurance benefts, subtractng taxes and based on mcro data, better fts our model. They estmate the ncome-shock processes for three educatonal categores separately. Ther results ndcate that the values of persstence, ρ l, are qute smlar among the three educatonal groups; the values range from to Addtonally, they fnd that the varance of ε lt decreases by educaton level, from to Based on ther estmatons, the ρ l s chosen to be and the varance of ε lt s set at as n the mddle-educaton group for the dosyncratc labor endowment shock. I then apply the procedure descrbed n Tauchen (1986) to approxmate ths AR(1) process usng a fve-state Markov chan, wth a maxmum and mnmum equal to plus and mnus 2.5 standard devatons of the uncondtonal dstrbuton. In order to examne the effects n dfferent ncome groups, I set fve permanent ncome groups wth weghts {.25,.25,.25,.20,.05} from the bottom to the top ncome level n the model economy, whch can be comparable wth the U.S. data shown n Table Because labor effcency λ represents permanent ncome level n the model, I set fve values of λ to create these fve permanent ncome groups. The λ for the mddle ncome group (also the mean) s normalzed at unty, and the values of labor effcency for the permanent ncome groups are {0.25, 0.75, 1, 1.8, 2.4} so that these ncome levels are roughly comparable wth those n Table Medcal expendture shock To characterze medcal expendture shock, I calbrate a Markov process drectly from the data nstead of estmatng an AR(1) process, as n some prevous studes. 40 The frst reason s that the dstrbuton of medcal expendture s very skewed and clearly not symmetrc. 70% of the workng-age populaton spent only $250 on health care n 1996, whch was equvalent to 1% of the average ncome n However, the top 1% of the populaton spent $73,197, whch was 298% of the average ncome. Secondly, the persstence of medcal expendture s not constant n dfferent states, but an AR(1) process needs to specfy a constant persstence for every state. Therefore, I use the report of persstence n health care expendture n Monhet (2003) to dentfy the transton probabltes for a four-state Markov chan of medcal expendture. 41 The 39 Actually, the ncome groups n Table 1 are defned by current ncome, because t s not easy to get true permanent ncome from the cross-secton data. However, each ncome group ncludes a range of ncome that can absorb some current ncome fluctuaton. Hence the data may stll reflect a pattern n dfferent permanent ncome levels. 40 For example, Freenberg and Sknner (1994) and Hubbard et al. (1995). 41 Monhet uses the data from 1996/97 Medcal Expendture Panel Survey (MEPS) to determne the persstence and provdes a detaled report. 21

22 Table 3: States of medcal expendture Expendture Average Exp. % of Average State Range ($ n 1996) Income (1996) Low bottom 70% 250 1% Far 70 95% 3,099 13% Hgh 95 99% 16,173 66% Very Hgh top 1% 73, % Source: MEPS, procedure s descrbed below, the defntons of states are summarzed n table 3 and the results are reported n table 4. I defne the 4 states of medcal expendture shock as: low, far, hgh and very hgh. People who have mdecal expendture n the bottom 70% are set n the state low, those n the range from 70 to 95% are set n the state far, those n the range from 95 to 99% are n the state hgh and the top 1% are n the state very hgh. In order to reflect the true costs for health care that unnsured households must face, I use the mean of total annual costs of health care n each range to represent the medcal expendture shock n each state. 42 In the U.S. workng-age populaton n 1996, the mean of medcal cost n the low state was $250, that n the far state was $3,099, that n the hgh state was $16,173 and that n the very hgh state was $73,197. These costs were 1%, 13%, 66% and 298% of the average ndvdual ncome n 1996, respectvely (from the Medcal Expendture Panel Survey, MEPS). So we set the medcal expendtures (X) n the four states as the above percentages of average labor ncome n the model. To calbrate the transton probabltes for the Markov chan, I use the data provded n Monhet (2003). Monhet reports households medcal-expendture rank n 1996 and ther rank n 1997 to study the persstence n medcal expendture. Ths nformaton s enough to calculate the transton probabltes for each state from 1996 to One thng that needs to be noted s that the data nclude all-age populaton, not only the workng-age populaton. Hence the probabltes mght overstate the persstence n hghexpendture states because the retred populaton s also ncluded. However, compared wth the persstence used n prevous studes for an AR(1) process (see Hubbard et al., 42 The costs nclude not only out-of-pocket expendture, but also nsurance payments. Thus the costs can represent the true costs that unnsured households must face. 22

23 Table 4: Transton probabltes of X Low Far Hgh Very Hgh Low Far Hgh Very Hgh Note: Identfed from Monhet (2003). 1995, for example), the persstence n hgh-expendture states here s stll lower. 43 The transton probabltes are lsted n Table 4. We can see that the matrx s not symmetrc and the persstence s not constant across states The socal welfare system We also need to measure the consumpton floor that the government tres to guarantee, above and beyond medcal expenses, through means-tested transfer payments (C). C s also the qualfcaton crteron for recevng socal welfare benefts: f a household s dsposable ncome and assets are less than C, the household can receve transfer payments to ensure the ablty of consumng C. However, measurng the means-tested consumpton floor C s dffcult, snce potental payments vary dramatcally by famly features, and even by the recpent s state or cty. In ths paper, I use the estmaton results reported n Hubbard et al. (1995) for the C, whch represents the consumpton floor provded n the U.S., but alternatve values of C wll be also revewed. Hubbard et al. (1995) make a frst approxmaton by calculatng the consumpton floor for representatve famles based on fgures n U.S. House (1991). Ther estmate of the consumpton floor ncludes only means-tested transfer payments such as AFDC, food stamps, and Secton 8 housng assstance for those under age 65. Unemployment nsurance s not ncluded because t s not means-tested, and t s ncluded n the measure of ncome. 44 Medcad s also not ncluded as a part of C because t s used exclusvely by the socal welfare system to pay for medcal expenses. 43 Feenberg and Sknner (1994) estmate the persstence coeffcent of the medcal-expense AR(1) process at 0.901, but ther sample s on households aged 55 and above. Hubbard et al. (1995) also use ths number (0.901) n ther medcal-expendture-shock process. Ths settng assumes that for all medcal expendture states, the persstence s always See the calbraton of labor endowment shock n the prevous subsecton. 23

24 Ther estmaton shows that, for a female-headed famly wth two dependent chldren and no outsde earnngs or assets, the medan AFDC and food stamp transfers ($5,764) plus expected housng subsdes ($1,173) were $6,937 n Ths amount s about 30% of the average permanent (annual) ncome level for hgh-school graduates reported n ther paper. However, as cautoned by Hubbard et al., the benefts mght be largely reduced f the father were present n the household or were marred to the mother, or f the household had fewer chldren or grown chldren. So the benefts mght be overestmated for a normal famly (wth both father and mother). Therefore, I assume C s 25% of average labor ncome to represent the sze of socal welfare system n the U.S., but a range from 15% to 30% wll also be revewed n the quanttatve analyses The employment-based health nsurance To characterze the employment-based health nsurance, I smplfy the feature of accessng prvate health nsurance n group and non-group markets as a probablty contngent on labor endowment status (equvalently, work status): f a person s labor endowment status s lower (representng a worse work status), that person wll have a lower probablty of beng covered by health nsurance. I set the probablty at 25% n the worst labor state, 60% n the second worst, 90% n the medan state (where labor endowment s unty), 95% n the second best and 99% n the best. Thus total unnsured households are 20% of total workng age populaton, whch s consstent wth the U.S. data n Numercal method The numercal method to solve ths model s based on the approach ntroduced by Ayagar (1994). I brefly descrbe the procedure below. In ths general equlbrum model, the equlbrum nterest rate (or rate of return of assets, r) s determned endogenously by households asset demand (equvalently, captal supply) and the frm s asset supply (equvalently, captal demand). Snce there are no producton shocks and the aggregate effectve labor supply s constant, t s clear that the frm s captal demand s a decreasng functon of r. 47 In addton, once r and 45 Ths settng, 25% of average labor ncome, s the total estmated benefts, subtractng the expected housng subsdes. 46 Ths smple settng assumes that households n every ncome group face the same dstrbuton of probablty of beng nsured. It may be more reasonable to set hgher probabltes for hgher permanent-ncomelevel households. However, t s dffcult to measure the dfference n the probablty dstrbuton. Moreover, ths complcated settng wll not change the results of ths paper. 47 There are only dosyncratc rsks n the model economy. Ths mples no aggregate uncertanty. That 24

25 equlbrum captal are determned, the wage rate w for each labor effcency level (λ ) s also determned. 48 For the asset demand, gven an r (so w s also gven) and a socal welfare parameter (C), I approxmate the asset holdng decson as a functon of dsposable ncome and wealth H (for each combnaton of the fve possble current labor endowment shocks l, the four possble current medcal expendture shocks x and the two possble nsurance coverage states α ) for each permanent ncome level (.e λ ). Because of the exstence of the socal welfare system, zero asset holdng s possble n some crcumstances. So t s necessary to consder the corner soluton when dong ths approxmaton. Usng the asset holdng polcy functon of each ncome level households, wth the smulaton of Markov chans for labor endowment shock, medcal expendture shock and nsurance status, we can calculate the sample mean of the smulated seres of asset holdngs as the aggregate assets demanded under the gven r. Addtonally, we can also obtan the assets suppled (captal demanded) from the frm under ths r. If the assets demanded exceed the assets suppled, ths mples the current r s hgher than the equlbrum; f the assets suppled exceeds the assets demanded, ths mples that t s lower. In ether case, we use the bsecton method as ntroduced n Ayagar (1994) to get a new r and re-do the the process untl demand and supply become equal. Once the steady state s approxmated, we can use the soluton to calculate the objects of our nterest. For the goal of ths paper, I calculate the medan asset holdngs of nsured and unnsured households n each permanent ncome level wth varous socal welfare systems and health nsurance systems, and compare these results wth the U.S. data. Furthermore, I wll use the data generated from the model to perform regresson s, n aggregate, the labor supply and the medcal expendture are both constant, although households do face uncertanty. 48 Because the mean of labor effcency λ and the mean of labor endowment l are both normalzed to one, we can defne the per capta captal k = K/N, and transform the total producton functon n equaton (10) to per capta producton functon: f(k) = k θ. (f3) The frst order necessary condton mples: k = ( ) 1 r+ δ 1 θ, θ (f4) whch s a decreasng functon of r. Gven an r, then wage rate w s also determned w = λ (1 θ)k θ, for each λ. (f5) The average wage rate s w = (1 θ)k θ. 25

26 analyss n order to compare wth the exstng emprcal studes. 6 Quanttatve Analyses and Results Ths secton provdes quanttatve analyses, and show that the model successfully explans the puzzlng correlaton between health nsurance coverage and savngs observed n the U.S. In addton, ths model suggests that countres wth dfferent szes of socal welfare systems and/or wth dfferent features of health nsurance systems may have very dfferent relatonshps between observed health nsurance coverage and savngs. In order to show the role of the socal welfare system clearly, I frst examne the dfference n savng behavor between (permanently) nsured and unnsured households n economes wth dfferent szes of socal welfare systems but wthout uncertanty of nsurance coverage. Then I add the employment-based (work-contngent) health nsurance system nto the model. Wth ths addton, the numercal results are mproved to ft the U.S. data not only n drecton, but also n magntude. 6.1 Baselne Model: Role of the Means-Tested Socal Welfare System In ths smpler case, health nsurance coverage status s gven permanently. The workcontngent health nsurance system s shut down. So we can examne clearly the devaton of household savng behavor from the standard model that s caused by the exstence of a means-tested socal welfare system. I compare two groups of households: households n one group are covered by health nsurance permanently and those n the other group are not. In order to better understand the role of the socal welfare system and to verfy the expectaton of the model, economes wth varous szes of socal welfare systems are smulated Comparson of two economes and the U.S. data Table 5 presents a comparson of ratos of medan asset holdngs between nsured and unnsured households (a n /a un ) n two economes: one wth a small socal welfare system and the other wth a large socal welfare system. For the small socal welfare system, C s set to be only 1% of average labor ncome, whle n the economy wth a large socal welfare system, C s set to be 25% of average labor ncome. Addtonally, n order to compare wth the pattern seen n the data, the ratos calculated wth three measures of assets from the U.S. data are also presented, whch were already reported n Table 1. The economy wth a small socal welfare system s expected to behave as the precautonary savng hypothess mples: households covered by health nsurance hold less 26

27 Table 5: Ratos of medan asset holdngs (a n /a un ) Model (permanent HI) U.S. Data* Income Small socal Large socal Measure 1 Measure 2 Measure 3 group welfare system welfare system (Lqud) (Fnancal) (Net worth) Bottom nd rd th Top Notes: (1) The rato s defned as medan asset holdngs of nsured households dvded by that of unnsured households. (2) Small socal welfare system: C =.01 w. Large socal welfare system: C =.25 w. *Calculated from the report n Starr-McCluer (1996). Orgnal source: Survey of Consumer Fnances (1989). assets than those that are uncovered. Ths s because ts socal welfare system only has slght effect on savng behavor. The smulated results confrm ths expectaton (the frst column n Table 5). We can clearly observe that the ratos of medan asset holdngs n all ncome groups are less than one, whch means the unnsured households hold more assets than the nsured households, as the standard precautonary savng model predcts. The second column presents the medan asset holdng ratos n the economy wth a large socal welfare system. The ratos are greater than one except those n the two hghest ncome groups (whch are stll close to one). Ths s not what the standard model predcts, but s expected by ths model. Because of the large means-tested socal welfare system (a greater C), the postve savng effect of health nsurance surpasses the negatve savng effect. In addton, we can also observe that the rato decreases as permanent ncome level ncreases. Ths also confrms our another expectaton of the model because the effect of the socal welfare system s smaller on the hgher permanent ncome households. Ths pattern shown n the model economy s consstent wth the U.S. data, presented n the last three columns and calculated from the report n Starr-McCluer (1996) wth dfferent measures of assets. However, the magntude of the ratos n ths model economy s less than that n the data. 27

28 Table 6: Ratos of medan asset holdngs (a n /a un ) wth varous values of C Sze of socal welfare system Income group C =.01 w C =.10 w C =.15 w C =.25 w C =.35 w Bottom nd rd th Top Is postve nsurance-savng correlaton puzzlng? Accordng to the prevous qualtatve analyss n Secton 4, we expect that the postve savng effect of health nsurance wll ncreases as the sze of the socal welfare system C ncreases (n a reasonable range). Furthermore, the net savng effect of health nsurance wll become postve f the sze of the socal welfare system s large enough. In order to verfy ths, economes wth varous szes of socal welfare systems are smulated here. We can do ths by varyng the value of C. A hgher C means that the economy provdes more protecton for households ncome and expendture uncertanty, and also provdes a hgher probablty of qualfcaton for the socal welfare system. Table 6 presents the smulaton results. C vares from 1% to 35% of average labor ncome (denoted by w) to represent dfferent szes of socal welfare systems. Ths table clearly shows the pattern that we expect: as C ncreases, the nsured households tend to hold more assets than the unnsured so that the ratos ncrease, and the ratos tend to be greater than one when C s hgh. Therefore, n an economy wth a means-tested socal welfare system, the exstence of precautonary savng motve does not necessarly lead to the concluson that nsured households should hold less assets than the unnsured even when there s no selecton problem. 6.2 Model wth Employment-Based Health Insurance System Now I examne an economy wth both socal welfare system and employment-based health nsurance system as the U.S. In ths economy, the health nsurance coverage s uncertan and contngent on work status. Ths s expected to further ncreases nsured households savngs. 28

29 Table 7: Ratos of medan asset holdngs (a n /a un ) Model (employment-based HI) U.S. Data* Income Small socal Large socal Measure 1 Measure 2 Measure 3 group welfare system welfare system (Lqud) (Fnancal) (Net worth) Bottom nd rd th Top Notes: Small socal welfare system: C =.01 w. Large socal welfare system: C =.25 w. *Calculated from the report n Starr-McCluer (1996) wth three measures of assets. To be comparable wth the prevous case, two economes wth small and large socal welfare systems are also smulated. Table 7 reports the smulaton results. The frst column n the table shows the asset holdng ratos (a n /a un ) generated from the economy wth a small socal welfare system (where C =.01 w). We can observe that the ratos are greater than one n all permanent ncome groups even wth such a low C, but the magntude s stll smaller than the data. The second column n Table 7 presents the asset holdng ratos n the economy wth a large socal welfare system (where C =.25 w). All of these ratos are greater than those n the small socal welfare system and close to the U.S. data. Moreover, we also fnd the rato decreases as ncome level ncreases, a fndng whch s also consstent wth the pattern shown n the data. 6.3 Senstvty Analyss In the prevous cases, the rsk averson coeffcent (µ) of the CRRA utlty functon s set at 3, as suggested by many emprcal studes. In order to show the robustness of the results, I provde a senstvty test for alternatve values of rsk averson µ. The results are reported n Table 8. We can see that the pattern of asset holdng ratos stll holds under each alternatve value of µ (from two to fve). In addton, snce precsely measurng the means-tested consumpton floor C s dffcult (see dscusson n Secton 3), I also perform a senstvty test for a range of reasonable socal welfare szes (C). The values of C from 15% to 30% of average labor ncome are tested. Table 9 presents the results. We can see that the pattern of asset holdng ratos under each value of C stll holds. 29

30 Table 8: Senstvty test rsk averson (µ) Ratos of medan asset holdngs (a n /a un ) Income group µ = 2 µ = 3 µ = 4 µ = 5 Bottom 2nd rd th Top Note: C =.25 w (large socal welfare system). Table 9: Senstvty test sze of the socal welfare system (C) Ratos of medan asset holdngs (a n /a un ) Income group C =.15 w C =.20 w C =.25 w C =.30 w Bottom , nd rd th Top Note: rsk averson µ = Dscusson: Why and When s the Standard Model Invald? Accordng to the above quanttatve analyss, the conventonally expected negatve correlaton between nsurance and savngs s observable only when the sze of the socal welfare system s small and the health nsurance coverage s stable. Otherwse, we are very lkely to observe a postve correlaton between health nsurance coverage and savngs, as observed n the U.S. However, ths postve correlaton does not necessarly mply the nonexstence of precautonary savng. There are at least two assumptons that guarantee the valdty of the standard model: Frst, no other factors affectng savng decsons vary wth the nsurance coverage. Second, the nsurance coverage s permanent, whch means no uncertanty of losng nsurance coverage. In ths paper, the ntroducton of the two nsttutons, the means-tested socal welfare system and the employment-based health nsurance system, volates these 30

31 two assumptons. Wth the means-tested socal welfare system, health nsurance not only decreases precautonary savngs, but also reduces the savng effects of the socal welfare system. Ths volates the frst assumpton. Wth the employment-based health nsurance, households currently n good employment states wth nsurance coverage wll save for future possble bad tmes wthout nsurance coverage. Therefore, n a real world economy (for example, the U.S.), a postve nsurancesavngs correlaton lkely reflects that some assumptons behnd the standard model do not hold, rather than mples the nonexstence of precautonary savng. 7 Implcatons for Emprcal Testng In addton to descrptve statstcs, emprcal studes also use econometrc methods that control for other characterstcs n order to more precsely analyze the effect of health nsurance on savngs. To compare wth these prevous emprcal studes and to evaluate current emprcal approaches, I also apply a smlar econometrc method to the model economy. We can see that the emprcal results from the model economy are consstent wth the exstng emprcal fndngs. However, nterpretaton of the results based on ths model greatly dffers from the conventonal nterpretaton. Moreover, I wll dscuss the mplcatons of the model for our assessment of the emprcal approaches to testng the precautonary savng hypothess. 7.1 Econometrc Method and Regresson Results I apply a conventon regresson model and use the data generated from smulaton of the model to examne whether ths emprcal approach can properly test for the exstence of precautonary savng. The model wth a means-tested socal welfare system and a employment-based health nsurance system (C =.25 w and rsk averson µ = 3), as analyzed n the prevous secton, s used to represent the U.S. economy. In the model economes, by constructon, every household has the same rsk averson and does have precautonary savng motvaton. Therefore, an approprate regresson model should correctly reveal the exstence of precautonary savng. The conventonal regresson model based on the standard precautonary savng model s set as follows: log(a,t ) = αhi,t + β Q,t + ε,t, (12) 31

32 Table 10: Comparson of results of regressons Starr-McClure (1996) Measure 1 Measure 2 Measure 3 Model Economy (Lqud) (Fnancal) (Net worth) HI coverage (0.027) (0.18) (0.16) (0.19) Perm t Income (0.016) (0.11) (0.11) (0.15) Medcal Shock (0.017) (0.10) (0.10) (0.12) Socal welfare system: C =.25 w. Rsk averson coeffcent: µ = 3. 1 Medcal expendture n the model; llness status n Starr-McClure (1996). where a,t s household s asset holdngs at tme t, HI,t s a dummy varable of health nsurance coverage, and Q,t s a vector of varables, whch control for all other drectly observable household characterstcs that also affect savng. 49 Accordng to the standard model, we would nterpret a negatve coeffcent of HI (.e. α) as verfcaton of the exstence of precautonary savng. By the same logc, a postve coeffcent would be nterpreted as rejecton of the exstence of precautonary savng. Ths knd of regresson model was also used n Starr-McClure (1996), where she reported a sgnfcantly postve coeffcent of HI usng U.S. cross-secton data of workng age populaton. 50 When Starr-McClure (1996) estmated the above asset holdng equaton (12), she controlled for selectvty, estmated permanent ncome, health problem, and other household characterstcs as ndependent varables n Q,t. Here I follow her approach, and because the model envronment s much smpler than the real world, I need only control for permanent ncome level and medcal expendture shock as household characterstcs. The regresson results are reported n Table 10. For comparson, the results n Starr- McClure (1996) are also presented n the table. 51 The frst column of the table presents 49 See the control varables used n Starr-McClure (1996) for example. 50 She used both ths smple regresson model and a maxmum lkelhood selecton model (to control for selectvty) to estmate the asset holdng equaton, but found the same results a sgnfcant and postve effect of health nsurance on savngs. In addton, there s no selecton problem n the model economy, so I do not need to control for selectvty. 51 Only varables related to the model are reported. Emprcal studes based on real data need to control for more varables, for example, age, educaton, sex, etc. In addton, Starr-McClure (1996) puts a lot of effort nto controllng for selectvty, whch does not exst n the model. 32

33 Table 11: Comparson of regresson results - controllng for ncome shock Model economy Wthout controllng for ncome shock Controllng for ncome shock HI coverage Permanent Income Medcal Expendture Shock Income Shock 1.82 Notes: (1) Socal welfare system: C =.25 w. Rsk averson coeffcent: µ = 3. (2) All coeffcents are sgnfcant at 5%. the ordnary least square regresson results usng data from the smulated model economy; we can see that the coeffcent of health nsurance coverage n the model economy s sgnfcantly postve and consstent wth Starr-McClure s fndngs. 7.2 Interpretaton and Implcatons for Emprcal Testng Based on the conventonal nterpretaton, a sgnfcant and postve coeffcent of health nsurance n the above regresson would lead to a rejecton of the exstence of precautonary savng. However, ths nterpretaton s not always correct. As shown by the above exercse, the conventonal regresson model s unable to truly reveal the exstence of precautonary savng n the model economy. In fact, ths regresson model can only capture the net savng effect of health nsurance, rather than exclusvely the effect on precautonary savng. As prevously analyzed, n an economy wth a means-tested socal welfare system and employment-based health nsurance, we are very lkely to observe a postve effect of health nsurance on savngs, even when there s no selecton bas. The followng provdes a dscusson about bases n the conventonal regresson model and some possbltes for mprovement Bas caused by employment-based health nsurance and possble correcton When health nsurance coverage s contngent on work status, households currently covered by health nsurance are very lkely also n full-tme and better job postons, whch mply hgher ncome. Therefore, the coeffcent of health nsurance coverage s combned wth the effect from an ncentve of consumpton smoothng: people n good tmes 33

34 wth nsurance coverage save for future possble bad tmes wthout coverage. Moreover, nsurance coverage assocated wth hgher ncome further reduces the negatve savng effect of the socal welfare system, and then further ncreases nsured household s savngs. To correct ths bas, we frst need to control for ncome shock (.e. the devaton from a household s permanent ncome level). In practce, ncome shock s not easy to dentfy, especally when usng cross-secton data. However, n the model, we can control for t to show the potental dfference n the regresson results. The comparson of regresson results are reported n Table 11. We can see that when controllng for the ncome shock (the second column), the coeffcent of health nsurance becomes smaller. Ths s because ncome shock (mplyng change n work status) explans some of the postve savng effect. Addtonally, the coeffcent of ncome shock s postve, as we expect Bas caused by the socal welfare system and possble correcton As explaned prevously, n an economy wth a means-tested socal welfare system, the ntroducton of health nsurance wll cause two opposng effects on savngs. The negatve effect s the reducton of precautonary savngs as n a standard model, and the postve effect s due to the exstence of the socal welfare system. However, we can only observe the net effect n data. To dentfy and separate these two combned savng effects through a regresson model s dffcult, even n the model economy. One possblty wll be that: f we can fnd an economy experencng regme changes n ts socal welfare system and health nsurance system, t wll be easer to observe these combned effects wth the regresson model. For example, f the economy s regme changes from a small socal welfare system (assume C =.01 w) and constant health nsurance coverage (Regme 1) to a large socal welfare system, where C =.25 w (Regme 2), and then the health nsurance system changes to an employment-based (work-contngent) health nsurance system (Regme 3), wth other thngs beng equal, we wll be able to observe that: (1) n Regme 1, the regresson coeffcent of health nsurance mostly reflects the reducton of precautonary savngs caused by health nsurance, and so t s expected to be negatve; (2) from Regme 1 to Regme 2, the dfference n the regresson coeffcent of health nsurance s due to the regme change n the socal welfare system (the postve savng effect of health nsurance becomes larger); 34

35 Table 12: Comparson of regresson results - regme change Model economes wth dfferent regmes Small SW system Large SW system Large SW system Constant HI Constant HI Employment-based HI HI coverage Permanent Income Medcal Shock Income Shock Notes: (1) Large socal welfare system: C =.25 w. Small socal welfare system: C =.01 w. (2) Rsk averson coeffcent: µ = 3. (3) All coeffcents are sgnfcant at 5%. (3) from Regme 2 to Regme 3, the dfference n the regresson coeffcent s nstead due to the regme change n the health nsurance system (the uncertan nsurance coverage strengthens the postve savng effect). I use the same model to smulate an economy wth the above regme changes, and run a regresson under each regme to show these effects. The regresson results are lsted n Table 12. Income shock s assumed to be observable when runnng the regressons. We can see that, under Regme 1, the coeffcent of health nsurance coverage s negatve (-0.50), whch s consstent wth the conventonal expectaton. When the economy changes to Regme 2, the coeffcent becomes postve (0.38) because of the regme change n the socal welfare system. When the economy changes to Regme 3, the coeffcent becomes larger (0.68) due to employment-based health nsurance. 8 Concluson and Possble Future Research From the vantage pont of a standard model of precautonary savng, there are two puzzlng household savng patterns observed n the U.S., (1) a postve correlaton between prvate health nsurance coverage and household asset holdngs, and (2) a shrnkng dfference n asset holdngs between nsured and unnsured households wth ncreased ncome level. Ths paper suggests that the puzzlng fndngs can be explaned by the exstence of two nsttutons: a szable asset-based means-tested socal welfare system and an employment-based health nsurance system. Ths paper develops a standard stochastc dynamc general equlbrum model ncorporatng these two nsttutons to analyze 35

36 the ssue, and shows that the model can generate a pattern of asset holdng ratos consstent wth the U.S. data. Ths result does not depend on the assumpton of heterogenety of rsk averson (.e. selectvty). Ths paper ponts out that the standard model of precautonary savng (or, the conventonal expectaton: a negatve correlaton between nsurance coverage and savngs) requres at least two assumptons to be vald: that the nsurance coverage s permanent (no uncertanty for the coverage); and that no other factors affectng savng decsons vary wth health nsurance coverage. The above two nsttutons volate these two assumptons. I have shown that only when the means-tested socal welfare system s small and the health nsurance coverage s constant, do we observe a negatve correlaton between nsurance coverage and savngs as the standard precautonary savng model suggests. As also shown n the analyss, health nsurance coverage reduces the negatve savng effect caused by the socal welfare system, and the employment-based health nsurance system ncreases the ncentve to save when nsured. Ths explans the postve correlaton observed n the U.S., and shows that the emprcal results do not therefore mply the nonexstence of a precautonary savng motve, when an economy has the nsttutons we observe n practce. Ths study also brngs nto queston the approprateness of the conventonal emprcal approaches to testng the precautonary savng hypothess. To show ths, I apply a regresson approach, as used n exstng emprcal studes, to the model economy. The regresson results show a sgnfcant postve coeffcent of health nsurance coverage n an economy wth the above two nsttutons confrmng the fndngs n exstng emprcal studes. However, the postve coeffcent does not mply the nonexstence of precautonary savng. In fact, people do have a precautonary savng motve n the model economy. Ths s because the conventonal regresson model captures the net savng effect of health nsurance, rather than exclusvely the effect on precautonary savngs. It s dffcult to dstngush the savng effect caused by the two nsttutons from the effect of health nsurance on precautonary savng. Ths paper has dscussed some probabltes for mprovng the emprcal tests, but more research s stll needed. Possble Future Research Unlke Starr-McClure (1996) and Guargla and Ross (2004), who focus on controllng for the effect of selectvty but stll fnd a sgnfcantly postve savng effect of health nsurance coverage, ths paper suggests that the above two nsttutons, the socal welfare system and the health nsurance system, also have strong effects on savng decsons. In order to more approprately test for the exstence of precautonary savng, we need 36

37 to control for the effects caused by these nsttutonal factors. In Secton 7, I have dscussed some possbltes for mprovng the emprcal test approach. I use smulaton to show that f we can observe an economy wth regme changes, we wll be able to dentfy the negatve effect of health nsurance on precautonary savngs and the postve savng effects of health nsurance caused by the nsttutons of the regmes. However, ths s largely relant on the avalablty of the data. Another possble approach s the comparson of cross-country data, but t requres a careful study of the consttutons n each country, and may requre controllng for other country-specfc characterstcs. Therefore, developng an approprate emprcal approach to dentfy those combned effects, as suggested by ths paper, stll requres more research, and s mportant for future studes on savng behavor. In addton to the ssues dscussed above, exstng emprcal studes show seemngly conflctng evdence about the effects of health nsurance on precautonary savng. For example, Starr-McClure (1996) and Guargla and Ross (2004) use the data of prvate health nsurance n the U.S. and the U.K., respectvely, and fnd a postve effect on savngs; Gruber and Yelowtz (1999) and Maynard and Qu (2005) use data of Medcad n the U.S., and fnd a negatve savng effect; Chou et al. (2003, 2004) use data on the unversal health nsurance program n Tawan and employ the dfferences-n-dfference approach. They also fnd the savng effect of ths health nsurance program s negatve. Snce they use dfferent data and varous approaches, t s not easy to nterpret these fndngs or even reconcle them. The model provded by ths paper may be useful n explanng why these studes have seemngly conflctng fndngs. Ths can help us to better understand the economcs behnd these emprcal approaches and fndngs. I expect that these seemngly conflctng fndngs are consstent n an envronment wth the two nsttutons ntroduced n ths paper. Because 20% of the workng-age populaton are consstently unnsured n the U.S., polces for ncremental health-nsurance expanson have attracted a large amount of attenton. The extenson of ths model s useful for evaluatng such alternatve approaches to ncremental health-nsurance expanson. Some alternatves that have been proposed nclude (1) expanson of publc health nsurance, (2) subsdy of non-group health nsurance purchases, and (3) reducton of the premum n the non-group market (wth the government as the rensurer for hgh medcal expenses). In order to analyze ths ssue, we can extend the model to allow the decson of health nsurance purchase, and then evaluate the effcency of these alternatve approaches n ncreasng health-nsurance access. The soluton of such an extended model wll be more complcated and may requre more advanced technques. 37

38 References Ayagar, S. Rao (1994), Unnsured dosyncratc rsk and aggregate savng. Quarterly Journal of Economcs,109:3, Carroll, C. D. (1992), The Buffer-Stock Theory of Savng: Some Macroeconomc Evdence. Brookngs Papers Econ. Actvty, 2: Chou, S., J. Lu and J. Hammtt (2003), Natonal Health Insurance and Precautonary Savng: Evdence from Tawan. Journal of Publc Economcs, 87: Chou, S., J. Lu and C. J. Huang (2004), Health Insurance and Savngs Over the Lfe Cycle A Semparametrc Smooth Coeffcent Estmaton. Journal of Appled Econometrcs, 19: Deaton, A. (1991), Savng and Lqudty Constrants. Econometrca, 59: Dynan, K. E., J. Sknner and S. P. Zeldes (2004), Do the Rch Save More? Journal of Poltcal Economy, 112:2, Feenberg, D. and J. Sknner (1994), The Rsk and Duraton of Catastrophc Health Care Expendtures. Revew of Economcs and Statstcs, 76:4, Gruber, J. and A. Yelowtz (1999), Publc Health Insurance and Prvate Savngs. Journal of Poltcal Economy, 107:6, Guargla, A. and M. Ross (2004), Prvate medcal nsurance and savng: evdence from the Brtsh Household Panel Survey. Journal of Health Economcs, 23:4, Herrng, B. and M. Pauly (2003), Incentve-Compatble Guaranteed Renewable Health Insurance. NBER Workng Paper #9888. Hubbard, R. G., J. Sknner and S. P. Zeldes (1995), Precautonary savng and socal nsurance. Journal of Poltcal Economy, 103, Huggett, M and G. Ventura (2000), Understandng why hgh ncome households save more than low ncome households. Journal of Monetary Economcs 45, Leland, H. E. (1968), Savng and Uncertanty: The Precautonary Demand for Savng. Quarterly Journal of Economcs, 82:3, Ljungqvst, L. and T. J. Sargent (2000), Recursve Macroeconomc Theory, Cambrdge, Mass: MIT Press. Kotlkoff, L. J. (1989), Health expendtures and precautonary savngs. What Determnes Savngs? by L. J. Kotlkoff, Cambrdge, Mass: MIT Press. Maynard, Alex and Japng Qu (2005), Publc nsurance and prvate savngs: who s affected and by how much? Workng Paper. McGarry, Kathleen (2002), Publc Polcy and the U.S. Health Insurance Market: Drect and Indrect Provson of Insurance. Natonal Tax Journal, LV:4,

39 Monhet, Alan C. (2003), Persstence n Health Expendtures n the Short Run: Prevalence and Consequences. Medcal Care, 41:7, Powers, E. T. (1998), Does Means-Testng Welfare Dscourage Savng? Evdence from a Change n AFDC Polcy n the Unted States. Journal of Publc Economcs, 68: Sandmo, A. (1970), The Effect of Uncertanty on Savng Decsons. Revew of Economc Studes, 37, Starr-McCluer, Martha (1996), Health Insurance and Precautonary Savngs. Amercan Economc Revew, 86:1, Swartz, K. (2003), Rensurng Rsk to Increase Access to Health Insurance. Amercan Economc Revew, 93:2,

40 Appendx A1. Why s a smple selectvty assumpton nsuffcent to explan the puzzlng savng patterns observed n the U.S.? Assume there are two types of households: more rsk averse and less rsk averse. Because asset holdngs are manly determned by the dfference n rsk averson, the asset holdng ratos (the more-rsk-averse relatve to the less-rsk-averse) n the low-ncome and hgh-ncome groups wll be smlar wthout nsurance coverage. Suppose that all the more-rsk-averse households get heath nsurance coverage, but the less-rsk-averse do not, whch s the selectvty. The nsurance coverage wll reduce uncertanty, and wll reduce a greater percentage of the orgnal asset holdngs n the low-ncome morersk-averse households than n the hgh-ncome more-rsk-averse households. Thus, the asset holdng dfference between the nsured and the unnsured n hgh ncome groups wll not change much, but n low ncome groups wll be much smaller. Ths s contrary to what we observe n U.S. data. The key pont s that, even wth selectvty, health nsurance wll stll only have a negatve effect on savngs, whch can not explan the dfference across ncome groups. I also provde a smple and standard two-perod model to llustrate ths. I construct a smple two-perod model wth two rsk averson types of households. Preference s stll assumed to be the CRRA utlty functon, as n the text. The rsk averson coeffcent µ m for the more rsk averse households s larger than the less rsk averse households, µ l. Household ncome s gven n perod 1, and there are two ncome levels: y r 1 and yp 1, where y r 1 > yp 1 wth no uncertanty n perod 1. Income n perod 2 (y 2) s uncertan, and households mght have a medcal expendture shock x. For smplcty, I assume there are only two possble states (good and bad) n the perod 2 wth the same probablty. For household j, n the good state, y j 2 = 1.5y j 1 and x = 0; n the bad state, y j 2 = 0.5y j 1 and x = 0.2ȳ, where ȳ = 0.5(y r 1 +yp 1 ). Ths means that the expected household ncome n the perod 2 s equal to ts ncome n perod 1, and that the medcal expendture s equal to 20% of average ncome. In addton, I smply assume the more rsk averse households have health nsurance coverage and the less rsk averse households do not. Ths smple assumpton represents the selectvty. The consurance rate ᾱ s set at 20%, and the nsurance premum q s assumed far, as n the text. A household n rsk averson group ( = m,l) and n ncome group j ( j = r, p) 40

41 Table 13: Asset holdng ratos n a smple selectvty model Income group Asset holdng rato (a m j /a l j ) Low ncome (wth y p 1 ) 1.14 Hgh ncome (wth y r 1 ) 1.37 Note: Assume µ m = 6 and µ l = 1. The more rsk averse households are covered by health nsurance, but the less rsk averse are not. chooses consumptons c j 1, c j 2 and asset holdngs a j to maxmze [ ] E 0 U (c j 1 )+βu (c j 2 ) subject to U (c j t ) = (c j t ) 1 µ /(1 µ ); t = 1,2; c j 1 + a j = y j 1 ; c j 2 = y j 2 +(1+r)a j α x j q ; α = ᾱ, q = q f = m; α = 1, q = 0 f = l; (y j 2,x j ) {(0.5y j 1,0.2ȳ),(1.5y j 1,0)}; Pr((y j 2,x j ) = (0.5y j 1,0.2ȳ)) = Pr((y j 2,x j ) = (1.5y j 1,0)) = 0.5. From the frst order necessary condtons, we can get the Euler equaton as: (y j 1 a j ) µ = β(1+r).5 [ (.5y j 1 +(1+r)a j α.2ȳ q ) µ +(1.5y j 1 +(1+r)a j q ) µ]. To llustrate the asset holdng decsons more clearly and make the case smpler, I assume β = 1 and r = 0. Then I specfy rsk averson levels µ m = 6 and µ l = 1, and specfy y r 1 = 2yp 1 as an example to examne the effects of selectvty. Usng the above Euler equaton and ths specfcaton, we can solve for the asset holdng decsons for households wth dfferent rsk averson and ncome levels. The computaton results are lsted n Table 13. In ths smple case, the asset holdng decson (a j ) for a household wth rsk averson µ and ncome level y j 1 s smply a functon of the frst perod ncome (y j 1 ). Although the nsured (more rsk averse) households ndeed hold more assets than unnsured (less rsk averse)households, we can see the pattern of asset holdng rato (a m j /a l j, the nsured 41

42 relatve to the unnsured) n ths case s opposng to the U.S. data n whch the rato decreases as ncome level ncreases. We observe a larger asset holdngs rato n the hgh ncome group n ths model wth selectvty (the thrd column n Table 13). Before the ntroducton of health nsurance, the asset holdng rato represents the dfference n rsk averson. The asset holdng rato (a mr /a lr ) n the hgh ncome group s qute smlar to the rato (a mp /a lp ) n the low ncome group (a mr /a lr = 1.5 and a mp /a lp = 1.4). However, after more rsk averse households buy heath nsurance, the low ncome nsured households wll reduce a larger porton of ther orgnal asset holdngs than the hgh ncome nsured households. Therefore, there wll be a larger percentage reducton n a mp than n a mr. Then the asset holdng rato n the hgh ncome group wll be even larger (a mr /a lr = 1.36 and a mp /a lp = 1.14). Ths llustrates that the smple assumpton of selectvty s able to explan the more asset holdngs for nsured households, but s not consstent wth the pattern of the decreasng asset holdng rato wth ncreased ncome level n the U.S. data. A2. Analyss of the negatve savng effects of the socal welfare system By comparng equaton (8) wth the Euler equaton n an economy wthout the socal welfare system, we can fnd that there are two channels through whch the socal welfare system can decrease households asset holdngs. 52 The frst channel s reducng the fluctuatons of ncome and expendture, because n ths economy, the dsposable resources (H ) n bad states (when H C) never falls below C. Hence, for each value of a (asset holdngs n the next perod), the badstate consumpton n ths economy (C a ) must be hgher than or equal to the badstate consumpton n the economy wthout the socal welfare system (H a ).53 Ths 52 Consder an economy wth the same settngs as descrbed n ths secton except the socal welfare system settngs (.e. no transfer payment, T R, n household s budget constrant). For comparson, I stll dvde the expectaton nto two parts, when H > C and when H C, and then can get the Euler equaton as follows: U c =β(1+r )E[U c ] { ]} =β (1+r ) {p [U E c H ]+(1 > C p [U )E c H C + p { [ E U(c a ) H > C ] E [ U(c ) H C ]}} ; (f3) where c = H a n all stuatons. In the economy wth the socal welfare system, consumpton s C mnus asset holdngs when household s qualfed for socal benefts. 53 That s C a H a. 42

43 ndcates that the utlty n bad states s ncreased by the socal welfare system (because E[U(C a ) H C] E[U(H a ) H C]), and mples that the utlty gap between good states and bad states (E[U(c ) H > C] E[U(c ) H C], n the RHS of equaton (8)) s reduced. The second channel comes from the asset-based means test. Asset holdngs wll be useless f households qualfy for the socal welfare program. The reason s when household qualfes, ts consumpton c = C a does not depend on current asset holdngs a n the economy wth a means-tested socal welfare system. Therefore, E [ U(c )/ a H C] = 0 wth the probablty (1 p ). Compared wth the Euler equaton n the economy wthout the socal welfare system, we can observe that ths decreases the expected margnal return of asset holdngs. Both above effects reduce the value of the RHS n equaton (8) and therefore, household needs to reduce asset holdngs, a, to ncrease the return and balance the equaton. As mentoned n the text, n some crcumstances, even when a = 0, the RHS (return) s stll lower than the LHS (cost). Then the household wll choose to hold zero assets. The socal welfare system also ncrease the probablty of zero asset holdng by reducng the value of the RHS generally. A3. The net effect of health nsurance n an economy wth the means-tested socal welfare system. The RHS of the Euler equaton (8) n ths paper represents the expected margnal return of asset holdngs: p (1+r )E(U c H > C)+ p { [ E U(c a ) H > C] E [ U(c ) H C]}. Let H represent the lower bound of household dsposable ncome and assets. If C s set below H, no households can qualfy for the socal programs. The effect of health nsurance s a reducton of precautonary savngs. When the C s set hgher than H, there exst two postve savng effects of health nsurance: an ncrease n p and an ncrease n the utlty gap between good and bad states. The followng analyzes the effects of health nsurance on the expected return of assets, and dscusses the determnaton of the net effect. 43

44 The frst postve effect Snce p s defned as the probablty that H > C for household, gven a C, p s determned by the dstrbuton of H. 54 We know that H s the dsposable ncome and assets for household. 55 So, gven other parameters, ts dstrbuton s determned by the dstrbutons of labor endowment shock l and medcal expendture shock x, when the household s unnsured (see Fgure 1, the Unnsured curve for example). The ntroducton of health nsurance to the household reduces the varance of medcal expendture shock, and thus reduces the varance of H. 56 Moreover, the health-nsurance effect s manly on the left tal of the H dstrbuton, where a large x and/or a small l occur, but the effect s much smaller on the rght tal because x s already small there. Therefore, the dstrbuton of H after beng nsured can be llustrated as the Insured curve n Fgure 1. The left tal s reduced much more by the health nsurance than the rght tal. Gven C equal to the C2 n Fgure 1 for example, the probablty that H > C when unnsured (.e. an unnsured household s p ) s the area rght of the C2 and under the Unnsured curve; that under the Insured curve s the p when nsured. Wth dfferent settngs of C (such as C1, C2 and C3 n fgure 1), the value of p wll be dfferent. 57 The effect of health nsurance on the probablty p (.e. the change n p ) when C s set at C2, s the area change from the Unnsured curve to the Insured curve on the rght sde of lne C2 n Fgure 1. Because the ncrease n p s equvalent to the reducton n (1 p ), the area change s equvalent to the shaded area on the left sde of lne C2 n Fgure Hence, we can observe that when C s set a bt hgher than H, for example C1, the effect of health nsurance on p s postve and equal to the shaded area left of 54 The dstrbuton of H must be tme nvarant (.e. statonary) f the equlbrum exsts. Hence, the dstrbuton of H s equal to the dstrbuton of H 55 The defnton s that H = (1 τ)w l +(1+r)a α x q. 56 Insured households pay ᾱx and a constant q nstead of x. 57 In Fgure 1, the Unnsured curve represents the probablty densty functon (pdf) of H when household s not covered by health nsurance (denoted by f un (H )), and the Insured curve represents the pdf of H when household s nsured (denoted by f n (H )). The probablty that H > C wthout nsurance (denoted by p un, ) s defned as: p un, (C) = C f un(h )dh ; and the probablty that H > C wth nsurance (denoted by p n, ) s defned as: p n, (C) = C f n(h )dh. 58 Ths shaded area represents the change n 1 p from Unnsured to Insured. 44

45 f(h) 1-p( C2) p( C2) Unnsured f (H) un Insured f (H) n H C1 C2 C3 H Fgure 1: The effect of health nsurance on the dstrbuton of H wth C lne C1. When C s ncreased to C2, the postve effect on p s largest (.e. the reducton n 1 p s largest). However, when C s set beyond C2, the postve effect starts to decrease and mght even become zero or negatve at some pont, lke C3. Nevertheless, C can not be set too hgh (for example, beyond the mean of H), or the socal welfare system wll not be sustanable. In ths economy, the government taxes labor ncome to fnance the socal welfare system, and thus the hghest sustanable C must be smaller than the mean of the H. Ths mples that the C3 (hgher than the mean H) n fgure 1 s unsustanable. Therefore, we can smply focus on sustanable settngs of C, and omt the cases of hgh C wthout loss n the analyss. The second postve effect The second postve savng effect of health nsurance s on the second part of the RHS n the Euler equaton, [U(c ) H > C] E [U(c ) H C]. It s due to the ncrease n the expected utlty gap between good states (defned as H > C, that s when not qualfed for the socal programs) and bad states (defned as H C). However, ths effect s expected to be slght regardless of the value of C for the followng reasons. Health nsurance s only effectve when household s H > C because the socal welfare system wll cover the medcal expendture when qualfed. In addton, health n- 45

46 surance has smaller effects on households n better states (wth a hgher H ) n whch the medcal expendture shock x that occurs must be smaller. Therefore, when C s set hgh, health nsurance must have less postve effect on the expected utlty n the states when H > C. Thus, the ncrease n the expected utlty gap wll be smaller. Moreover, when C s low, the multpler n the second part of the RHS n the Euler equaton (8), p / a, s largely reduced by health nsurance.59 Ths reducton offsets the ncrease n the expected utlty gap. Because asset holdngs are a substtute for health nsurance, n terms of ncreasng the probablty of beng n good states (.e. p ), the ntroducton of health nsurance reduces the room for asset holdngs to ncrease p (when C s set below some threshold). 60 We can also use fgure 1 to llustrate ths. The effect of an asset-holdngs (a ) ncrease on H, gven other thngs beng equal, s to shft the whole dstrbuton of H to the rght (one unt ncrease n a leads to (1 + r)-unt ncrease n H n all states). Ths shft then ncreases the p because there s a larger area to the rght of C. Therefore, the margnal ncrease n p s that the probablty densty at C tmes (1+r),.e. (1 + r) f un, (C) f unnsured and (1 + r) f n, (C) f nsured. In fgure 1, we can observe that as long as C s below C2, health nsurance reduces the margnal ncrease of a n p. In addton, ths reducton s larger when when C s not hgh (on the other hand, when health nsurance has a larger effect on ncreasng p ). In concluson, when C s not hgh (n the case of fgure 1, that s below C2), the effect of the ncrease n the expected utlty gap mght be largely offset by the reducton of ( p / a ); when C s hgh, the ncrease n the expected utlty gap tself s small. Therefore, we expect that ths postve effect of health nsurance on savng s slght, regardless of the value of C. 59 p / a represents the margnal ncrease n p of assets. 60 Ths means: ( p n, (C)/ a ) ( p un, (C)/ a ) > 0 when C < C; where p n, s the probablty that H > C when household s nsured, p un, s the probablty that H > C when unnsured, and C s some value n the possble range of H. At C, the probablty densty of H wth health nsurance coverage starts to surpass the probablty densty of H wthout nsurance coverage. By the defnton of p, we can get: ( p n, (C)/ a ) = (1+r ) f n, (C); ( p un,(c)/ a ) = (1+r ) f un, (C). Usng fgure 1 for llustraton, we see that the threshold C s C2 because f n, (C) s smaller than f un, (C) when C s set below C2. 46

47 The negatve effect Besdes the postve savng effects, health nsurance also causes a decrease n the condtonal expected margnal utlty (E[U c H > C]).61 The decrease s due to the assumpton of convexty of margnal utlty (to ensure precautonary savng). Health nsurance reduces the fluctuaton of medcal expendture shock x as well as the fluctuaton of H, and consequently reduces the expected margnal utlty, E[U c H > C]. However, ths reducton, caused by health nsurance, decreases wth ncreased C. Ths s because health nsurance can only work n the states n whch household s not qualfed for the socal programs (H > C). When C s set hgher, a large part of the medcal expendture uncertanty x has already been reduced by the socal welfare system and health nsurance has less ablty to reduce the fluctuaton of H (the x fluctuaton s already smaller n ths crcumstance). Fgure 2 uses a smple example to llustrate that the reducton of E[U c H > C] decreases wth ncreased C. Suppose that there are only two states of H, low and hgh. Low H mples that low l and hgh x occur, and hgh H mples that hgh l and low x occur. In Fgure 2, the curve U c represents the (convex) margnal utlty functon of consumpton. C un 1 and C un 2 represent the consumptons wth low H and hgh H respectvely and E[U cun H > C] s the expected margnal utlty when households are unnsured. When households are covered by health nsurance, the fluctuaton of H s reduced and so s the fluctuaton of consumpton. C n 1 and C n 2 represent the consumptons wth low H and hgh H respectvely and E[U cn H > C] s the expected margnal utlty when households are nsured. C H = C represents the consumpton when households are qualfed for the socal welfare programs. Part (A) s the case of low C (denoted by C low ) and part (B) s the case of a hgher C (denoted by C hgh ). In (A), the C s too low to have any effects on H or consumpton (because (C H = C low ) s lower than C un 1). In (B), the C s set hgher than the low H when households are unnsured but lower than the low H when households are nsured. Thus, under ths socal welfare system, the unnsured household consumpton n the bad state (C un 1) s ncreased to the lne (C H = C hgh ). Compared wth (A) and (B), we can clearly observe that health nsurance has a larger reducton n the condtonal expected margnal utlty n (A) than n (B), because the dfference between E[U cun H > C] and E[U cn H > C] s larger when C s smaller p (1+r )E(U c H > C) s the frst part of the expected return of assets n the Euler equaton (8). 62 The convexty of margnal utlty and the medcal expendture shock x are the key reasons. 47

48 (A) Low C Uc C 1 un C 1 n E[Uc un H> C] E[Uc n H> C] 0 C (H= C low ) C 2 n C 2 un Uc C Uc (B) Hgher C C 1 un C 1 n E[Uc un H> C] C (H= C hgh ) E[Uc n H> C] C 2 un C 2 n Uc C Fgure 2: The reducton of the condtonal expected margnal utlty wth dfferent values of C 48

49 Table 14: The effects of health nsurance on expected return of asset holdngs Effects of health nsurance on the expected return of assets* Effects on the frst part Effects on the second part Value p E(U c H > C) E[U(c ) H > C] p / a of C E[U(c ) H C] below H n.a. negatve effect n.a. n.a. (largest) H to C2 postve effect, negatve effect, postve effect, negatve effect, ncreasng decreasng wth decreasng wth decreasng untl C2 ncreased C ncreased C untl C2 at C2 postve effect same as above same as above zero (largest) C2 to C m postve effect, same as above same as above postve decreasng from C2 from C2 Notes: *That s the RHS of the Euler equaton (8): β { (1+r )p E(U c H > C)+ p a {E [U(c ) H > C] E [U(c ) H C]}}. (1) C represents the sze of socal welfare system; (2) H s the lower bound of H ; (3) C2 s the threshold as n fgure 1; (4) C m s the maxmum sustanable C. The net effect Table 14 s the summary of the above analyss. A household s asset-holdng decson s determned by the expected return of assets, whch s represented by the RHS n the Euler equaton (8). Accordng to the above analyss, health nsurance has both postve and negatve effects on asset-holdng decson, and the effects change wth dfferent settngs of C. When C s low, the postve effect on p s small, but the negatve effect on E(U c H > C) s large (largest when C H), and the effects on the second part of the RHS n the Euler equaton are largely offset by each other. As a result, the net effect of health nsurance on asset holdngs s expected to be negatve, whch s what the tradtonal precautonary savng hypothess predcts. In contrast, when C s hgh (around the threshold, whch s denoted by C2 n table 14), the postve effect on p s large (largest at C2), and the negatve effect on E(U c H > C) s largely decreased by the hgh C. In addton, the net effect on the second part of the RHS n the Euler equaton s small and must be postve when C C2. 49

50 Table 15: The net effects of health nsurance on asset holdngs Settng of C The net (observed) effect Comparson wth on asset holdngs the tradtonal PSH (1) Low negatve as the tradtonal PSH predcts Hgh (2) postve unlke what the tradtonal PSH predcts Notes: (1) PSH stands for the precautonary savng hypothess; (2) In a reasonable range. If C s set very hgh, the nsured wll stll have a very hgh probablty of qualfyng for the socal welfare benefts. Then the net effect mght be negatve. And so, when C s hgh, the net effect of health nsurance on asset holdngs s expected to be postve, whch does not follow the expectaton of the tradtonal precautonary savng hypothess. Table 15 s the summary of the net effect of health nsurance. Ths paper extends the mplcaton of precautonary savng n an economy wth a socal welfare system, and these results are verfed n the quanttatve analyss n the text. 50

51 f(h) 1-p( C2) p( C2) ginsured f g n (H) Insured f (H) n Unnsured f (H) un H C1 C2 C3 H Fgure 3: The effect of work-contngent health nsurance system A4. The mechansm of the work-contngent health nsurance system Because hgher labor endowment (equvalently, labor ncome) usually comes together wth health nsurance coverage, households can consume more when they are nsured. Thus, ther current margnal utlty of consumpton (the LHS of the Euler equaton) s lower than unnsured households (they on average have lower labor endowment). So nsured households have hgher ncentve to save, gven the expected margnal return of assets (the RHS of the Euler equaton) remanng constant. Moreover, because the dfference n nsurance status s now assocated wth a dfference n ncome, nsurance coverage wth hgher ncome further reduces the mportance of the socal welfare system and further ncreases the postve savng effects of health nsurance. Comparng the low-labor-state unnsured household s dstrbuton of H wth the hgh-labor-state nsured household s, the latter not only has a smaller varance (as n fgure 1), but also has a hgher mean. Usng fgure 3 for llustraton, f orgnally unnsured household becomes nsured wthout any change n labor state, the dstrbuton of H wll change from the Unnsured curve to the Insured curve, as shown n fgure 1. However, f household becomes nsured wth a better labor state, the H dstrbuton wll change to the Insured curve and then shft rght to the ginsured curve. The change from Unnsured to Insured s caused by health nsurance, as explaned n fgure 1. The shft from Insured to ginsured s due to the ncrease n labor endowment, whch represents the change from unemployment/part-tme job to full tme and 51

52 good job postons. 63 Hence, we can observe that the ncrease n p from an unnsured state wth low labor endowment to an nsured state wth hgh labor endowment s greater than that ncrease n fgure 1. Suppose that the C s set at C2 n Fgure 3, the ncrease n p ncludng the change n labor endowment s the two shaded area left of lne C2, and the ncrease wthout s the left shaded area alone, whch s the same as n fgure 1. In addton, we can see that the threshold of C (at whch the pdf of nsured H surpasses the pdf of unnsured H)s now hgher than C2. Other effects of health nsurance on the expected return of assets (the RHS of Euler equaton) are smlar to how they work before, and these effects become small when C s set around the threshold. Therefore, we can expect that, under the work-contngent health nsurance system, the nsured households wll hold more assets than the unnsured, and the dfference s larger than n an economy wthout ths health nsurance system. 63 Hgher labor endowment makes H hgher n every state. Thus, ths makes the dstrbuton shft to the rght. 52