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: mnchung@ucla.edu. 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