HEALTH ECONOMICS Health Econ. 25: 357–371 (2016) Published online 16 January 2015 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/hec.3142
HEALTH INSURANCE, HEALTH SAVINGS ACCOUNTS AND HEALTHCARE UTILIZATION RICHARD PETERa , SEBASTIAN SOIKAa and PETRA STEINORTHb a Institute b School
for Risk Management and Insurance, Ludwig-Maximilians-Universität München, Germany
of Risk Management, Insurance and Actuarial Science, St. John’s University New York, NY, USA
SUMMARY Assuming symmetric information, we show that a high-deductible health plan (HDHP) combined with a tax-favored health savings account (HSA) induces more savings and less treatment compared with a full coverage plan under reasonable risk preferences. Furthermore, a higher tax subsidy increases savings in any case but decreases medical utilization if and only if treatment expenses are above the deductible. A larger deductible increases savings but does not necessarily decrease healthcare utilization. Whether an HDHP/HSA combination is preferred over a full coverage contract depends on absolute risk aversion. A higher tax advantage increases the attractiveness of an HDHP/HSA combination, whereas the effects of changes in the deductible are ambiguous. The paper shows that a potential regulator needs to carefully set the size of the deductible as only in a certain corridor of the probability of sickness, its effect on aggregate healthcare costs are unambiguously favorable. Copyright © 2015 John Wiley & Sons, Ltd. Received 18 March 2014; Revised 15 October 2014; Accepted 7 December 2014 JEL classification:
D14, H24, H31, I11, I12
KEY WORDS: health risk; healthcare utilization; health insurance; health savings accounts; medical savings; high-deductible health plans; consumer driven health care
1. INTRODUCTION Medical costs and medical inflation are a major concern to any society aiming to provide comprehensive medical care to its members. According to the World Bank data, the portion of the gross domestic product used for medical costs increased from 13.8% in 1995 to 17.9% in 2011 in the US, from 6.8% to 9.3% in the UK and from 8.3% to 12.0% in the Netherlands.1 Several culprits for this increase have been discussed among which an increasing healthcare utilization has been mentioned repeatedly. In our paper, we investigate how health plans influence medical utilization under symmetric information. We are particularly interested to see whether a highdeductible health plan (HDHP) combined with a health savings account (HSA) can actually create incentives to reduce the consumption of medical services and therefore reduce medical costs. In our paper, we confirm the previous result that the existence of tax-preferred savings for medical costs implies a first best solution with a positive deductible even at actuarially fair insurance prices (Steinorth, 2011). Of course, this changes the demand for medical services and medical savings. Accordingly, it seems natural to
1
Correspondence to: Institute for Risk Management and Insurance, Ludwig-Maximilians-Universität München, Germany. E-mail: [email protected]
See http://data.worldbank.org/indicator/SH.XPD.TOTL.ZS/countries/1W?display=default.
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compare healthcare utilization between a traditional full coverage health plan and an HDHP/HSA combination under symmetric information. We investigate how HSAs affect the trade-off between cost efficiency and medical service provision under symmetric information. It is very important to understand the impact of the given choice of coverage on the demand for medical treatment in order to set the right incentives when designing plans.2 We believe that carefully analyzing incentives when consumers’ decisions are observable is a necessary step before potential information problems as adverse selection and moral hazard are tackled.3 Health savings accounts have been introduced in the US in 2004 and have been attracting a growing number of insured since then covering 15.5 million US Americans as of January 2013.4 A qualifying HDHP allows the insured to open up an HSA where he or she can contribute savings for medical purposes on a tax-free basis within certain limits. The aim of introducing these accounts was to create incentives for individuals to save more for medical purposes and to reduce medical utilization. HDHPs increase consumers’ financial responsibilities, which can reduce medical utilization in several ways. Individuals may forgo medical treatment, which is not absolutely necessary5 as well as shop around for better prizes for the same procedures. In our paper, we focus on the first aspect and leave the second for future research. In addition to the US, forms of HSAs have also been introduced in South Africa, Canada, Singapore and China. Our model can also be applied to medical savings accounts (MSAs) in South Africa, while the accounts in other countries differ in their setup and consequently may provide different incentives.6 A lot of the literature on HSAs focuses on the take up after their introduction (Feldman et al., 2005; Glied and Remler, 2005; Cardon and Showalter, 2008) and discusses potential adverse selection (Cardon and Showalter, 2007) and favorable selection issues (McDevitt et al., 2014). It has also been investigated how much more cost-sharing these health plans will bring (Remler and Glied, 2006; Cardon and Showalter, 2008). Borah et al. (2011) investigate empirically whether HSAs reduce medical expenditures. Steinorth (2011) shows that in the context of ex-ante moral hazard (primary prevention), HSAs do not necessarily accomplish the political goals as their introduction does not decrease ex-ante moral hazard and increase savings at the same time. In addition, Zabinski et al. (1999), Pauly and Herring (2000) and Greene et al. (2006) investigate the related concept of MSAs, while Cardon and Showalter (2001, 2003), Hamilton and Marton (2008) and Cardon (2012) analyze flexible spendings accounts, which require individuals to use their balance at the end of the year. Yet, there is relatively little theoretical literature on medical utilization in the context of HSAs. In general, medical utilization has been studied from a variety of different angles in the theoretical literature before. Pauker and Kassirer (1975) are the first to analyze treatment decisions within an expected utility paradigm. Under diagnostic uncertainty, the probability of the patient having the disease should exceed a critical threshold for treatment to be applied. Bleichrodt et al. (2003) analyze patients’ willingness to pay for health improvements when comorbidities matter, and Berger et al. (2013) study the impact of ambiguity on treatment decisions. In contrast to these papers, we focus on how financial arrangements determine how much 2
Incentives to utilize medical services cautiously are expected to work in favor of the insured because of lower premiums. If, however, these incentives are misaligned, patient satisfaction may suffer, and insured may want to switch plans. The rise and later fall of health maintenance organizations (HMOs) in the US in the 1980s and 1990s illustrates this trade-off between cost control and the provision of medical services nicely. For a more detailed discussion on this, please refer to Mechanic (2004) and Coombs (2005). In the beginning, HMOs were very successful because of competitive insurance pricing. But, over time, many insured shied away from HMOs, where one of the major reasons was that the insured felt that they did not receive sufficient treatment. 3 These two phenomena are often mentioned in the discusion of HDHP/HSAs. 4 This constitutes an increase of 15% since 2012, see the AHIP 2013 HSA census available at http://www.ahip.org/HSA2013/. 5 Yet, this may also negatively impact health outcomes. 6 Medical savings accounts are very popular in South Africa where about half of the private health insurance contracts include one. A highdeductible insurance plan is combined with an MSA where individuals and employers can contribute into. The main difference to the US concept, however, is that the deductible does only apply to outpatient services. This is consistent with our model when considering outpatient services only. In Canada, individuals can open a health spending account regardless of their initial health plan, and they can also purchase supplemental health insurance with the account balance. Medisave accounts in Singapore are mandatory, and income ranges determine the out-of-pocket expenses under the public healthcare system, which can be covered from the accounts. As modeled in our paper, individuals have to cover a nonmarginal part of their health expenses themselves, they are, however, not free to choose how much they want to save in their accounts. MSAs in China also come with a catastrophic health insurance plan as modeled in our paper, yet the level of savings is set by regulation as in Singapore. Copyright © 2015 John Wiley & Sons, Ltd.
Health Econ. 25: 357–371 (2016) DOI: 10.1002/hec
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treatment is optimal for the consumer. The empirical literature on treatment choices is huge too. The seminal paper by Manning et al. (1987) investigates the impact of deductible choice on healthcare utilization in the RAND Health Insurance Experiment. Numerous papers have since investigated the impact of plan design changes on the demand for medical services in different countries around the globe, see Mello et al. (2002), Gerfin and Schellhorn (2006), Sarma and Simpson (2006) and Zhong (2011) just to name a few. Our paper adds to the existing literature by adding HSAs to the utilization literature. We are particularly interested in seeing how HSAs impact ex-post medical expenses and whether they ensure higher savings for individuals at the same time. Our main findings are that the tax advantage within an HSA seems to be the most effective parameter to create unambiguous and well-directed incentives. It implies more medical savings and increases the favorability of an HDHP/HSA combination under relatively mild conditions. Manipulating the deductible does not have such straightforward effects. The incentives are sensitive to the probability of sickness, that is, the individual’s risk type. Utilization only decreases with a higher deductible when the probability of sickness is sufficiently high, while a lower deductible does not automatically imply that an HDHP/HSA combination becomes more favorable over a traditional coverage plan. After the introduction, the paper structure is as follows: We first introduce a general model of medical care utilization and savings. In the third section, we discuss the impact of HSAs. We also provide comparative statics with respect to the deductible and the tax favor. In the fourth section, we study consumer expected utility under both alternatives to finance medical care and develop a sufficient condition for the superiority of the HDHP/HSA combination. Furthermore, we develop comparative statics for this condition. The paper ends with some concluding remarks and public policy recommendations. 2. THE MODEL SETUP In this section, we introduce our model to investigate savings and treatment decisions as related to different health plans. We consider two points in time, t1 and t2 , now and then. At t1 , individuals are young and healthy and earn w1 on the labor market. At t2 , individuals are older and might fall ill. They receive earnings w2 . If individuals become ill when old, treatment is necessary. Illness occurs with probability p, and treatment comes at a cost of T .7 Individuals derive utility from consumption and from health at both points in time. Following Kifmann (2001), we assume separability across points in time and between financial and nonfinancial consumption.8 Hence, at t1 , utility derived from consumption c1 and health h1 is given by u.c1 / C kh1 , where u is a twice continuously differentiable utility function that is increasing and concave (u0 > 0; u00 < 0) and k measures the importance of utility from health compared with consumption utility. As individuals are assumed to be healthy at t1 , we normalize h1 to 0. In the second period, utility derived from consumption c2 and health h2 is given by v.c2 / C kh2 . Again, v is a twice continuously differentiable utility function that is increasing and concave (v 0 > 0; v 00 < 0). Note that preferences of financial consumption at t1 and t2 need not be identical.9 As the aim of the paper is to focus on saving and treatment decisions and how they depend on different forms of financing medical consumption, individuals have to determine how much wealth to transfer from t1 to t2 , that is, they have to decide about their optimal savings denoted by s. We normalize the interest rate to 0. Furthermore, given individuals fall ill, they have to determine how much treatment T to consume. Treatment is productive in the sense that the health state when ill is increasing in the level of treatment. Yet, treatment
7
Our modeling of disease is closely related to Cardon and Showalter (2001), Cardon (2010) and Steinorth (2011). The question how changes in health status affect marginal utility of consumption is still not completely resolved. Viscusi and Evans (1990) use job injuries to infer that marginal utility drops because of decreased health, whereas Evans and Viscusi (1991) find that decreased health corresponds to a drop in income and does not affect the structural form of consumption utility. 9 Our specification includes the discounted expected utility model, in which consumption utility at t2 is given by ıu.c2 / with ı 2 .0; 1 being the rate of pure preference for the present. 8
Copyright © 2015 John Wiley & Sons, Ltd.
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expenditures incur. Formally, we assume the healthcare production function h.T / to be twice continuously differentiable with h0 > 0 and h00 < 0. We already mentioned that the focus in our paper is on the effects of plan design on how much treatment and savings will be demanded. To this end, we assume the per unit cost of treatment and the productivity to be exogenous to the consumer. For instance, technologies and prices could be uniform in the market meaning that they do not differ among providers. An alternative way to reduce healthcare cost would be the use of cost sharing to provide incentives to the consumer to shop for prices. This is not covered in our model. Using the notation developed earlier, the individual’s expected utility without insurance writes as V 0 D u.w1 s/ C pŒv.w2 C s T / C kh.T / C .1 p/Œv.w2 C s/ C kh2 :
(1)
The optimal solution .s 0 ; T 0 / for the endogenous saving and treatment decision is determined via the first-order conditions10 Vs0 D u0 .w1 s/ C pv 0 .w2 C s T / C .1 p/v 0 .w2 C s/ D 0; VT0 D pv 0 .w2 C s T / C pkh0 .T / D 0:
(2)
The first equation is the usual consumption smoothing condition over the life cycle, that is, (expected) marginal consumption utility across points in time needs to be equalized, whereas the second equation describes the trade-off between costs incurred by consuming treatment and health benefits resulting from it.11 We can see that the demand for treatment depends on several factors: the production technology for health (h0 ), the relative importance of utility from health and consumption (k) and the cost of treatment (v 0 ). Observe that without insurance the probability of sickness is irrelevant for treatment consumption because treatment is only purchased when ill. With insurance, it impacts the premium calculation. If individuals take up health insurance with a deductible, D 0, this yields expected utility of V D Du .w1 s p max.T D; 0// C pŒv.w2 C s min.D; T // C kh.T / C .1 p/Œv.w2 C s/ C kh2 :
(4)
In t2 , the individual incurs the minimum of his or her treatment cost T and the deductible. Therefore, the corresponding fair insurance premium in t1 is given by the portion of the treatment cost that exceeds the deductible times the probability of sickness.12 This formulation contains the case with treatment exceeding the deductible and treatment falling below it. In both cases, the objective function is concave, and the optimal s and T can be characterized via the respective first-order conditions. Let us extend a classical result to our case. Proofs are in the appendix. Remark 1 If insurance is actuarially fair, full insurance .D D 0/ is optimal.
Subscripts of model parameters denote partial derivatives, that is, Vs0 is shorthand for @V 0 =@s and VT0 is shorthand for @V 0 =@T . This notation is used throughout the paper. 11 0 Note that V 0 is concave, because Vss < 0 and the determinant of the Hessian is positive, 10
0 0 2 00 00 VT0 T .VsT / D .u00 C .1 p/vnt /pvt00 C .u00 C pvt00 C .1 p/vnt /pkh00 > 0: Vss
(3)
Subscript t denotes consumption with treatment and nt consumption without treatment. As a result, .s 0 ; T 0 / is a maximizer of expected intertemporal utility over consumption and health. 12 We model the insurance premium p max.T D; 0/ to be paid only in the first period. As individuals save endogenously, they can smooth their income according to their preferences. Therefore, it makes no difference if the insurance premium is paid only in the first period, only in the second period or is split over both periods. This holds true for all variations of the model presented in this paper. Copyright © 2015 John Wiley & Sons, Ltd.
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This is reminiscent of a result obtained by Dionne and Eeckhoudt (1984) who show that in a two-period model with nonseparable preferences and endogenous consumption, financial risks should be fully insured if and only if the premium is actuarially fair.13 This is known as Mossin’s Theorem (1968), which carries over to our setup with a health risk. Hence, the resulting expected utility with the optimal level of coverage D D 0, that is, full coverage, is given by V F C D u.w1 s pT / C v.w2 C s/ C pkh.T / C .1 p/kh2 ;
(5)
so that the financial risk is completely eliminated. We use superscript F C to denote this situation with a standard full coverage health insurance contract. The optimal saving and treatment decision .s F C ; T F C / under full coverage are characterized by the first-order conditions VsF C D u0 .w1 s pT / C pv 0 .w2 C s/ C .1 p/v 0 .w2 C s/ D 0; VTF C D pu0 .w1 s pT / C pkh0 .T / D 0:
(6)
Note that V F C is concave, which can be shown analogously to Footnote 11. Next, we compare saving and treatment decisions when uninsured to a situation with full insurance coverage. To this end, we need to compare the endogenously determined values s 0 and T 0 with s F C and T F C , which are also endogenous. As we will make frequent use of the technical procedure to do that, we state it in the following Lemma. Lemma 1 2 Let f W R2 ! R be a concave function in the variables .x; y/, that is, fxx < 0 and fxx fyy fxy > 0, which is maximal at .x ; y /. Let ˛ 2 R be a value we want to compare x with. Then, x > ˛ if and only if fx .˛; y/ O > 0 where yO is the value that maximizes f .˛; y/. This approach is due to Gollier (2001, p. 151) and has recently been applied in Hofmann and Peter (2014) to study prevention and saving decisions. For our problem, results are summarized in the following proposition. Proposition 1 A standard full coverage health insurance contract leads to more treatment and less savings than in the absence of insurance. Figure 1 graphically illustrates Proposition 1. It demonstrates that the existence of a health insurance contract implies that individuals utilize more medical treatment. This is in line with empirical observations (Kondo and Shigeoka, 2013). As the health insurance company pays the treatment cost, the individual has to accumulate less savings. Furthermore, we know from earlier that health insurance coverage is utility enhancing, and the model predicts that realized health status should be uniformly larger for health insured individuals than for individuals without coverage everything else equal. This is due to the fact that T F C > T 0 results in h.T F C / > h.T 0 /. Note that we assume individual treatment decisions to be observable by the insurance company so there is no ex-post moral hazard. This is to focus on the redistributive effects of different health plans. Health insurance stipulates more treatment which is, however, not to be confused with overutilization as discussed by Pauly (1974). In our model, individuals pay the fair premium for the medical services they demand later in life. The effect observed here can be attributed to the fact that redistribution between different states of the world and different points in time is beneficial for a risk-averse individual. In the absence of insurance, saving is the only instrument to redistribute between today and tomorrow. Actuarially fair insurance coverage is a second way to optimize an endowed consumption stream by redistributing from today to a specific future state. Hence, the set
13
Menegatti and Rebessi (2011) confirm this result in a two-period prevention model with separable preferences.
Copyright © 2015 John Wiley & Sons, Ltd.
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Figure 1. Relationship between .s 0 ; T 0 / and .s F C ; T F C /
of consumption possibilities is enlarged, which increases expected utility and gives more freedom to consume additional treatment. 3. INTRODUCTION OF HEALTH SAVINGS ACCOUNTS Let us now investigate how individual saving and treatment decisions are altered when HSAs are introduced. HSAs are only available in combination with an HDHP. HSAs benefit from two preferential tax treatments. Contributions are tax exempt and will remain tax exempt if used for defined medical services. Furthermore, interest earned in the account is also tax exempt. All withdrawals that are not used for qualified medical services are considered nonqualified withdrawals. Individuals need to pay income taxes on these withdrawals, and there is also a 20% tax penalty for individuals younger than 65 years in order to offset previous tax favors and interest earned on saved tax dollars. Because of the tax benefits, we model that individuals will put their savings for medical purposes into the HSA, which we denote by . The preferential tax treatment is modeled in a way that, when savings are used for treatment in t2 , the individual receives .1 C / from their HSA. This seems a little different than the original tax treatment at first glance. As everything else is after tax, we assume that .1C/ are the combined tax benefits of HSAs, which we express as a form of interest on after tax dollars.14 If individuals remain healthy, they can withdraw the money from their HSA but do not receive any tax benefits. In this sense, we implicitly assume that the tax penalty exactly removes any tax benefits for unqualified withdrawals. Consequently, the amount withdrawn is as if the money was saved outside the HSA because we set the interest rate to 0.15 We also assume that the individual’s savings for medical services plus the tax benefit from the HSA do not exceed the deductible, .1 C / D, which seems reasonable for practical purposes. Our approach follows Steinorth (2011). Without an HSA, the individual would still choose D D 0, which yields expected utility V F C as in (5) with the optimal solution .s F C ; T F C / for saving and treatment. Using superscript HSA to denote the situation for 14 15
This interpretation allows us to draw on the concept of partial risk aversion later. The health savings account balance bears tax exempt interest, and the individuals also receive interest on the part of the tax exempt contributions, which would have usually been subject to income tax. Assuming equal marginal tax rates upon contribution and nonqualified withdrawal, there would still be a tax advantage because of accrued tax-free interest. The penalty aims at correcting this. Yet, it still depends on the individual situation whether unqualified withdrawals are beneficial or not from a tax perspective as marginal tax rates upon contribution and withdrawal, the timing of contributions and withdrawals in the health savings account and the interest rate environment matter. As all these factors are not of first concern to our study, we decided to assume that the penalty exactly removes the previous tax advantage for nonqualified withdrawals.
Copyright © 2015 John Wiley & Sons, Ltd.
Health Econ. 25: 357–371 (2016) DOI: 10.1002/hec
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an HDHP/HSA combination, the insured have expected utility of V HSA D u.w1F C p max.T D; T F C // C p v.w2F C C .1 C / min.D; T C T F C // Ckh.T C T F C / C .1 p/Œv.w2F C C / C kh2 ;
(7)
with w1F C WD w1 s F C pT F C , w2F C WD w2 C s F C and T WD T T F C . Note that our formulation of the objective function contains the case in which treatment exceeds the deductible as well as the case in which it does not. Similar to V D as in (4), the individual incurs the minimum of his or her treatment cost, T C T F C and the deductible, D, at t2 . The corresponding fair insurance premium in t1 is again given by the portion of the treatment cost that exceed the deductible times the probability of sickness.16 In this sense, we investigate the change T of treatment consumption induced by the HDHP/HSA combination compared with a conventional full coverage health insurance plan. We also study the amount of additional savings induced by the alternative financing regime. As a full coverage plan eliminates the financial risk, it is appropriate to interpret s F C as consumption smoothing savings. An HDHP leaves part of the financial risk with the insured, and the HSA stimulates further wealth accumulation through the tax advantage in the sick state such that we call the medical savings. With treatment above the deductible, the optimal solution . HSA ; THSA / is given by the first-order conditions VHSA D u0 .w1F C p.T D// C p.1 C /v 0 .w2F C C .1 C / D/ C .1 p/v 0 .w2F C C / D 0; VTHSA
D pu
0
.w1F C
(8) 0
p.T D// C pkh .T C T
FC
/ D 0;
and again it can be shown as in Footnote 11 that . HSA ; THSA / is the maximum. The case with treatment below the deductible is analogous. Note that the insured’s expected utility under a standard full coverage health insurance contract is nested within V HSA by setting the tax favor and the deductible equal to 0. In this case, the optimal medical savings are 0, and the optimal treatment is given by the treatment under full insurance so that also the change in treatment consumption is 0. As in Steinorth (2011), we now show that the preferential tax treatment of medical savings makes partial insurance coverage desirable for the individual. Consequently, if the individual was free to choose the deductible, he or she would select a positive one. This is important because we can then compare a conventional full coverage plan to an HDHP/HSA plan, which are both first best efficient. Hence, the tax advantage alone implies the optimality of cost sharing, and we do not have to bring forward any arguments of asymmetric information. Remark 2 With a positive tax favor, the optimal deductible is positive. A positive tax favor incentivizes medical savings such that second-period wealth in the sick state is larger. Consequently, it becomes easier for the individual to bear out-of-pocket expenses, which save premium money on the health insurance contract. This explains the utility-enhancing effect of cost sharing in our setup. In the following proposition, we characterize treatment and saving decisions with an HDHP/HSA combination in the general case. We utilize the concept of partial risk aversion (PRA) as introduced by
16
The term T F C in the maximum function in (7) compensates the premium payment that is already included in w1F C , as it is not needed in case of treatment lower than the deductible.
Copyright © 2015 John Wiley & Sons, Ltd.
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Menezes and Hanson (1970). It is defined as PRA.xI y/ WD xv 00 .x C y/=v 0 .x C y/ and measures the elasticity of the risk premium with respect to the size of the risk at a given wealth level. PRA complements the concept of relative risk aversion (RRA), which measures how the risk premium is affected at the margin when wealth and risk change by the same proportion, and the concept of absolute risk aversion (ARA), which has a similar interpretation when only wealth is changed, but the risk remains unaffected. A classical application is the standard portfolio problem where a decision-maker allocates his orher wealth between a risky and a safe asset. PRA allows signing the comparative statics of changes in risk of the return distribution of the risky asset. It is particularly relevant to our analysis as we model the tax incentive as some form of interest payment that only applies in the sick state. In this sense, the return on savings in the HSA is risky. Eeckhoudt and Gollier (1996) and Chiu et al. (2012) discuss PRA in relation to RRA. Technically, PRA is otained by multiplying RRA with the risky part of someone’s wealth, x, divided by overall wealth, x C y. Proposition 2 Under prudence and PRA less than 1, an HDHP/HSA combination leads to positive medical savings and to less treatment consumption compared to full insurance.17 This shows that the HDHP/HSA combination should work in the direction as intended by regulation. Individuals save more compared with a traditional full coverage health plan and, at the same time, utilize medical services less. Critics of HSAs argue that any observed differences in the medical utilization are the result of favorable selection, which is shown to exist in HSAs (McDevitt et al., 2014). We show in Proposition (2) that, other things being equal, different health plan designs lead to different treatment choices too.18 In our framework, this also implies that health outcomes should be expected to be worse with the HDHP/HSA plan, because the underlying healthcare production function is strictly increasing. In a model where consumers can shop around for better prices, this is not necessarily the case. We discuss how overall expected utility is affected later in the paper. For Proposition 2, the required assumption of prudence is relatively common in economics. It ensures that individuals have a precautionary saving motive, that is, save more when we add a 0-mean risk to future income (Kimball, 1990). Measures for PRA naturally vary depending on how much wealth is actually at risk. In our case, the risky part of wealth is the HSA balance compared with the overall wealth in case of requiring medical treatment. If we assume w2F C to be sufficiently large compared with the HSA balance, PRA below 1 does not impose a very strict boundary.19 Another interpretation is that the threshold on PRA simply provides an (endogenous) upper limit to RRA, in our case that RRA should not exceed overall wealth in the second period divided by the portion of wealth in the HSA. This upper limit is itself larger than 1. The intuition of this assumption is that it ensures that the marginal benefit of medical saving increases in the tax advantage. This is not automatically satisfied as there are two competing effects: First, a higher tax advantage increases the return on medical saving when sick, which is a positive first-order effect. However, marginal utility is diminishing so with a higher tax advantage additional medical savings are valued less. This is a negative second-order effect. The first-order effect predominates if and only if PRA is bounded by 1.20 If an HDHP/HSA combination is purchased, the contract design, that is, the tax advantage and the deductible D, influence the amount of medical savings HSA and the treatment THSA . We examine the 17
More specifically, when treatment exceeds the deductible, medical savings are positive in any case, and when treatment falls below the deductible, treatment decreases in any case. So the assumptions of prudence and partial risk aversion less than 1 are only needed for treatment in the first case and for saving in the second. 18 Critics of health savings accounts have also argued that individuals will cut back small expenses today but that this will lead to even larger expenses in the future because of forgone prevention or surveillance. Hence, the net effect on healthcare costs could be positive. Although interesting, this is beyond the scope of the current analysis because we aggregate treatment in one variable T and do not decompose it into early and late medical expenditures. This is left for future research. 19 Binswanger (1981) and Bar-Shira et al. (1997) find PRA to be less than 1 for the majority of their subjects. 20 This parallels results on the comparative statics with respect to the interest rate in canonical consumption-saving models, see Gollier (2001), Chapter 16 for a summary. With HSAs, however, the ‘interest rate’ is state-dependent because in the healthy state the tax benefit is lost. This is why we obtain PRA and not RRA in the sufficient condition. Copyright © 2015 John Wiley & Sons, Ltd.
Health Econ. 25: 357–371 (2016) DOI: 10.1002/hec
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marginal effects of these parameters on saving and treatment decisions undertaken by individuals. The following proposition holds: Proposition 3 - For treatment above the deductible, a higher tax advantage leads to more tax-favored medical savings and less treatment if and only if the individual has PRA less than 1. - Below the deductible, PRA less than 1 is sufficient for a higher tax advantage to imply more tax-favored medical savings and more treatment. - A higher deductible leads to more medical savings in any case. Under prudence, it leads to less treatment if and only if the probability of sickness is smaller than a threshold, p. O The results of Proposition 3 can be interpreted in the same way as those of Proposition 2. A higher tax advantage leads to more savings, but only if PRA is not too high because of the additional risk in t2 , where the tax subsidy is only paid when sick. Note that the assumption of prudence is not needed for this result because it is a direct wealth effect. With a larger tax subsidy saving via the HSA is more attractive and should be increased. For treatment, however, the effect depends on whether we are above or below the deductible. Above the deductible, there is no direct effect of the tax subsidy on treatment as the premium is paid in period 1. This implies that there is no indirect effect on medical saving so that individuals will save more because of the first HSA effect described earlier. This in turn implies a substitution effect on treatment because VT < 0. Intuitively, higher savings reduce consumption in the first period so that the marginal cost resulting from the payment of the insurance premium are higher. Consequently, treatment is reduced, and we should expect worse health outcomes in this case. Below the deductible things are quite different. If the individual pays treatment costs out of her own pocket, there will be a direct effect of the tax advantage on treatment because the marginal cost of this out-of-pocket payment is lower when the tax advantage is larger (and saving is fixed). Furthermore, there is complementarity between saving and treatment. When saving increases at a given tax advantage, there is more wealth in the second period. This reduces the marginal cost of out-of-pocket expenses implying more treatment. As before, PRA less than 1 ensures that saving reacts positively to an increase in the tax advantage, and the complementarity implies that indirect effects confirm the direct effects, and both decisions move in the same direction. As a consequence, we should expect better health outcomes when treatment is below the deductible. Note that the incentives arising from the tax advantage depend on the relationship between treatment and deductible. For the deductible, we can safely focus on the case with treatment exceeding it because otherwise optimal decisions will not depend on it. When the deductible is increased, the individual has to save more to be able to pay the higher out-of-pocket expenses in case of illness in the second period. In other words, the marginal benefit of medical savings is larger if the deductible is increased. This is in line with Proposition 2, where without health insurance (which can be seen as an increase of the deductible from D D 0 to D D T ) savings are larger: More of the financial risk has to be borne by the insured, which stimulates further savings. The net effect on the treatment variable is, however, ambiguous because of the fact that there are two competing effects. On the one hand, there is a positive direct effect. The reason is that with a higher deductible the premium is lower so that the marginal cost of additional treatment, that is, the increase in insurance premium, is smaller. This effect is more pronounced with a higher probability of sickness p as increasing the deductible decreases the insurance premium by more.21 However, there is a negative substitution effect arising from the increase in savings. Similar to aforementioned, an increase in savings reduces first-period consumption and hence raises the marginal cost of additional premium expenditures for more treatment. Again, the size of this effect is increasing in the probability of sickness, or said differently, it is more negative for a larger probability of sickness.22 21 22
Furthermore, this effect is convex in the probability of sickness. Obviously, it is nil for the limit case where the probability of sickness is 0. Moreover, one can demonstrate that it is concave in the probability of sickness and nil for the limit case where the probability of sickness is 0. In addition, the absolute value of the curvature is larger for the substitution effect than for the direct effect. This explains why the net effect is first negative then positive.
Copyright © 2015 John Wiley & Sons, Ltd.
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Figure 2. Effect of changes in the deductible on treatment
Our proposition illustrates that for a small probability of sickness, the first effect is less significant, and the second effect prevails so that larger deductibles lead to less treatment consumption. With a large probability of sickness, however, the opposite is true. The direct effect prevails, and individuals consume more treatment. Consequently, we should expect better health outcomes if and only if the probability of sickness exceeds p. O We provide a graphical illustration in Figure 2.
4. CHOICE BETWEEN A HEALTH SAVINGS ACCOUNT AND A TRADITIONAL PLAN We now investigate how a traditional full coverage health plan compares with an HDHP/HSA combination. As a first step, we develop a sufficient condition that implies the superiority of the latter health plan. Then, we investigate how changes of the tax advantage and the deductible affect the desirability of the HDHP/HSA combination. We will use this later to develop public policy implications. For the first purpose, we nest expected utility with a conventional plan and with an HDHP/HSA combination into one objective function. For ease of exposition, we focus on the case with treatment exceeding the deductible, but note that the other case is analytically very close. We control the parameters and D, which characterize the HDHP/HSA plan, by introducing variable l between 0 and 1. The insured’s expected utility can be written as
V .lI .l/; T .l// D u.w1F C p.T lD// C pŒv.w2F C C .1 C l/ lD/ C kh.T C T F C / C .1 p/Œv.w2F C C / C kh2
(9)
with and T endogenous to l. The case l D 0 corresponds to the case of full insurance with .0/ D 0; T .0/ D 0, so that expected utility becomes V F C .s F C ; T F C /; the case l D 1 corresponds to the case with an HDHP/HSA combination, where .1/ D HSA ; T .1/ D THSA and expected utility becomes V HSA . HSA ; THSA /: We can derive the following proposition, which relies on a first-order Taylor series expansion. Copyright © 2015 John Wiley & Sons, Ltd.
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Proposition 4 An HDHP/HSA combination is preferred to a full insurance contract if the individual’s ARA is below the threshold ‚.l/,23 that is, if
8l 2 Œ0; 1 W
ARA.w2F C C .1 C l/ lD/ < ‚.l/ D
.lpD C / : l.D /D.1 p/
(10)
It seems natural that the degree of risk aversion determines whether an individual chooses the HDHP/HSA combination or the full coverage contract as the first contract imposes the financial risk of the deductible. In addition, it also depends on contract design, that is, on the tax advantage and the deductible D and on the optimal amount of medical savings given a certain l whether (10) is satisfied. Medical savings in turn depend on the tax favor and the deductible. This illustrates that condition (10) is rather complex. To gain intuition on the threshold, we can look at the case of constant absolute risk aversion (CARA) utility, for which the lefthand side of (10) is constant. We find that an increase in the tax advantage or in the probability of sickness p increases the threshold so that the HDHP/HSA combination becomes more attractive, whereas an increase of the deductible D has ambiguous effects. To obtain general findings for generic utility, we now evaluate directly how changes in the tax advantage and the deductible affect consumer well-being as arising from the HDHP/HSA combination. Because expected utility with the traditional plan does not depend on these parameters, an increase in expected utility with the HDHP/HSA combination directly increases the likelihood of this health plan being preferred to the other. The following proposition summarizes our results. Proposition 5 - A higher tax advantage increases consumer expected utility with the HDHP/HSA plan. - With treatment above the deductible, consumer expected utility with the HDHP/HSA plan decreases in the deductible if and only if the probability of sickness is smaller than a threshold, p. The intuition is fairly straightforward. For a given treatment and saving strategy, a higher tax advantage increases expected utility as utility at t2 in the sick state is larger with the higher tax favor. Note that is always positive because individuals will not want to borrow against the HSA. If individuals then adjust their treatment and savings portfolio, this increases overall expected utility even more. This is in line with the behavior of the threshold ‚ from Proposition 4 under CARA utility, where a higher tax advantage increases the threshold ‚ and therefore makes more individuals prefer the HDHP/HSA combination over full insurance. Proposition 5 implies that, if the individual’s propensity to develop a disease is sufficiently high, a lower deductible decreases their expected utility under the HDHP/HSA combination. A lower deductible actually has two effects that inversely impact the individual’s well-being, which is also the reason why the threshold ‚ from Proposition 4 reacts ambiguously to changes in the deductible. With a lower deductible, the premium for the HDHP is more expensive, which lowers the consumer’s expected utility. But, at the same time, out-of-pocket expenditures are smaller when the deductible is decreased, which increases the individual’s expected utility. The net effect depends on the probability of sickness again: if it is small enough, the positive effect from lower out-of-pocket expenditures dominates, and the net effect is positive.
23
It can easily be seen that ‚.l/ is well defined for all l 2 .0; 1. For l D 0, however, both the nominator and the denominator are 0 because of .0/ D 0. Utilizing l’Hopital’s rule, we obtain that lim .l/= l D lim @=@l. As in the proof of Proposition 3, we apply the l#0
l#0
implicit function theorem to show that .@=@l/ det H D VT VT l VT T Vl is positive, so @=@l is positive and therefore ‚.l/ is also well defined for l D 0. Copyright © 2015 John Wiley & Sons, Ltd.
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Note that this reasoning only applies when treatment exceeds the deductible. With treatment below the deductible, the fair premium is nil because the consumer does not receive any payment from the health insurance company. Consequently, changes of the deductible above the level of treatment have no effect on consumer expected utility.24 Finally, we can compare the two thresholds, pO and p. This sheds light on the question how treatment consumption and consumer expected utility relate to each other. We summarize our results in the following remark. Remark 3 Under prudence, it holds that pO > p (D;
HEALTH INSURANCE, HEALTH SAVINGS ACCOUNTS AND HEALTHCARE UTILIZATION RICHARD PETERa , SEBASTIAN SOIKAa and PETRA STEINORTHb a Institute b School
for Risk Management and Insurance, Ludwig-Maximilians-Universität München, Germany
of Risk Management, Insurance and Actuarial Science, St. John’s University New York, NY, USA
SUMMARY Assuming symmetric information, we show that a high-deductible health plan (HDHP) combined with a tax-favored health savings account (HSA) induces more savings and less treatment compared with a full coverage plan under reasonable risk preferences. Furthermore, a higher tax subsidy increases savings in any case but decreases medical utilization if and only if treatment expenses are above the deductible. A larger deductible increases savings but does not necessarily decrease healthcare utilization. Whether an HDHP/HSA combination is preferred over a full coverage contract depends on absolute risk aversion. A higher tax advantage increases the attractiveness of an HDHP/HSA combination, whereas the effects of changes in the deductible are ambiguous. The paper shows that a potential regulator needs to carefully set the size of the deductible as only in a certain corridor of the probability of sickness, its effect on aggregate healthcare costs are unambiguously favorable. Copyright © 2015 John Wiley & Sons, Ltd. Received 18 March 2014; Revised 15 October 2014; Accepted 7 December 2014 JEL classification:
D14, H24, H31, I11, I12
KEY WORDS: health risk; healthcare utilization; health insurance; health savings accounts; medical savings; high-deductible health plans; consumer driven health care
1. INTRODUCTION Medical costs and medical inflation are a major concern to any society aiming to provide comprehensive medical care to its members. According to the World Bank data, the portion of the gross domestic product used for medical costs increased from 13.8% in 1995 to 17.9% in 2011 in the US, from 6.8% to 9.3% in the UK and from 8.3% to 12.0% in the Netherlands.1 Several culprits for this increase have been discussed among which an increasing healthcare utilization has been mentioned repeatedly. In our paper, we investigate how health plans influence medical utilization under symmetric information. We are particularly interested to see whether a highdeductible health plan (HDHP) combined with a health savings account (HSA) can actually create incentives to reduce the consumption of medical services and therefore reduce medical costs. In our paper, we confirm the previous result that the existence of tax-preferred savings for medical costs implies a first best solution with a positive deductible even at actuarially fair insurance prices (Steinorth, 2011). Of course, this changes the demand for medical services and medical savings. Accordingly, it seems natural to
1
Correspondence to: Institute for Risk Management and Insurance, Ludwig-Maximilians-Universität München, Germany. E-mail: [email protected]
See http://data.worldbank.org/indicator/SH.XPD.TOTL.ZS/countries/1W?display=default.
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compare healthcare utilization between a traditional full coverage health plan and an HDHP/HSA combination under symmetric information. We investigate how HSAs affect the trade-off between cost efficiency and medical service provision under symmetric information. It is very important to understand the impact of the given choice of coverage on the demand for medical treatment in order to set the right incentives when designing plans.2 We believe that carefully analyzing incentives when consumers’ decisions are observable is a necessary step before potential information problems as adverse selection and moral hazard are tackled.3 Health savings accounts have been introduced in the US in 2004 and have been attracting a growing number of insured since then covering 15.5 million US Americans as of January 2013.4 A qualifying HDHP allows the insured to open up an HSA where he or she can contribute savings for medical purposes on a tax-free basis within certain limits. The aim of introducing these accounts was to create incentives for individuals to save more for medical purposes and to reduce medical utilization. HDHPs increase consumers’ financial responsibilities, which can reduce medical utilization in several ways. Individuals may forgo medical treatment, which is not absolutely necessary5 as well as shop around for better prizes for the same procedures. In our paper, we focus on the first aspect and leave the second for future research. In addition to the US, forms of HSAs have also been introduced in South Africa, Canada, Singapore and China. Our model can also be applied to medical savings accounts (MSAs) in South Africa, while the accounts in other countries differ in their setup and consequently may provide different incentives.6 A lot of the literature on HSAs focuses on the take up after their introduction (Feldman et al., 2005; Glied and Remler, 2005; Cardon and Showalter, 2008) and discusses potential adverse selection (Cardon and Showalter, 2007) and favorable selection issues (McDevitt et al., 2014). It has also been investigated how much more cost-sharing these health plans will bring (Remler and Glied, 2006; Cardon and Showalter, 2008). Borah et al. (2011) investigate empirically whether HSAs reduce medical expenditures. Steinorth (2011) shows that in the context of ex-ante moral hazard (primary prevention), HSAs do not necessarily accomplish the political goals as their introduction does not decrease ex-ante moral hazard and increase savings at the same time. In addition, Zabinski et al. (1999), Pauly and Herring (2000) and Greene et al. (2006) investigate the related concept of MSAs, while Cardon and Showalter (2001, 2003), Hamilton and Marton (2008) and Cardon (2012) analyze flexible spendings accounts, which require individuals to use their balance at the end of the year. Yet, there is relatively little theoretical literature on medical utilization in the context of HSAs. In general, medical utilization has been studied from a variety of different angles in the theoretical literature before. Pauker and Kassirer (1975) are the first to analyze treatment decisions within an expected utility paradigm. Under diagnostic uncertainty, the probability of the patient having the disease should exceed a critical threshold for treatment to be applied. Bleichrodt et al. (2003) analyze patients’ willingness to pay for health improvements when comorbidities matter, and Berger et al. (2013) study the impact of ambiguity on treatment decisions. In contrast to these papers, we focus on how financial arrangements determine how much 2
Incentives to utilize medical services cautiously are expected to work in favor of the insured because of lower premiums. If, however, these incentives are misaligned, patient satisfaction may suffer, and insured may want to switch plans. The rise and later fall of health maintenance organizations (HMOs) in the US in the 1980s and 1990s illustrates this trade-off between cost control and the provision of medical services nicely. For a more detailed discussion on this, please refer to Mechanic (2004) and Coombs (2005). In the beginning, HMOs were very successful because of competitive insurance pricing. But, over time, many insured shied away from HMOs, where one of the major reasons was that the insured felt that they did not receive sufficient treatment. 3 These two phenomena are often mentioned in the discusion of HDHP/HSAs. 4 This constitutes an increase of 15% since 2012, see the AHIP 2013 HSA census available at http://www.ahip.org/HSA2013/. 5 Yet, this may also negatively impact health outcomes. 6 Medical savings accounts are very popular in South Africa where about half of the private health insurance contracts include one. A highdeductible insurance plan is combined with an MSA where individuals and employers can contribute into. The main difference to the US concept, however, is that the deductible does only apply to outpatient services. This is consistent with our model when considering outpatient services only. In Canada, individuals can open a health spending account regardless of their initial health plan, and they can also purchase supplemental health insurance with the account balance. Medisave accounts in Singapore are mandatory, and income ranges determine the out-of-pocket expenses under the public healthcare system, which can be covered from the accounts. As modeled in our paper, individuals have to cover a nonmarginal part of their health expenses themselves, they are, however, not free to choose how much they want to save in their accounts. MSAs in China also come with a catastrophic health insurance plan as modeled in our paper, yet the level of savings is set by regulation as in Singapore. Copyright © 2015 John Wiley & Sons, Ltd.
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treatment is optimal for the consumer. The empirical literature on treatment choices is huge too. The seminal paper by Manning et al. (1987) investigates the impact of deductible choice on healthcare utilization in the RAND Health Insurance Experiment. Numerous papers have since investigated the impact of plan design changes on the demand for medical services in different countries around the globe, see Mello et al. (2002), Gerfin and Schellhorn (2006), Sarma and Simpson (2006) and Zhong (2011) just to name a few. Our paper adds to the existing literature by adding HSAs to the utilization literature. We are particularly interested in seeing how HSAs impact ex-post medical expenses and whether they ensure higher savings for individuals at the same time. Our main findings are that the tax advantage within an HSA seems to be the most effective parameter to create unambiguous and well-directed incentives. It implies more medical savings and increases the favorability of an HDHP/HSA combination under relatively mild conditions. Manipulating the deductible does not have such straightforward effects. The incentives are sensitive to the probability of sickness, that is, the individual’s risk type. Utilization only decreases with a higher deductible when the probability of sickness is sufficiently high, while a lower deductible does not automatically imply that an HDHP/HSA combination becomes more favorable over a traditional coverage plan. After the introduction, the paper structure is as follows: We first introduce a general model of medical care utilization and savings. In the third section, we discuss the impact of HSAs. We also provide comparative statics with respect to the deductible and the tax favor. In the fourth section, we study consumer expected utility under both alternatives to finance medical care and develop a sufficient condition for the superiority of the HDHP/HSA combination. Furthermore, we develop comparative statics for this condition. The paper ends with some concluding remarks and public policy recommendations. 2. THE MODEL SETUP In this section, we introduce our model to investigate savings and treatment decisions as related to different health plans. We consider two points in time, t1 and t2 , now and then. At t1 , individuals are young and healthy and earn w1 on the labor market. At t2 , individuals are older and might fall ill. They receive earnings w2 . If individuals become ill when old, treatment is necessary. Illness occurs with probability p, and treatment comes at a cost of T .7 Individuals derive utility from consumption and from health at both points in time. Following Kifmann (2001), we assume separability across points in time and between financial and nonfinancial consumption.8 Hence, at t1 , utility derived from consumption c1 and health h1 is given by u.c1 / C kh1 , where u is a twice continuously differentiable utility function that is increasing and concave (u0 > 0; u00 < 0) and k measures the importance of utility from health compared with consumption utility. As individuals are assumed to be healthy at t1 , we normalize h1 to 0. In the second period, utility derived from consumption c2 and health h2 is given by v.c2 / C kh2 . Again, v is a twice continuously differentiable utility function that is increasing and concave (v 0 > 0; v 00 < 0). Note that preferences of financial consumption at t1 and t2 need not be identical.9 As the aim of the paper is to focus on saving and treatment decisions and how they depend on different forms of financing medical consumption, individuals have to determine how much wealth to transfer from t1 to t2 , that is, they have to decide about their optimal savings denoted by s. We normalize the interest rate to 0. Furthermore, given individuals fall ill, they have to determine how much treatment T to consume. Treatment is productive in the sense that the health state when ill is increasing in the level of treatment. Yet, treatment
7
Our modeling of disease is closely related to Cardon and Showalter (2001), Cardon (2010) and Steinorth (2011). The question how changes in health status affect marginal utility of consumption is still not completely resolved. Viscusi and Evans (1990) use job injuries to infer that marginal utility drops because of decreased health, whereas Evans and Viscusi (1991) find that decreased health corresponds to a drop in income and does not affect the structural form of consumption utility. 9 Our specification includes the discounted expected utility model, in which consumption utility at t2 is given by ıu.c2 / with ı 2 .0; 1 being the rate of pure preference for the present. 8
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expenditures incur. Formally, we assume the healthcare production function h.T / to be twice continuously differentiable with h0 > 0 and h00 < 0. We already mentioned that the focus in our paper is on the effects of plan design on how much treatment and savings will be demanded. To this end, we assume the per unit cost of treatment and the productivity to be exogenous to the consumer. For instance, technologies and prices could be uniform in the market meaning that they do not differ among providers. An alternative way to reduce healthcare cost would be the use of cost sharing to provide incentives to the consumer to shop for prices. This is not covered in our model. Using the notation developed earlier, the individual’s expected utility without insurance writes as V 0 D u.w1 s/ C pŒv.w2 C s T / C kh.T / C .1 p/Œv.w2 C s/ C kh2 :
(1)
The optimal solution .s 0 ; T 0 / for the endogenous saving and treatment decision is determined via the first-order conditions10 Vs0 D u0 .w1 s/ C pv 0 .w2 C s T / C .1 p/v 0 .w2 C s/ D 0; VT0 D pv 0 .w2 C s T / C pkh0 .T / D 0:
(2)
The first equation is the usual consumption smoothing condition over the life cycle, that is, (expected) marginal consumption utility across points in time needs to be equalized, whereas the second equation describes the trade-off between costs incurred by consuming treatment and health benefits resulting from it.11 We can see that the demand for treatment depends on several factors: the production technology for health (h0 ), the relative importance of utility from health and consumption (k) and the cost of treatment (v 0 ). Observe that without insurance the probability of sickness is irrelevant for treatment consumption because treatment is only purchased when ill. With insurance, it impacts the premium calculation. If individuals take up health insurance with a deductible, D 0, this yields expected utility of V D Du .w1 s p max.T D; 0// C pŒv.w2 C s min.D; T // C kh.T / C .1 p/Œv.w2 C s/ C kh2 :
(4)
In t2 , the individual incurs the minimum of his or her treatment cost T and the deductible. Therefore, the corresponding fair insurance premium in t1 is given by the portion of the treatment cost that exceeds the deductible times the probability of sickness.12 This formulation contains the case with treatment exceeding the deductible and treatment falling below it. In both cases, the objective function is concave, and the optimal s and T can be characterized via the respective first-order conditions. Let us extend a classical result to our case. Proofs are in the appendix. Remark 1 If insurance is actuarially fair, full insurance .D D 0/ is optimal.
Subscripts of model parameters denote partial derivatives, that is, Vs0 is shorthand for @V 0 =@s and VT0 is shorthand for @V 0 =@T . This notation is used throughout the paper. 11 0 Note that V 0 is concave, because Vss < 0 and the determinant of the Hessian is positive, 10
0 0 2 00 00 VT0 T .VsT / D .u00 C .1 p/vnt /pvt00 C .u00 C pvt00 C .1 p/vnt /pkh00 > 0: Vss
(3)
Subscript t denotes consumption with treatment and nt consumption without treatment. As a result, .s 0 ; T 0 / is a maximizer of expected intertemporal utility over consumption and health. 12 We model the insurance premium p max.T D; 0/ to be paid only in the first period. As individuals save endogenously, they can smooth their income according to their preferences. Therefore, it makes no difference if the insurance premium is paid only in the first period, only in the second period or is split over both periods. This holds true for all variations of the model presented in this paper. Copyright © 2015 John Wiley & Sons, Ltd.
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This is reminiscent of a result obtained by Dionne and Eeckhoudt (1984) who show that in a two-period model with nonseparable preferences and endogenous consumption, financial risks should be fully insured if and only if the premium is actuarially fair.13 This is known as Mossin’s Theorem (1968), which carries over to our setup with a health risk. Hence, the resulting expected utility with the optimal level of coverage D D 0, that is, full coverage, is given by V F C D u.w1 s pT / C v.w2 C s/ C pkh.T / C .1 p/kh2 ;
(5)
so that the financial risk is completely eliminated. We use superscript F C to denote this situation with a standard full coverage health insurance contract. The optimal saving and treatment decision .s F C ; T F C / under full coverage are characterized by the first-order conditions VsF C D u0 .w1 s pT / C pv 0 .w2 C s/ C .1 p/v 0 .w2 C s/ D 0; VTF C D pu0 .w1 s pT / C pkh0 .T / D 0:
(6)
Note that V F C is concave, which can be shown analogously to Footnote 11. Next, we compare saving and treatment decisions when uninsured to a situation with full insurance coverage. To this end, we need to compare the endogenously determined values s 0 and T 0 with s F C and T F C , which are also endogenous. As we will make frequent use of the technical procedure to do that, we state it in the following Lemma. Lemma 1 2 Let f W R2 ! R be a concave function in the variables .x; y/, that is, fxx < 0 and fxx fyy fxy > 0, which is maximal at .x ; y /. Let ˛ 2 R be a value we want to compare x with. Then, x > ˛ if and only if fx .˛; y/ O > 0 where yO is the value that maximizes f .˛; y/. This approach is due to Gollier (2001, p. 151) and has recently been applied in Hofmann and Peter (2014) to study prevention and saving decisions. For our problem, results are summarized in the following proposition. Proposition 1 A standard full coverage health insurance contract leads to more treatment and less savings than in the absence of insurance. Figure 1 graphically illustrates Proposition 1. It demonstrates that the existence of a health insurance contract implies that individuals utilize more medical treatment. This is in line with empirical observations (Kondo and Shigeoka, 2013). As the health insurance company pays the treatment cost, the individual has to accumulate less savings. Furthermore, we know from earlier that health insurance coverage is utility enhancing, and the model predicts that realized health status should be uniformly larger for health insured individuals than for individuals without coverage everything else equal. This is due to the fact that T F C > T 0 results in h.T F C / > h.T 0 /. Note that we assume individual treatment decisions to be observable by the insurance company so there is no ex-post moral hazard. This is to focus on the redistributive effects of different health plans. Health insurance stipulates more treatment which is, however, not to be confused with overutilization as discussed by Pauly (1974). In our model, individuals pay the fair premium for the medical services they demand later in life. The effect observed here can be attributed to the fact that redistribution between different states of the world and different points in time is beneficial for a risk-averse individual. In the absence of insurance, saving is the only instrument to redistribute between today and tomorrow. Actuarially fair insurance coverage is a second way to optimize an endowed consumption stream by redistributing from today to a specific future state. Hence, the set
13
Menegatti and Rebessi (2011) confirm this result in a two-period prevention model with separable preferences.
Copyright © 2015 John Wiley & Sons, Ltd.
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Figure 1. Relationship between .s 0 ; T 0 / and .s F C ; T F C /
of consumption possibilities is enlarged, which increases expected utility and gives more freedom to consume additional treatment. 3. INTRODUCTION OF HEALTH SAVINGS ACCOUNTS Let us now investigate how individual saving and treatment decisions are altered when HSAs are introduced. HSAs are only available in combination with an HDHP. HSAs benefit from two preferential tax treatments. Contributions are tax exempt and will remain tax exempt if used for defined medical services. Furthermore, interest earned in the account is also tax exempt. All withdrawals that are not used for qualified medical services are considered nonqualified withdrawals. Individuals need to pay income taxes on these withdrawals, and there is also a 20% tax penalty for individuals younger than 65 years in order to offset previous tax favors and interest earned on saved tax dollars. Because of the tax benefits, we model that individuals will put their savings for medical purposes into the HSA, which we denote by . The preferential tax treatment is modeled in a way that, when savings are used for treatment in t2 , the individual receives .1 C / from their HSA. This seems a little different than the original tax treatment at first glance. As everything else is after tax, we assume that .1C/ are the combined tax benefits of HSAs, which we express as a form of interest on after tax dollars.14 If individuals remain healthy, they can withdraw the money from their HSA but do not receive any tax benefits. In this sense, we implicitly assume that the tax penalty exactly removes any tax benefits for unqualified withdrawals. Consequently, the amount withdrawn is as if the money was saved outside the HSA because we set the interest rate to 0.15 We also assume that the individual’s savings for medical services plus the tax benefit from the HSA do not exceed the deductible, .1 C / D, which seems reasonable for practical purposes. Our approach follows Steinorth (2011). Without an HSA, the individual would still choose D D 0, which yields expected utility V F C as in (5) with the optimal solution .s F C ; T F C / for saving and treatment. Using superscript HSA to denote the situation for 14 15
This interpretation allows us to draw on the concept of partial risk aversion later. The health savings account balance bears tax exempt interest, and the individuals also receive interest on the part of the tax exempt contributions, which would have usually been subject to income tax. Assuming equal marginal tax rates upon contribution and nonqualified withdrawal, there would still be a tax advantage because of accrued tax-free interest. The penalty aims at correcting this. Yet, it still depends on the individual situation whether unqualified withdrawals are beneficial or not from a tax perspective as marginal tax rates upon contribution and withdrawal, the timing of contributions and withdrawals in the health savings account and the interest rate environment matter. As all these factors are not of first concern to our study, we decided to assume that the penalty exactly removes the previous tax advantage for nonqualified withdrawals.
Copyright © 2015 John Wiley & Sons, Ltd.
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an HDHP/HSA combination, the insured have expected utility of V HSA D u.w1F C p max.T D; T F C // C p v.w2F C C .1 C / min.D; T C T F C // Ckh.T C T F C / C .1 p/Œv.w2F C C / C kh2 ;
(7)
with w1F C WD w1 s F C pT F C , w2F C WD w2 C s F C and T WD T T F C . Note that our formulation of the objective function contains the case in which treatment exceeds the deductible as well as the case in which it does not. Similar to V D as in (4), the individual incurs the minimum of his or her treatment cost, T C T F C and the deductible, D, at t2 . The corresponding fair insurance premium in t1 is again given by the portion of the treatment cost that exceed the deductible times the probability of sickness.16 In this sense, we investigate the change T of treatment consumption induced by the HDHP/HSA combination compared with a conventional full coverage health insurance plan. We also study the amount of additional savings induced by the alternative financing regime. As a full coverage plan eliminates the financial risk, it is appropriate to interpret s F C as consumption smoothing savings. An HDHP leaves part of the financial risk with the insured, and the HSA stimulates further wealth accumulation through the tax advantage in the sick state such that we call the medical savings. With treatment above the deductible, the optimal solution . HSA ; THSA / is given by the first-order conditions VHSA D u0 .w1F C p.T D// C p.1 C /v 0 .w2F C C .1 C / D/ C .1 p/v 0 .w2F C C / D 0; VTHSA
D pu
0
.w1F C
(8) 0
p.T D// C pkh .T C T
FC
/ D 0;
and again it can be shown as in Footnote 11 that . HSA ; THSA / is the maximum. The case with treatment below the deductible is analogous. Note that the insured’s expected utility under a standard full coverage health insurance contract is nested within V HSA by setting the tax favor and the deductible equal to 0. In this case, the optimal medical savings are 0, and the optimal treatment is given by the treatment under full insurance so that also the change in treatment consumption is 0. As in Steinorth (2011), we now show that the preferential tax treatment of medical savings makes partial insurance coverage desirable for the individual. Consequently, if the individual was free to choose the deductible, he or she would select a positive one. This is important because we can then compare a conventional full coverage plan to an HDHP/HSA plan, which are both first best efficient. Hence, the tax advantage alone implies the optimality of cost sharing, and we do not have to bring forward any arguments of asymmetric information. Remark 2 With a positive tax favor, the optimal deductible is positive. A positive tax favor incentivizes medical savings such that second-period wealth in the sick state is larger. Consequently, it becomes easier for the individual to bear out-of-pocket expenses, which save premium money on the health insurance contract. This explains the utility-enhancing effect of cost sharing in our setup. In the following proposition, we characterize treatment and saving decisions with an HDHP/HSA combination in the general case. We utilize the concept of partial risk aversion (PRA) as introduced by
16
The term T F C in the maximum function in (7) compensates the premium payment that is already included in w1F C , as it is not needed in case of treatment lower than the deductible.
Copyright © 2015 John Wiley & Sons, Ltd.
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Menezes and Hanson (1970). It is defined as PRA.xI y/ WD xv 00 .x C y/=v 0 .x C y/ and measures the elasticity of the risk premium with respect to the size of the risk at a given wealth level. PRA complements the concept of relative risk aversion (RRA), which measures how the risk premium is affected at the margin when wealth and risk change by the same proportion, and the concept of absolute risk aversion (ARA), which has a similar interpretation when only wealth is changed, but the risk remains unaffected. A classical application is the standard portfolio problem where a decision-maker allocates his orher wealth between a risky and a safe asset. PRA allows signing the comparative statics of changes in risk of the return distribution of the risky asset. It is particularly relevant to our analysis as we model the tax incentive as some form of interest payment that only applies in the sick state. In this sense, the return on savings in the HSA is risky. Eeckhoudt and Gollier (1996) and Chiu et al. (2012) discuss PRA in relation to RRA. Technically, PRA is otained by multiplying RRA with the risky part of someone’s wealth, x, divided by overall wealth, x C y. Proposition 2 Under prudence and PRA less than 1, an HDHP/HSA combination leads to positive medical savings and to less treatment consumption compared to full insurance.17 This shows that the HDHP/HSA combination should work in the direction as intended by regulation. Individuals save more compared with a traditional full coverage health plan and, at the same time, utilize medical services less. Critics of HSAs argue that any observed differences in the medical utilization are the result of favorable selection, which is shown to exist in HSAs (McDevitt et al., 2014). We show in Proposition (2) that, other things being equal, different health plan designs lead to different treatment choices too.18 In our framework, this also implies that health outcomes should be expected to be worse with the HDHP/HSA plan, because the underlying healthcare production function is strictly increasing. In a model where consumers can shop around for better prices, this is not necessarily the case. We discuss how overall expected utility is affected later in the paper. For Proposition 2, the required assumption of prudence is relatively common in economics. It ensures that individuals have a precautionary saving motive, that is, save more when we add a 0-mean risk to future income (Kimball, 1990). Measures for PRA naturally vary depending on how much wealth is actually at risk. In our case, the risky part of wealth is the HSA balance compared with the overall wealth in case of requiring medical treatment. If we assume w2F C to be sufficiently large compared with the HSA balance, PRA below 1 does not impose a very strict boundary.19 Another interpretation is that the threshold on PRA simply provides an (endogenous) upper limit to RRA, in our case that RRA should not exceed overall wealth in the second period divided by the portion of wealth in the HSA. This upper limit is itself larger than 1. The intuition of this assumption is that it ensures that the marginal benefit of medical saving increases in the tax advantage. This is not automatically satisfied as there are two competing effects: First, a higher tax advantage increases the return on medical saving when sick, which is a positive first-order effect. However, marginal utility is diminishing so with a higher tax advantage additional medical savings are valued less. This is a negative second-order effect. The first-order effect predominates if and only if PRA is bounded by 1.20 If an HDHP/HSA combination is purchased, the contract design, that is, the tax advantage and the deductible D, influence the amount of medical savings HSA and the treatment THSA . We examine the 17
More specifically, when treatment exceeds the deductible, medical savings are positive in any case, and when treatment falls below the deductible, treatment decreases in any case. So the assumptions of prudence and partial risk aversion less than 1 are only needed for treatment in the first case and for saving in the second. 18 Critics of health savings accounts have also argued that individuals will cut back small expenses today but that this will lead to even larger expenses in the future because of forgone prevention or surveillance. Hence, the net effect on healthcare costs could be positive. Although interesting, this is beyond the scope of the current analysis because we aggregate treatment in one variable T and do not decompose it into early and late medical expenditures. This is left for future research. 19 Binswanger (1981) and Bar-Shira et al. (1997) find PRA to be less than 1 for the majority of their subjects. 20 This parallels results on the comparative statics with respect to the interest rate in canonical consumption-saving models, see Gollier (2001), Chapter 16 for a summary. With HSAs, however, the ‘interest rate’ is state-dependent because in the healthy state the tax benefit is lost. This is why we obtain PRA and not RRA in the sufficient condition. Copyright © 2015 John Wiley & Sons, Ltd.
Health Econ. 25: 357–371 (2016) DOI: 10.1002/hec
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marginal effects of these parameters on saving and treatment decisions undertaken by individuals. The following proposition holds: Proposition 3 - For treatment above the deductible, a higher tax advantage leads to more tax-favored medical savings and less treatment if and only if the individual has PRA less than 1. - Below the deductible, PRA less than 1 is sufficient for a higher tax advantage to imply more tax-favored medical savings and more treatment. - A higher deductible leads to more medical savings in any case. Under prudence, it leads to less treatment if and only if the probability of sickness is smaller than a threshold, p. O The results of Proposition 3 can be interpreted in the same way as those of Proposition 2. A higher tax advantage leads to more savings, but only if PRA is not too high because of the additional risk in t2 , where the tax subsidy is only paid when sick. Note that the assumption of prudence is not needed for this result because it is a direct wealth effect. With a larger tax subsidy saving via the HSA is more attractive and should be increased. For treatment, however, the effect depends on whether we are above or below the deductible. Above the deductible, there is no direct effect of the tax subsidy on treatment as the premium is paid in period 1. This implies that there is no indirect effect on medical saving so that individuals will save more because of the first HSA effect described earlier. This in turn implies a substitution effect on treatment because VT < 0. Intuitively, higher savings reduce consumption in the first period so that the marginal cost resulting from the payment of the insurance premium are higher. Consequently, treatment is reduced, and we should expect worse health outcomes in this case. Below the deductible things are quite different. If the individual pays treatment costs out of her own pocket, there will be a direct effect of the tax advantage on treatment because the marginal cost of this out-of-pocket payment is lower when the tax advantage is larger (and saving is fixed). Furthermore, there is complementarity between saving and treatment. When saving increases at a given tax advantage, there is more wealth in the second period. This reduces the marginal cost of out-of-pocket expenses implying more treatment. As before, PRA less than 1 ensures that saving reacts positively to an increase in the tax advantage, and the complementarity implies that indirect effects confirm the direct effects, and both decisions move in the same direction. As a consequence, we should expect better health outcomes when treatment is below the deductible. Note that the incentives arising from the tax advantage depend on the relationship between treatment and deductible. For the deductible, we can safely focus on the case with treatment exceeding it because otherwise optimal decisions will not depend on it. When the deductible is increased, the individual has to save more to be able to pay the higher out-of-pocket expenses in case of illness in the second period. In other words, the marginal benefit of medical savings is larger if the deductible is increased. This is in line with Proposition 2, where without health insurance (which can be seen as an increase of the deductible from D D 0 to D D T ) savings are larger: More of the financial risk has to be borne by the insured, which stimulates further savings. The net effect on the treatment variable is, however, ambiguous because of the fact that there are two competing effects. On the one hand, there is a positive direct effect. The reason is that with a higher deductible the premium is lower so that the marginal cost of additional treatment, that is, the increase in insurance premium, is smaller. This effect is more pronounced with a higher probability of sickness p as increasing the deductible decreases the insurance premium by more.21 However, there is a negative substitution effect arising from the increase in savings. Similar to aforementioned, an increase in savings reduces first-period consumption and hence raises the marginal cost of additional premium expenditures for more treatment. Again, the size of this effect is increasing in the probability of sickness, or said differently, it is more negative for a larger probability of sickness.22 21 22
Furthermore, this effect is convex in the probability of sickness. Obviously, it is nil for the limit case where the probability of sickness is 0. Moreover, one can demonstrate that it is concave in the probability of sickness and nil for the limit case where the probability of sickness is 0. In addition, the absolute value of the curvature is larger for the substitution effect than for the direct effect. This explains why the net effect is first negative then positive.
Copyright © 2015 John Wiley & Sons, Ltd.
Health Econ. 25: 357–371 (2016) DOI: 10.1002/hec
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Figure 2. Effect of changes in the deductible on treatment
Our proposition illustrates that for a small probability of sickness, the first effect is less significant, and the second effect prevails so that larger deductibles lead to less treatment consumption. With a large probability of sickness, however, the opposite is true. The direct effect prevails, and individuals consume more treatment. Consequently, we should expect better health outcomes if and only if the probability of sickness exceeds p. O We provide a graphical illustration in Figure 2.
4. CHOICE BETWEEN A HEALTH SAVINGS ACCOUNT AND A TRADITIONAL PLAN We now investigate how a traditional full coverage health plan compares with an HDHP/HSA combination. As a first step, we develop a sufficient condition that implies the superiority of the latter health plan. Then, we investigate how changes of the tax advantage and the deductible affect the desirability of the HDHP/HSA combination. We will use this later to develop public policy implications. For the first purpose, we nest expected utility with a conventional plan and with an HDHP/HSA combination into one objective function. For ease of exposition, we focus on the case with treatment exceeding the deductible, but note that the other case is analytically very close. We control the parameters and D, which characterize the HDHP/HSA plan, by introducing variable l between 0 and 1. The insured’s expected utility can be written as
V .lI .l/; T .l// D u.w1F C p.T lD// C pŒv.w2F C C .1 C l/ lD/ C kh.T C T F C / C .1 p/Œv.w2F C C / C kh2
(9)
with and T endogenous to l. The case l D 0 corresponds to the case of full insurance with .0/ D 0; T .0/ D 0, so that expected utility becomes V F C .s F C ; T F C /; the case l D 1 corresponds to the case with an HDHP/HSA combination, where .1/ D HSA ; T .1/ D THSA and expected utility becomes V HSA . HSA ; THSA /: We can derive the following proposition, which relies on a first-order Taylor series expansion. Copyright © 2015 John Wiley & Sons, Ltd.
Health Econ. 25: 357–371 (2016) DOI: 10.1002/hec
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Proposition 4 An HDHP/HSA combination is preferred to a full insurance contract if the individual’s ARA is below the threshold ‚.l/,23 that is, if
8l 2 Œ0; 1 W
ARA.w2F C C .1 C l/ lD/ < ‚.l/ D
.lpD C / : l.D /D.1 p/
(10)
It seems natural that the degree of risk aversion determines whether an individual chooses the HDHP/HSA combination or the full coverage contract as the first contract imposes the financial risk of the deductible. In addition, it also depends on contract design, that is, on the tax advantage and the deductible D and on the optimal amount of medical savings given a certain l whether (10) is satisfied. Medical savings in turn depend on the tax favor and the deductible. This illustrates that condition (10) is rather complex. To gain intuition on the threshold, we can look at the case of constant absolute risk aversion (CARA) utility, for which the lefthand side of (10) is constant. We find that an increase in the tax advantage or in the probability of sickness p increases the threshold so that the HDHP/HSA combination becomes more attractive, whereas an increase of the deductible D has ambiguous effects. To obtain general findings for generic utility, we now evaluate directly how changes in the tax advantage and the deductible affect consumer well-being as arising from the HDHP/HSA combination. Because expected utility with the traditional plan does not depend on these parameters, an increase in expected utility with the HDHP/HSA combination directly increases the likelihood of this health plan being preferred to the other. The following proposition summarizes our results. Proposition 5 - A higher tax advantage increases consumer expected utility with the HDHP/HSA plan. - With treatment above the deductible, consumer expected utility with the HDHP/HSA plan decreases in the deductible if and only if the probability of sickness is smaller than a threshold, p. The intuition is fairly straightforward. For a given treatment and saving strategy, a higher tax advantage increases expected utility as utility at t2 in the sick state is larger with the higher tax favor. Note that is always positive because individuals will not want to borrow against the HSA. If individuals then adjust their treatment and savings portfolio, this increases overall expected utility even more. This is in line with the behavior of the threshold ‚ from Proposition 4 under CARA utility, where a higher tax advantage increases the threshold ‚ and therefore makes more individuals prefer the HDHP/HSA combination over full insurance. Proposition 5 implies that, if the individual’s propensity to develop a disease is sufficiently high, a lower deductible decreases their expected utility under the HDHP/HSA combination. A lower deductible actually has two effects that inversely impact the individual’s well-being, which is also the reason why the threshold ‚ from Proposition 4 reacts ambiguously to changes in the deductible. With a lower deductible, the premium for the HDHP is more expensive, which lowers the consumer’s expected utility. But, at the same time, out-of-pocket expenditures are smaller when the deductible is decreased, which increases the individual’s expected utility. The net effect depends on the probability of sickness again: if it is small enough, the positive effect from lower out-of-pocket expenditures dominates, and the net effect is positive.
23
It can easily be seen that ‚.l/ is well defined for all l 2 .0; 1. For l D 0, however, both the nominator and the denominator are 0 because of .0/ D 0. Utilizing l’Hopital’s rule, we obtain that lim .l/= l D lim @=@l. As in the proof of Proposition 3, we apply the l#0
l#0
implicit function theorem to show that .@=@l/ det H D VT VT l VT T Vl is positive, so @=@l is positive and therefore ‚.l/ is also well defined for l D 0. Copyright © 2015 John Wiley & Sons, Ltd.
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Note that this reasoning only applies when treatment exceeds the deductible. With treatment below the deductible, the fair premium is nil because the consumer does not receive any payment from the health insurance company. Consequently, changes of the deductible above the level of treatment have no effect on consumer expected utility.24 Finally, we can compare the two thresholds, pO and p. This sheds light on the question how treatment consumption and consumer expected utility relate to each other. We summarize our results in the following remark. Remark 3 Under prudence, it holds that pO > p (D;
