Optimal Health Insurance for Multiple Goods and Time Periods


 Barbra Hutchinson
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1 Preliminary draft for review only 0R.P. Ellis, S. Jiang, and W.G.Manning MultipleGood_ dox June 30, 0 Optimal Health Insurane for Multiple Goods and Time Periods Randall P. Ellis a,, Shenyi Jiang b, Willard G. Manning a Department of Eonomis, Boston University, b Hanqing dvaned Institute of Eonomis and Finane, Shool of Finane, Renmin University of China, Beijing, P.R. China, Harris Shool of Publi Poliy Studies, The University of Chiago, knowledgements: We have benefited from omments during presentations at Boston University, Harvard and Renmin University, The University of Chiago, the E annual meeting, and the merian Soiety of Health Eonomists onferene (Duke. We also thank David Bradford, Kate Bundorf, lbert Ma, Tom MGuire, David Meltzer, Joe Newhouse, Kosali Simon, and Frank Sloan for their useful omments. The opinions expressed are those of the authors, and not neessarily those of Boston University, Renmin University, or The University of Chiago.
2 bstrat This paper reexamines the effiienybased arguments for optimal health insurane with multiple treatment goods and multiple time periods. Substitutes and positively orrelated demands shoks aross health are goods have redued optimal patient ost sharing. In a multiperiod model, savings ompliate optimal insurane rules, but positively serially orrelated errors imply lower ost sharing is desirable. Positively orrelated unompensated osts also redue optimal ost sharing on the overed servies. Empirial results using insurane laims data examining inpatient, outpatient, and pharmaeutial spending provide a rationale for overing pharmaeutials and outpatient spending more fully than is implied by stati, one health are good models.  
3 Introdution One of the major themes in health eonomis sine its ineption has been the behavior of patients and providers in the presene of health insurane or sikness funds that over part or all of the ost of health are. The entral eonomi motivation for suh arrangements is that riskaverse individuals an redue their finanial risk by pooling the risks through insurane that effetively shifts funds from the (ex post well individual paying premiums to the (ex post sik individual reeiving reimbursement for health are servies. theoretial and empirial onern has been the adverse effets of moral hazard that arise from the inentives in suh health plans when the marginal ost of an insured servie to the onsumer/patient at the point of servie is less than the soial osts of produing it. To the extent that patients respond to lower outofpoket pries of health are, health insurane will inrease the amount and quality of the are purhased, generating an exess burden from the inreased use. Empirial support for the law of demand applying to health are is substantial from the literature on observational studies, natural experiments, and the randomized experiments suh as the RND Health Insurane Experiment (Newhouse, 98; Newhouse et al., 993; Zweifel and Manning, 000. Muh of the eonomi literature on optimal health insurane fouses on the fundamental tradeoff of risk spreading and appropriate inentives (Cutler and Zekhauser, 000, p Speifially, it examines either the dead weight losses from moral hazard, the tradeoff between moral hazard and the gains from insuring against finanial risk, or the differential overage of multiple goods with varying degrees of risk. Muh of this work employs a oneperiod, one health are good model with unertainty about health states or unertainty about levels of health are expenditure [rrow, 963, 97, 976; Besley, 988; Cutler and Zekhauser, 000; Dardanoni and Wagstaff, 990; Pauly, 968, 974; Spene and Zekhauser, 97; Zekhauser, 970]. Papers thatderive the optimal insurane strutures using this framework have employed variants of the tradeoff between the risk premium (as refleted by the rrowpratt approximation and the deadweight loss from moral hazard (as refleted in Harberger loss or related measures or the ompensating variation (Manning and Marquis, 996. See Feldstein (973, Feldstein and Freedman (977, Buhanan and Keeler (99, Manning and Marquis (996, Newhouse et al. (993, Feldman and Manning (997 for other theoretial and empiriallybased examinations of optimal insurane. Blomqvist (997 extends the theory to nonlinear insurane shedules. lmost all of this literature has been based on either a oneperiod model or a twoperiod model where the onsumer selets the oinsurane before knowing his or her realized state of  3 
4 health, but health are expenditures are hosen onditional on the state of the world that ours. The ommon result in this literature is that one should selet the optimal overage in a plan with a onstant opayment or oinsurane rate suh that the marginal gains from risk redution from a hange in the oinsurane (opayment rate just equal the marginal osts of inreasing moral hazard. The onsensus of the empirial strain of this literature is that optimal levels of ost sharing usually involve neither full insurane (zero outofpoket ost, nor being uninsured. Depending on the formal model approah and the data employed, optimal oinsurane rates range from the perent range (Feldstein and Friedman, 977; Manning and Marquis, 996 down to values that are in the mid 0 perent range or lower, possibly with a dedutible and/or stoploss (Blomqvist, 997; Buhanan et al., 99; Feldman and Dowd, 99; Feldman and Manning, 997; Newhouse et al., 993. Our primary interest in this paper is optimal insuranefor health are in markets where there are two or more health are goods either two or more ontemporaneous health are goods or health are goods in two or more periods.we are aware of only three papers that have onsidered a multigood or multiperiod framework for health are goods.besley (988 provides a multigood extension to this literature in whih demands for health are goods are stohasti, but does not model either omplementarity or substitutability between goods or onsider orrelated shoks in demand. Goldman and Philipson (007 model two goods in one period to show how omplementarity and substitutability of health are servies affets optimal ost sharing in an expeted utility format. Ellis and Manning (007 model the ase of one treatment good and one prevention good to highlight how optimal insurane rules differ for the two types of goods, but inlude only one health demand shok and do not model multiple treatment goods, or orrelations over time. To keep the model tratible, we take a step bakwards from Ellis and Manning and do not onsider preventive are in this analysis.using the same utility struture as in the earlier paper, weexaminehow optimal ost sharing is affeted by the orrelation struture of random shoks affeting demand for health are both aross goods and over time. We also examine how savings and unompensated health losses affet the optimal insurane alulations. The important insight here is that, all other things being equal, health are goods that are positively orrelated should be more generously insured than those that are negatively orrelated or unorrelated. This holds both for ontemporaneously orrelated health are treatment goods and serially orrelated shoks over time: health are treatment goods that have more positive  4 
5 orrelations should have more generous overage. The basi logi is that if the demand for two goods or over two periods are unertain, then their ex ante ombined variane is larger if they are positively orrelated than if there is no orrelation or a negative one. Risk averse individuals will prefer more generous insurane (lower oinsurane rates to redue their finanial risk than if they ignored the orrelation or treated them as independent. One very speifi ase of this is when some aspets of health are are overed whereas others are not; for example onsider time osts of are. In that ase, apositive orrelation between the unompensated and overed loss leads to a redution in the optimal oinsurane rate. Thus, unompensated health losses provide a new rationale for reduing ost sharing for health are treatment goods beause of the positive orrelation in unompensated are and insured are for those health events. nother ase of ontinuing interest is the treatment of aute versus hroni are. Our findings indiate that hroni are servies should have better overage than aute are servies, all other things equal, beause hroni are servies are more highly positively serially orrelated. Following rrow (963, Pauly (968, and Zekhauser (970, we fous on only the demandside while examiningoptimal ost sharing, a topi that has already reeived enormous attention in the literature. We fous on risk aversion, moral hazard, insurane loading osts and unompensated losses without attempting to model other onerns that influene optimal insurane overage: orreting for externalities, suh as those that an our with ommuniable diseases (Hofmann, 007; altruism or publi good arguments for insurane overage (Coate, 995; Rask and Rask, 000 ; distributional onerns (suh as goals of elimination of poverty, or ahieving soial solidarity (ndrulis, 998;tim, 999; Maarse and Paulus, 003; Sin, et al., 003orretions of informational problems (i.e. uninsured onsumers make the wrong deisions (Doherty and Thistle, 996; or insurane that fosters more omplete oordination among health are providers (Duggan, 005; Lihtenberg, 00; and Newhouse, 006. Without denying the relevane of these other fators affeting optimal insurane, we reexamine the effiienybased This finding runs ounter to the ommon experiene that inpatient overage is more generous than that for most other health servies. The more generous overage of inpatient are is motivated by the high variane of inpatient are as well as by the fat that inpatient are is less responsive to ost sharing than other servies (Newhouse et al., 993. s we show below, the details are more ompliated beause the optimal oinsurane also depends on the variability in demand, own and ross prie effets, and (in the ases of multiple periods the disount rate
6 arguments for insurane with multiple goods and periods, and derive new results whih refine our understanding of the value of generous insurane overage from the onsumer s point of view. We first study the optimal insurane overage for health are treatment when there are two health are treatment goods. fter developing a general analytial model (with mathematial derivations in the ppendix whih we all our base ase, we examine a series of speial ases and do omparative statis. key attration of our speifiation is that we are able to solve for the optimal ost share as a losedform solution, and to rereate the results from the basi model that involves one health are treatment good. We also derive new results involving unompensated health are losses, orrelated health are shoks, and ross prie elastiities of demand with multiple goods. Our seond set of analytial results onsiders a twoperiod model in whih health are shoks in one period persist over time due to hroni onditions. In a multiperiod ontext, if a onsumer s savings reat to healthare shoks, then this hanges both the ost of risk as well the optimal ost sharing rates. Positively serially orrelated shoks imply that healthare should be more generously overed (lower ost sharing than when shoks are independent or negatively orrelated aross periods. Our third set of analytial results onsiders the two good, two period model, in whih goods may differ both their ontemporaneous and serial orrelation. This model provides insights into optimal overage for aute versus hroni onditions. The onluding setion of the paper disusses a few empirial results that have a bearing on our analytial findings. We briefly disuss empirial estimates of the variane of three broad sets of servies, and the magnitudes of ontemporaneous and intertemporal orrelations that shed light on the empirial relevane of our findings.. Model assumptions We examine a series of models that involve two health goods within one period and then over two time periods. The individual s utility funtion is defined over health states (or health status and the onsumption of other goods, Y, and a vetor of health servies, i, where the i in P indiates the i th healthare good. In the oneperiod model, onsumers have inome, I, i and fae pries P and P. In the underlying behavioral model, there is a health prodution i Y funtion that transforms health are into health status. For simpliity, we ignore the possibility of death, and assume that the moments of health are shoks do not depend on the level of ost  6 
7 sharing or inome, onsidering only briefly the ase where these variables might also affet the distribution of health shoks, not just onsumer hoies. Following muh of the literature (for example, Cutler and Zekhauser, 000, we examine only health insurane plans with a onstant oinsurane rate 0 for both treatment. We briefly introdue loading osts of insurane, whih are assumed to be a onstant proportion δ of insurane payments. Insurane premiums,, are ompetitively determined and depend on the opayment rates and the demand struture, but do not vary aross individuals. The insurane poliy is thus a pure oinsurane plan with no dedutible, stoploss, or limit on the maximum expenditure or level(s of overed servies. We use the following sequene of steps in our full model.. The insurer hooses the premium and oinsurane rates i for health are treatment.. Nature deides on the onsumer s state of illness as a vetor of random health shoks i that affet the demand for the vetor of health are goods. 3. The onsumer hooses quantities and Y to maximize utility in Period. 4. If a twoperiod model, repeat steps and 3 for Period. The demand for medial are servies has been shown by many empirial studies to be very inome inelasti for generously insured onsumers. For simpliity, we assume that is perfetly inome inelasti. Hene, I 0. While this inome elastiity assumption is strong and unrealisti, it buys us a great deal of simpliity that enables many losed form solutions for ases with multiple health treatment goods. We make a strong assumption about inome elastiities, but weaker assumptions about other parameters below. We avoid onern about orner solutions by further assuming that inome is always suffiient to pay for at least some of all other goods Y after paying for. Utility in every period is separable in health status and the utility of onsumption. orollary of this is that health are shoks do not have any effet on the marginal utility of inome, other than through their effet on medial expenditures. Health shoks affet spending on medial are and hene the marginal utility of inome, but do not diretly affet this marginal utility of inome. This is a ommon theoretial assumption and is also assumed in many empirial studies (e.g., rrow, 963; Zekhauser, 970; Manning and Marquis,
8 . One period model In an earlier version of the paper, we examined more general notation that allowed for an arbitrary number of health are goods, but we find that all of the relevant intuition is obtained using only two health are goods. Demand urves for eah good are assumed to be linear.  BP / P G P / P  B P / P G P / P Y Y Y Y (( For simpliity, we normalize the marginal osts of all goods to be one, and express all pries in terms of the share of this marginal ost paid by the onsumer. Hene PY and P i i, and the ost share is the onsumer prie of the i th health are good i.s in Ellis and Manning (007, we also inorporate the idea that illness may involve other uninsured, unompensated losses. These losses an be of two types: unompensated outof poket osts whih we assume are proportional to overed osts (denoted as i benefit funtion for eah medial servie L and unompensated health shok losses L.The marginal i is assumed to be a linear demand urve with a onstant slope Bi on the prie regardless of the realization of the random health shok i. Stohasti health treatment demand is introdued by letting, where ~ F, with 0 i i E. ssumptions about the variane ovariane vetor of are made below but throughout we assume that the distribution does not depend on the outofpoket prie or inome. This orresponds to the horizontal interept of the demand urves having a mean of when the full onsumer prie i L i is zero. Using this onvention, a single onsumer s demand urve for eah medial servie an be written as + B L. (3 i i i i i i In order to introdue risk aversion, we apply a monotonially inreasing onave funtion V to the indiret utility funtion onsistent with the demand funtion in Equation(. In an earlier version of the paper we onsidered a more general version of our model allowing n goods, but we find that all of the ritial intuition is seen with two goods. Using this notation, we write the one period indiret utility funtion with two treatment good as  8 
9 B( L B ( L J ( L V(, I C, V L ( L ( ( L G ( L ( L where (4 J I, Using the linear demand equation for, the insurer s breakeven ondition for the insurane premium is ( B ( L G ( L ( ( B ( L G ( L where is the administrative loading fator suh that insurane osts proportion more than atuarially fair insurane. In a twoperiod model, we assume that the same premium is harged in both periods. While we have used the somewhat restritive assumptions of linear demand, additive errors, and zero inome effets, this speifiation has two attrative features. The error terms only interat with ost shares in a simple multipliative form. This failitates introduing multiple goods and multiple periods. The linear speifiation also allows us to onsider ross prie elastiities in a natural way. We now turn to the soial planner s problem of hoosing the optimal oinsurane rates when there are two health are treatment goods ( and, and a omposite allothergoods ommodity, Y.We develop the model using a general speifiation, and then derive various ases of interest as speial ases. The optimal ost sharing rates i (5 for health are treatment will maximize the expetation of Equation(4. Taking its partial derivative with respet to and setting equal to zero yields an i equation that haraterizes the soial optimum. Sine this expression will not in general have a simple losed form solution, we take a Taylor series approximation of the partial derivative V I, evaluated around the nonstohasti arguments of the utility funtion. This solution an be written as  9 
10 VI J K VII J K L L E V 0 E B ( L G ( L where J I, K ( L ( L B ( L / B ( L / G ( L ( L ( B ( L G ( L (, ( B ( L G ( L ( B B BL G G GL, (6 Defining the absolute risk aversion parameter R II, E, E E, and using E E 0 V V I, we show in the ppendix that this result an be rearranged to obtain first order onditions haraterizing the optimal insurane rates follows. ( B B BL G G G L B( L G R L L and 0 ( B B BL G G G L B( L G R L L 0 (7 (8(9 as We onsider a variety of ways of interpreting these two equations below. In broad terms the first line of eah equation gives the marginal ost of inreasing the ost share, while the seond line, whih involves R gives the marginal benefit in terms of redued ost of risk. 3. One health are good, base ase L L B G 0 We first examine what we all the Base Case whih is the ase of a single good in a single period with no unompensated osts, and no loading fee. In terms of the notation of our model, this orresponds to L L B G 0. This yields the wellknown result  0 
11 from the literature haraterizing the seond best optimal insurane for onstant oinsurane rate plans when there is a simple tradeoff between moral hazard and the ost of risk: B R 0 where B 0, and the first term (the marginal osts due to moral hazard are inreasing in the prie response B. The gains from risk pooling are inreasing in the variane in health are demand. Solving for the optimal oinsurane rate yields Equation(0, where the optimal base ase oinsurane rate variane in demand. is inreasing in the prie response B and dereasing in the underlying B BaseCase B R Intuitively, there is a tradeoff in hanging to inrease risk bearing as the patient pays more of the prie, and reduing moral hazard. The optimal oinsurane rate is the ratio of the loss from moral hazard to the net loss from hanging risk bearing plus moral hazard. If the demand is perfetly inelasti ( B 0 (0, then the optimal oinsurane rate is zero if there is any risk( 0 at all. If there is no variane or the onsumer is risk neutral ( R 0, then the optimal oinsurane rate BaseCase should be. The optimal oinsurane rate lies between 0 and, inlusive: One health are good, with insurane loading ( 0 Starting with the base ase, we next relax the assumption of no insurane loading fator. The new expression for the optimal oinsurane rate beomes B B B R B ( s long as the insurane loading fator δ is not prohibitively large, then 0, and 0. Moreover, the mean expeted level of spending (with a marginal outofpoket prie of zero,, enters in the numerator suh that as osts go up, it is desirable to inrease the ost share to redue the ineffiieny due to the insurane loading fator ( δ> 0. Having a positive loading fee inreases the osts of moral hazard relative to the gains from reduing risk. We leave the loading fee rate at zero for the remainder of our results.  
12 3.3 dding unompensated healthrelated losses in the one good ase L 0 Inorporating unompensated health loss so that L 0 also affets optimal ost sharing for overed treatment goods, inreasing the overage (dereasing the ostsharing desired that we found in our analysis of prevention versus treatment (Ellis and Manning, 007. The expression for the optimal ost share on health good beomes R L BaseCase B R ( Sine all of the terms after the minus sign are positive, it is straightforward to see that the optimal oinsurane rate is dereasing in the size of the unompensated loss L. With unompensated losses, it is also possible for to be negative or to reah a orner solution where 0 for either large L or small B. Equation( provides an effiienybased rationale for why full insurane an be seond best optimal: the absene of omplete insurane markets to fully transfer inome into highost ill health states of the world means that oinsurane rates are set at or loser to zero than they would have been if the alternative insurane markets were omplete and onsumers were able to insure against all health are losses. There are many health servies and onditions whih have substantial unompensated health are related losses. This is partiularly true in developing ountries where disability and unemployment insurane is rare and produtivity losses from ill health are often large. Wagstaff (007 provides doumentation of the large magnitudes of inome losses from illness in Vietnam. Thus, inomplete insurane markets provide a rationale for more generous insurane overage of health are treatment, even when welfare losses due to moral hazard and insurane loading may be important 3.4 Multiple health are treatment goods 0, 0, 0, B 0, L L 0 n important motivation in this paper for modeling two rather than one health are treatment goods is to be able to examine the role of ross prie elastiities and orrelated health demands. For ease of exposition, we now assume there are no unompensated health losses and no insurane loading fator, 0, but explore the general ase for demand parameters with two goods. In ontrast to the situation with two goods in a one period but no unompensated losses, there are now shifts in both the marginal osts from the dead weight loss of, and shifts in the  
13 risk bearing. If L L 0, then (7 and (9 above an be solved for the optimal ost share for health good and simplified to: G R Base Case BR BR (3 with a similar expression for the seond health are good. Sine all of the terms multiplying and G are nonnegative, the negative sign means that the first order effet of inreasing either term is to redue the optimal ost share relative to the base ase. Sine G B R 0, reduing reinfores the effet that G 0. Hene, when two health servies beome stronger gross substitutes in the sense that G is inreased, then both servies should have lower ost shares. This finding repliates the finding of Goldman and Philipson (007 that as goods beome stronger omplements ( G 0 they should have higher ost sharing. The impliations of the ovariane between health goods on optimal insurane are more omplex. Beause of the seond order effet of on we annot unambiguously sign for all possible values of G and. We an evaluate this term for ertain speial ases. One speial ase is that the partial derivative an be signed as negative for the limiting ase where approahes zero. nother set of speial ases orrespond to when G takes on ertain values, whih is easiest to see graphially. Figures a show the effet of hanges in the optimal oinsurane rate for three ranges of G as the ovariane term inreases from zero to a positive level, where in eah figure the solid line orresponds to the equations (7 and(9 with 0. The figures examine for the three possible ases for G aording to whether goods and are prie neutralg 0, omplementsg 0 or substitutesg 0. In the first two ases, the optimal oinsurane rate for both health are goods will tend to fall as goods ovariane inreases. In eah ase, as the ovariane beomes more positive, then Line (Equation(9 pivots lokwise around the origin, while Line (Equation(7 rotates ounterlokwise, ensuring that both oinsurane rates fall
14 But when the two goods are substitutesg 0, it is possible for one line to rotate further than the other. This depends on the magnitudes of Bi andg. good with a higher prie response B i or the greater variane i will rotate less. In this situation, it is possible for one good to have its oinsurane rate fall while the other stays the same or inreases. Figure illustrates the ase where the two goods are substitutes, G 0, and 0 but 0. The opposite omparative stati ould our if Line swings substantially and Line moves a little, with optimal oinsurane rate falling for health are good and possible for good as the health are goods beome more positively orrelated. It is straightforward to show that ertain elements of the onventional results for insurane still apply, even if the overall level depends on omplementarity in demand or ovariane information. It is shown in the appendix that < 0, < 0, > 0, and < 0. B B 3. Multiple period model Our framework an also address the ase of multiple periods with orrelated health are demands. We fous here on the ase where there are no unompensated osts of illness, so that L 0, there are only two periods (indexed by and, and one health treatment good in eah i period where i is health are demand in period i 3. We only allow one ost share, and hene, and fous on the ase where demand is the same in eah period exept for the health shoks i,  B +. i i We fous on the ase where the parameters and prie struture are onstant over time: I I I,, and. To allow for the possibility that the health are demands in the two periods are orrelated, we assume that the seond period health shok is, where In a dynami model, we need to introdue savings, whih we assume to be optimally hosen. In our twoperiod model, net saving is deided in period after is known, and spent 3 Later in the paper and in the ppendix, we allow for more time periods
15 entirely in period.in general, the optimal level of savings will depend on the all of the parameters of the model. Of speial interest is that savings will depend on the ost share and the first period health shok, S (, with S / 0. In the ppendix, we show that the objetive funtion to be maximized through the hoie of an be written as follows V J K S(, L ( EV E, E V J K ( r S(, ( L where J I B ( K ( L B( (4 Exept for the savings funtion and the introdution of disounting,, this formulation is very similar in struture to what was used above for the ase with one period with multiple states of the world. The solution for the optimal hoie of is derived in the ppendix. We make the following three further assumptions in deriving our solution: Savings is optimal so that for all, V... ( r E V... rate. I I where r is the interest The utility funtion an be approximated using a seond order approximation with onstant relative risk aversion. Consumers an earn a return on savings ( r that is the inverse to their disount rate, so that ( r. In the ase of the quadrati utility funtion that we have used in our analysis, the optimal saving is approximated by S (, S s (shown in ppendix, where the expeted (ex ante savings ( r are S and the optimal savings (ex post are redued by the proportion R ( r ( r s multiplied by the outofpoket health payments and unompensated osts in time ( r  5 
16 period.the term s is the marginal propensity to save. In partiular, if ( r (as assumed, then S(, is redued to a simple funtional form S. 4 Under these assumptions, r when n the optimal ost share is B B. (5 ( BR s BR ( r( This result is very similar to the Base Case equation (0 for the ase of a single period, one health are good without any unompensated lossesl 0. Equation (9 differs from the ( base ase by the addition of a new savingsrelated term ( r( in the denominator. This term is a funtion of the orrelation oeffiient between period and period health shoks,,the interest rate r and the onsumer disount rate.note that is nondereasing in, r, and, and that is dereasing in. s before, we interpret the optimal ost sharing result for a variety of speial ases. First, onsider the ase where the marginal propensity to save s 0, so that savings does not respond to health shoks. In this ase,, and the one period model results remain orret even with two periods. The onsumer must absorb all health shoks fully in the first period, so there is no differene between the stati and dynami hoies of. Seond, onsider the ase where the period and period shoks are perfetly orrelated, so that.one again and the one period model results hold. lthough savings is possible, there are none beause the onsumer knows exatly what the shok will be in period. 4 In the more general ase of any n periods, where the autoorrelation terms are allowed to have an arbitrary pattern rather than a first order autoorrelation, we show in the ppendix that the optimal saving funtion still has a losed form S n i ( r ( n i i P n i i ( r
17 There is no diversifiation aross periods in the burden of health shoks. In this limiting ase, insurane should be the same as with no savings. Third, onsider the ase where health shoks are unorrelated over time, so that 0. The disount rate is a number lose to one, and it is onvenient to onsider the ase where and r=0 so that there is no interest or disounting. The plausible result in this two period model is that s will be lose to one half, and half of the burden of a health shok is born in period and half is deferred to period.sine the ost of risk goes up with the square of the deviation from ertainty, the savings redues riskiness in the first period to onequarter of the one period value and hene the ost of first period risk (proportional to the variane is redued to one quarter of the one period model results. Sine this burden is shared between two periods, the net redution in risk is by one half of the variane. The reason that only delines to 0.75 is that in a two period model there is no opportunity to redue the burden of shoks in the seond period. So while savings an redue the burden of first period shoks to one quarter of their unertainty ost, savings annot redue the burden of seond period shoks. This result with no disounting or interest rates losely approximates the result with disounting, sine the two terms ( r( will be approximately 4 if onsumers use the market interest rate for disounting. 4. Twoperiods with twogoods So far, we have addressed the ases of multiple treatment goods and multiple periods separately. It is also important to onsider the ase of multiple treatment goods with more than one period beause this allows us to address issues suh as the differential overage of aute and hroni health are. We fous on the ase where there are two goods and two periods. The onsumption of two goods in two periods is indexed by,,, with subsripts indiating goods, and supersripts time periods. Following the notation in the onegood multiple period ase, we assume, i,, I I I, and. i i i To make this more onrete, we onsider as a medial treatment good for an aute ondition and as a medial treatment good for a hroni ondition. Then, we would expet that the amount of good one in the first period is not orrelated with its own future amount, while the amounts of good two in both periods are positively orrelated, where, 0<.We  7 
18 assume loading fees and other unompensated osts are zero: L L 0. We also assume E( E( E( E( 0 and that VR The objetive funtion and first order onditions haraterizing the optimum are shown in the ppendix. The main new result of interest relates to how onsumers optimally hange savings in response to aute rather than hroni onditions. If disounting exatly equals +r, so that ( r. Then, we show in the appendix that the optimal saving funtion is approximated by S (,,,, r r sine s r and s. The expeted (ex ante savings S are the same with the twoperiod r onegood ase. The marginal propensity to save s is lower than s beause good is hroni are and patients who reeive a large shok for hroni are antiipate a orrelated shok the following period and hene do not adjust their savings as muh as for an aute are shok. Results for the general ase are presented in the appendix. For the speial ase where demand shoks have zero ovariane in the eah period, and good are neither omplements nor substitutes, so that 0, G 0, the twogoods twoperiods ases redue to onegood twoperiods ase. nd the optimal ost shares are approximated by B BR r B B R r The optimal ost shares are onsistent to equation (5 with 0 for the aute are. So the orrelation of spending for hroni are in two periods does not affet the optimal ost share of the aute ase. Thus, our findings indiate that hroni are should have better overage than aute, all  8 
19 other things equal, beause hroni are is positively serially orrelated, while aute are is unorrelated over time. 5. Theoretial Summary We have extended the theoretial literature on effiienybased models of optimal insurane to address issues that arise from orrelated soures of unertainty, whether the soure of the orrelation is due to orrelated demands aross different health are goods at a point in time, orrelated demands over time, or the orrelated losses that arise from unompensated losses that aompany the losses overed by the insurane plan. By using a quadrati indiret utility funtion and, hene, a linear demand speifiation with zero inome effets on the demand for health are treatment, we have been able to derive losed form expressions haraterizing optimal ost sharing on health are treatment when there are multiple health are goods or where there are multiple time periods. Table summarizes the omparative statis findings in this paper for the various parameters onsidered for health are treatment goods and multiple time periods. In some of the ases that we have onsidered, we an only sign the effets of a parameter on optimal ost shares for ertain parameter values. These ases have the omparative stati results in parentheses. The parameters in the first four rows of Table reaffirm onventional results found in the previous literature, while the terms at the bottom reflet our new results that extend the previous literature. It is well established that optimal ost sharing on health are treatment should be higher as demand beomes more elasti, onsumers beome less risk averse, or the variane of spending dereases. Our findings are onsistent with the findings from Besley (988 and others. Our theoretial findings also are onsistent with those of Goldman and Philipson (007 on omplements and substitutes that all other things equal, ost sharing should be higher for omplements than substitutes. Our new finding is that positively orrelated losses aross health are goods or over time should lead to more generous overage (lower ostsharing than unorrelated or negatively orrelated losses
20 6. Empirial Relevane. In this setion, we briefly examine the empirial magnitudes of two innovation of our model: the role of ontemporaneous orrelations aross multiple goods and of autoorrelations over time. We use data from the ThomsonReuters MarketSan data base from the period on a population of nonelderly (age <65 enrollees in employmentbased ommerial plans. We have restrited the sample to FFS, HMO, PPO and POS plans whih overed outpatient pharmay servies, in addition to outpatient physiian and inpatient servies. We inluded only those individuals who were ontinuously enrolled for the full five year period; this yields a sample of,335,448 individuals. Besides it size, these data have two major advantages. The first is that the enrollees are followed for several years, allowing us to study orrelations by type of health are over time. Seond, all of the enrollees had pharmay overage, thus allowing us to ontrast pharmay expenditure patterns with those of both inpatient and outpatient are. Table summarizes key means, standard deviations and orrelations from our five year sample, deomposed into three broad types of servies inpatient (faility, not inpatient physiian payments, all outpatient servies, and pharmay servies. 5 The first two olumns in the top half of Table reaffirms that inpatient spending, while not the largest expeted ost, is by far the most risky in a one year framework. Table also shows the autoorrelation oeffiients for spending for eah of the three servies differ meaningfully, with pharmaeutial spending having the highest autoorrelations and inpatient spending the lowest. The autoorrelations also reveal that spending is muh slower to return to normal levels following a health shok than a simple autoregressive R( pattern would indiate. Chroni onditions obviously explain this pattern. The larger orrelations for pharmay than outpatient are and for outpatient than inpatient suggest a larger orretion for pharmay than outpatient from the usual results for a one period model. The final olumn summarizes the impliations of using multiple years of spending to alulate finanial risk by presenting the standard deviation of five year sums of spending rather than one year spending. 5 Some researhers may expet a higher proportion of spending to be on inpatient are. The MEDSTT ommerial laims do not inlude Mediaid or Mediare enrollees, who have higher hospitalization rates. MEDSTT laims haves 4% of all overed harges for inpatient are in 004. Our sampling frame of using only people with five onseutive years of insurane overage has somewhat lower proportion of perent of spending in inpatient servies
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