Journal of Health Economics journal homepage: www.elsevier.com/locate/econbase

Moral hazard in insurance, value-based cost sharing, and the benefits of blissful ignorance
Mark V. Pauly ∗ , Fredric E. Blavin
Health Care Systems Department, The Wharton School, University of Pennsylvania, 3641 Locust Walk, Philadelphia, PA 19104-6218, United States

a r t i c l e

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a b s t r a c t
The conventional theory of optimal coinsurance rates for health insurance with moral hazard indicates that coinsurance should vary with the price responsiveness or price-elasticity of demand for different medical services. An alternative theory called “value-based cost sharing” indicates that coinsurance should be lower for services with higher (marginal) benefits relative to costs. This paper reconciles the two views. It shows that, if patient demands are based on correct information, optimal coinsurance is the same under either theory. If patient demands differ from informed demands, optimal coinsurance depends both on information imperfection and price responsiveness. Value-based cost sharing can be superior to providing information (even if the cost of information is minimal) when patient demands fall short of informed demands. An extended numerical example illustrates these points. © 2008 Published by Elsevier B.V.

Article history: Received 17 August 2007 Received in revised form 20 June 2008 Accepted 8 July 2008 Available online 18 July 2008 JEL classification: I11 Keywords: Health insurance Moral hazard Value-based cost sharing

1. Introduction Almost all private health insurances in the United States leave some fraction, usually modest, of spending on covered services unreimbursed. While the proportion of personal health expenditures American consumers pay out of pocket has fallen over time, the absolute dollar amount of such cost sharing has risen, along with unit prices and total spending (Catlin et al., 2008). The advent of tax-subsidized high deductible insurance (also known as “consumer-directed insurance”) linked to health savings accounts has heightened interest in many observers, and concern in some, about the effects of cost sharing in voluntary insurance. Compared to full coverage of expenses, out of pocket payments have two effects, one certain and one highly likely. Such payments expose the insured to higher financial risk (compared to full coverage) if spending on a costly illness occurs. They also generally provide incentives for lower levels of medical care use and spending than if coverage were complete. The basic theory of insurance design when there is a tradeoff between risk protection and moral hazard is well known (Pauly, 1968; and specifically, Zeckhauser, 1970). The fundamental idea is that consumers should be willing to sacrifice a modest amount of financial protection in order to experience stronger incentives to avoid the use of care of low value relative to its price. So, according to the moral hazard framework, there will generally be overuse of care relative to the optimum because the level of use chosen by an informed buyer facing a user price below the full cost will be too high.

∗ Corresponding author. Tel.: +1 215 898 2838; fax: +1 215 573 7025. E-mail addresses: Pauly@wharton.upenn.edu (M.V. Pauly), fblavin@wharton.upenn.edu (F.E. Blavin). 0167-6296/$ – see front matter © 2008 Published by Elsevier B.V. doi:10.1016/j.jhealeco.2008.07.003

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However, recent research and policy discussions have raised anew the question of whether and how the still-limited but growing evidence base on medical care effectiveness might be used in the design of insurance. Specifically, is evidence of medical effectiveness and/or cost-effectiveness useful for the design of efficient levels of cost sharing? An interesting and important recent analysis suggests an affirmative answer. First under the label of “benefits-based cost sharing” and then under the new term “value-based insurance design,” Fendrick and Chernew have proposed “tailoring copayments at the point of service to the evidence-based value of specific services for targeted groups of patients” (2006, p. 17; see also Fendrick et al., 2001). A more recent elucidation says that coverage design should depend on costs as well as medical benefits, that is, on the cost–quality tradeoff in a particular set of clinical interventions (Chernew et al., 2007). Even though the link between theory and practice is never exact, there has been some interest in and even some use of the benefits or value-based cost sharing concept in insurance design by some employers and insurers. Most of the discussion of this concept is linked to examples of specific clinical services; there is no complete model of how such evidence of effectiveness should be used in all medical settings. While some special (though clinically important) examples and anecdotes have been discussed, to our knowledge there is no careful and detailed specification of the ideal level of value-based cost sharing. More importantly, no one has spelled out the differences, if any, between the levels of cost sharing that would be specified as (second-best) optimal under each of the two theories in a wide variety of plausible circumstances. That is, no one has answered the question of whether and when the optimal level of coinsurance indicated by the theory of moral hazard differs from that under a value-based approach. In this paper we explore the question of whether and when taking such value into account might make a difference for efficient insurance design. We specifically consider as well the tradeoff between using cost sharing and using better and more persuasive patient information as instruments to deal with non-optimality. Reconciling the two approaches will require describing more carefully the theory and measurements of “evidence-based value” for evidence-based cost sharing, and will require specifying more precisely the information about marginal benefits and the likely behavior of patients and physicians. We first show that if consumer-patients and the physicians they consult both base demand and advice on the same information about marginal medical effectiveness, ideally as accurate as the general state of knowledge and evidence allows, the two approaches (properly defined) give identical messages about optimal benefit design. In these symmetric information models, coinsurance levels depend on more than evidence about medical benefit and, moreover, to the extent that they do depend on such evidence, it is not the typical single summary measure of average effectiveness (or cost-effectiveness) for a treatment of a given illness often invoked by informal evidence-based guidelines. Rather, the key knowledge is the full schedule of marginal benefits for a particular type of care, and, to the extent that this schedule varies across consumers based on measurable health status, income, or health and risk preferences, on that variation as well. More specifically, the primary determinant of the optimal level of coinsurance in the presence of moral hazard is, in the words of Phelps (2003, p. 328), the person-level “price responsiveness” of the demand or marginal benefit curve under full information, or, more elaborately, the configuration of the curve, and not any specific single measure or concept about the marginal cost-effectiveness of a given type of medical care.1 However, if information used by consumers (or individual physicians) is less perfect than that available to experts and planners, optimal cost sharing will be affected. As noted by Chernew et al. (2007), evidence on value will be relevant to setting coinsurance rates if one does not assume that “consumers will balance cost [user price] and clinical value optimally,” because consumer-patients have not received or have not believed evidence from physicians or others. Here we extend this insight about incomplete or imperfect information to show how coinsurance should vary based on the difference between patients’ beliefs about marginal benefit curves and expert knowledge about their true value. When moral hazard combines with imperfect information, we show that optimal coinsurance depends on the extent and nature of the imperfect information and on the price responsiveness of demand. When all agents have perfect knowledge of patient illness states and benefits of care, optimal coinsurance should be zero; insurance should take the form of a fixed dollar (indemnity) payment to cover the full cost of care when care is cost-effective, and should pay nothing in circumstances in which care has benefits that fall short of cost. When there is insurer ignorance about the patient’s true health state – the key necessary condition for moral hazard to occur – optimal coinsurance depends on both the extent and form of that imperfection, and patient demand responsiveness. More generally, if actual demand or patient-perceived marginal benefit curves fall moderately short of true curves, coinsurance should be lower than under fully informed demand, but should still vary with price responsiveness. As the shortfall between patient demand and true benefit gets larger, optimal coinsurance falls to zero if insurance markets work perfectly. However, we show that modest administrative costs associated with higher premiums can cause positive coinsurance to be optimal for some patients. If actual demand curves exceed fully informed curves, optimal coinsurance should generally be higher than in the informed patient case, and price responsiveness still matters. So, evidence on medical effectiveness or cost-effectiveness alone is not sufficient to determine the optimal design; information on the extent and value of financial risk protection matters as well. That is, the two theories are not mutually exclusive, but rather both should be used in insurance design when patients are imperfectly informed.

1 If the demand or marginal benefit curve is linear, the measure of price responsiveness is its slope; if the curve is constant elasticity, it is the elasticity. In the diagrammatic and numerical analysis that follows we will use linear demand curves, but the conclusions will carry over to constant elasticity curves as well.

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2. Some important preliminaries We want to focus on insurance design in situations where the main effect of care is to provide medical benefits, in the sense of better health per se, at higher cost. Consequently, the question of optimal cost sharing when a particular type of care produces cost offsets, which has been treated at length elsewhere (Pauly and Held, 1990), will be ignored. Of course, virtually any care that improves subsequent health probably also impacts subsequent medical spending, but that type of spending effect will not be considered. This also means that the empirical experience of some benefits managers who have emphasized the alleged ability to selectively lower cost sharing for certain services to lower overall spending (Newhouse, 2006) is not directly relevant to the contrast between theories that we want to draw. For simplicity we will ignore variations in prices of care (so that cost sharing can either be represented as a monetary copayment or proportional coinsurance). We also ignore the possibility that some coinsurance designs might affect insurance selection when insurers cannot distinguish among different initial risk levels (but we do assume that insurers cannot perfectly tell a person’s true health severity once illness has occurred). We will also assume that “optimal insurance” means insurance that maximizes consumer welfare (generally in the sense of expected utility), so we will only be concerned about apparently large effects of insurance or care on physical measures of health as they affect expected utility. We will not attach a separate value to maximizing health, “containing cost growth” or lowering cost levels per se. Finally, and more substantively, we will interpret the Chernew–Rosen–Fendrick definition of “the cost–quality tradeoff” (2007) as equivalent to basing (relative) value on marginal cost-effectiveness measures. As they note, what is potentially relevant is not just a comparison of “interventions with the highest value or effectiveness” with interventions with “little or no proven health benefit.” Cost needs to be considered also. If an intervention of the first sort is very expensive and one of the second sort is very cheap but has a little health benefit, value-based coinsurance might be the same for both. Moreover, if costs change (e.g., if an effective medical device becomes more expensive), then value-based cost sharing should probably change as well.2 What are alternative ways in which evidence on effectiveness and cost might influence insurance design? We can distinguish three possible cases, two of which have already been specified by Chernew, Rosen, and Fendrick. At one end of the spectrum, they describe a “first approach” to value-based design where the insurer only has enough information to “target clinically valuable services” that will be used by a broad population in response to patient and physician choices. In this case, targeting is based on the general literature about which services are cost-effective and which are not. At the other end of the spectrum, they suggest a different situation in which the insurer has enough knowledge to determine precisely which patients will benefit to differing extents from various treatments (because the insurer can know which patients have which illness severities and other patient characteristics which cause marginal effectiveness to vary) and therefore can determine the optimal quantity for each kind of patient. In this case, there will be no moral hazard and therefore no need for fractional patient cost sharing. Rather, when the insurer can know patient severity precisely (as well as other patient characteristics that can affect marginal benefit), optimal insurance should cover the cost of treatment in full for those with severities high enough to yield benefits greater than costs, and should not pay at all for treatment of people with lower levels of severity. There is obviously an intermediate case (which Chernew, Rosen, and Fendrick do not discuss): the insurer knows the marginal benefit curve for each level of illness severity, but cannot tell which patients are at which levels of severity. Insurer inability to determine illness severity, and therefore inability to bar less ill patients who get positive but smaller (than cost) marginal benefits from care is what causes moral hazard (Pauly, 2008). That is, insurer knowledge of illness states is generally not precise, even if the physician and/or patient might know the patient’s illness state. Although insurers may be able to some extent distinguish among people in different illness states, there is certain to be some unobserved variation in illness levels. As a result, there can be patient and agent-physician incentives to prescribe and render treatments in quantities or for illness levels in which the benefit, though positive and larger than the out-of-pocket cost, is small relative to the total cost. This case of partial insurer knowledge and positive but potentially small marginal benefit characterizes the situation in which moral hazard arises and needs to be controlled. In what follows, we first summarize the conventional treatment of moral hazard, and show that it potentially differs from judgments about improved or optimal coinsurance based on the informal “first approach” that uses broad characterizations of services in terms of relative effectiveness or cost-effectiveness. Then we explore how patient knowledge or beliefs about marginal medical effectiveness affects optimal cost sharing. We deal with cases of no moral hazard, but we focus on cases in which moral hazard occurs. We close with some observations on the relative importance of cases in which moral hazard matters, compared to those in which it does not. 3. Most of the Zeckhauser Model in a picture The classic statement of optimal coinsurance in health insurance was provided by Zeckhauser in 1970, in a paper that emphasized the desirability of departing from uniform cost sharing (ignoring the administrative costs of doing so) in cases in

2 In principle, any income effects on the valuation of marginal benefit in different health states arising from the combination of insurance premiums and benefit payments might matter. However, we will assume that all agents in the model have at least catastrophic coverage so there should not be large income effects.

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Fig. 1. Full information demand curves and optimal coinsurance rates, illness A and illness B.

which care for different illnesses provided different configurations of marginal benefit. Fig. 1 shows two “expected demand” or marginal benefit curves for two different illnesses (each with potentially varying levels of severity), A and B, for a representative consumer.3 The expected demand curve for a particular type of illness with varying levels of severity is obtained by multiplying the demand curve for each severity level by the probability of that level occurring. We assume that the unit price of care is normalized to $1, and that the initial coinsurance is set at the uniform level C1 . Now suppose we consider lowering the coinsurance rate to C2 . Because this reduction triggers a larger increase in quantity for illness A than for illness B, it is clear that the addition to welfare cost is larger for illness A (at MWCA ) than for illness B (at MWCB ). Lowering the coinsurance rate will also increase the risk protection provided by the insurance, and the consumer will also place a positive value on this. Assume that the variance of medical expenses is the same for both illnesses, so that the marginal risk premium associated with the increase in coverage is the same for both. The initial coinsurance rate will be the optimal rate if the increase in welfare cost just equals the marginal risk premium. It is clear that if this equality was satisfied for illness B, it will not be satisfied for illness A. The optimal coinsurance rate for illness A, the more price responsive demand or marginal benefit curve, will be higher than that for illness B. The reason why this is so is because the higher marginal welfare cost for A compromises the consumer’s desire for risk protection; alternatively, we could say that B can be better covered in terms of risk protection because an increase in coverage causes less of an increase in use of care whose marginal benefit is less than its marginal cost. This is the standard economic model of optimal coinsurance; the moral hazard approach is intended to specify coinsurance that will lead to the second-best optimal quantity of care for a population whose individual members’ precise illness levels are not known to any insurer. We specify the value-based cost sharing approach as one in which, among a variety of influences, the true marginal benefit curve for different quantities of some treatment is used to determine the optimal level of cost sharing. Would the optimal coinsurance rate determined by accurate physician and patient knowledge of marginal benefit correspond to the coinsurance rate that is optimal based on cost-benefit analysis? The answer is obviously affirmative: at the optimal coinsurance rate, the health benefit from an incremental change in that rate plus the risk reduction benefit equals the marginal premium cost associated with this change in coverage. That premium grows both because cost sharing is reduced at any given level of expense and because the expense level grows when coinsurance is reduced. Note, however, that this formulation adds to the evidence-based marginal medical benefit model consideration of the value of additional risk protection, something usually lacking in evidence of clinical effectiveness. For example, at uniform coinsurance C1 in Fig. 1 and the equilibrium quantity associated with it, the marginal effectiveness of care (per dollar of cost) is the same for both illnesses when the consumer is in well-informed equilibrium; cost-effectiveness will not differ. Care may not have the same average effectiveness (per dollar of cost) across illnesses under uniform coinsurance, but it will have the same marginal effectiveness. Moreover, at coinsurance rates which are not trivially low, informed patients will not be consuming care of “little or no proven health benefit.” If the coinsurance rates are set to optimally deal with moral hazard, however, the marginal clinical cost-effectiveness ratios will differ across these two illnesses. With higher coinsurance rates for illness A, its marginal cost-effectiveness ratio will be lower than that for care for illness B. But this divergence in cost-effectiveness ratios will be optimal as far as consumers are concerned, since lowering the coinsurance rate for A would cause a larger welfare cost than for illness B; cost-effectiveness information alone is not sufficient to specify optimal insurance coverage. One might also informally look at the average cost-effectiveness ratios for care for illnesses A and B; with the demand curves as drawn the ratio is higher for illness B than for A. In this example it would be correct to say that the optimal

3 The classic treatment of moral hazard envisions the effect of different levels of cost sharing as changing the quantity of care patients use, and we follow that approach here. There is an alternative model in which the quantity of care is predetermined (“take this pill once a day”) and the choice is which severity levels will use this quantity. If the marginal benefit from care falls as severity falls, this can also yield an expected demand curve.

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Fig. 2. Full information demand curves and optimal coinsurance rates, illness B and illness C.

coinsurance rate should, as the informal value-based criterion says, be lower for care for the illness with the higher average cost-benefit or cost-effectiveness ratio. But this correlation need not always occur. Consider the demand or marginal benefit curve for care for illness C in Fig. 2; this curve is drawn as having the same slope as the demand curve for illness B. But note that marginal benefit is higher at every quantity for illness C than for illness B. However, as can easily be seen, the marginal welfare cost that would be associated with care for this illness at C1 is the same as that for illness B. If the marginal risk premium was also the same, the optimal coinsurance rate would be the same for both illnesses even though the average benefit of (a dollar’s worth of) treatment at any quantity is higher for one illness than for the other. Of course, at the same coinsurance rate the marginal benefits per dollar would be equalized. In effect, optimality requires using a larger quantity of care in the case where it is more beneficial—but uniform coinsurance in this case would bring that about. To sum up: if patient demand curves, as influenced by physician advice, coincide with true evidence-based marginal benefit curves, the standard economic theory of moral hazard will still be correct even when value-based considerations are incorporated. A model that says that coinsurance should be determined solely by marginal (or average) evidence-based medical benefit or by the relationship between the marginal benefit and cost will, in contrast, not be correct—both because the value of risk reduction needs to be taken into account and because the shape of the marginal benefit curve (and not just its value at any quantity or its average value over some range of quantities) and the value the consumer attaches to reduction in financial risk will be relevant to efficiency. 4. Adding imperfect information Now we assume that patient-consumers do not have demand curves that map the marginal benefit curves generated by clinical evidence. Patients might not have been fully informed by their physicians, they might not have been given understandable information, or their views might be biased by prior beliefs or outside influences. Such imperfection in patient (and physician) behavior might lead patients to use demand or marginal benefit curves that either fall short of or exceed the true demand curves. In contrast to the traditional moral hazard model, where there can only be overuse of care relative to the optimum, in this case there can be underuse as well.4 There are obviously many ways in which imperfect information can affect demand or perceived marginal value curves. Assume that the full information optimal coinsurance rate is C1 in Fig. 3. Also assume that the uninformed curves have the same slopes as the true curves, but are shifted by imperfect information to the right or left. So in Fig. 3 assume that the illness A and B demand curves are shifted to the left to D A and D B ; patients underestimate marginal benefit for both illnesses at every quantity. Consider illness B. At the full information optimal level of coinsurance C1 , but with the new imperfect information demand curve, the marginal welfare cost would now be less than before, at MWC B . If we assume that the marginal risk premium remains the same, it is clear what ought to happen: the optimal level of coinsurance for both illnesses should fall. However, because demand shortfall leads to smaller quantities demanded (and therefore lower losses, relative to the perfect health state), the marginal risk premium will also be lower. It is plausible (though not guaranteed) that the marginal welfare cost will fall by more than the marginal risk premium. We will assume here that this is so.

4 The bulk of the deficiencies in clinical quality identified by McGlynn et al. (2003) represent underuse of clinically relevant therapies. Some examples of clinical underuse of particular therapies (especially in asymptomatic illnesses where benefits occur in the future) have been identified, and the exacerbation of this underuse by non-zero (or high) cost sharing has been documented (Goldman et al., 2004; Trivedi et al., 2008). However, the RAND experiment found that higher user prices reduced the use of emergency rooms to a greater extent for those with conditions for which emergency care was less necessary, and it also found that cost sharing did not appreciably affect overall health outcomes for all but low income high risk populations (even as cost sharing did reduce some process measures thought to have clinical benefit [Newhouse and the Insurance Experiment Group, 1993]). The empirical evidence thus suggests the view that there is both overuse and underuse in the health system, strengthening the case for a more nuanced use of cost sharing than at present.

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Fig. 3. Imperfect information demand curves and coinsurance rates for patients who underestimate marginal benefits, illness A and illness B.

The reason why coinsurance can be lowered if patients underestimate benefits is that doing so now causes less marginal welfare cost (e.g., MWC B ) at every quantity than in the full information case. In effect, the reason to lower coinsurance is not just to move use to a particular marginal cost-effectiveness level, but also because lower welfare cost at the margin allows insurance to do a better job at its main function, spreading risk. How far should coinsurance fall? Not by as much as would push medical care use up to the level that would have been demanded with informed demand and optimal coinsurance, or level C3 , but somewhere in between C3 and C1 , depending on the marginal risk reduction function. For illness A (in results not shown), it is easy to conclude that its optimal coinsurance rate should probably also move down. But the new coinsurance rate for illness A should remain higher than the new rate for illness B as long as imperfect information shifts both marginal benefit curves to the same extent. So, as before, marginal medical effectiveness relative to cost only provides part of the information needed to determine the appropriate value-based coinsurance rate. That rate also depends on the slope or elasticity of the marginal benefit curves, and the form of the person’s utility function for disposable income, which determines marginal risk reduction. The argument about how coinsurance should change in situations in which information imperfection leads to a demand in excess of the perfectly informed demand is symmetric. If patients, despite information provided by physicians, believe that care for some illness is more effective than it really is, cost sharing (other things equal) should be higher than if patients were convinced of the true marginal benefit curve. To be sure, the patient who wants too much to be done will generally require a physician’s prescription in order to do so, but some physicians will acquiesce in prescribing care. Coinsurance rates should rise to choke off welfare cost, but should still depend on slopes and utility functions as well as on the extent of information imperfection. Here overuse from moral hazard is amplified by imperfect information. These are both cases in which the imperfectly informed demand curves differ moderately from the true curves. It is easy to see that if the degree of information imperfection is much larger—if patients substantially underestimate the marginal benefits from the care they are prescribed and physicians have been unable to change their judgments, then cost sharing ceases to be generally useful as a device to affect the use of care. Suppose, for example, that underestimation of benefits leads to a curve far to the left like D B . Even at a zero user price, this patient so underestimates the marginal benefit for care that he or she views the marginal benefit from care as zero, even though it is in reality highly effective relative to its cost. Even negative coinsurance might be appropriate here (but it will reintroduce financial risk). This is the case apparently envisioned by many of the strongest examples of value-based cost sharing. If we add an administrative cost to raising premiums, then demand responsiveness in the form of the Pauly–Held “benign moral hazard” now may be relevant. If administrative costs keep premiums below the level needed to provide coverage of the costs of all such “high ignorance” services, and it was therefore necessary to decide which services should get a value-based reduction in its coinsurance, paradoxically it may now be the more price responsive service that should get the cost reduction, since lowering its cost sharing will have a larger effect in terms of moving it closer to the ideal level than would be the case for a less responsively demanded service. That is, if patient ignorance resulting in underestimation of benefits from care in a given setting is severe enough, the direct relationship between price responsiveness and optimal cost sharing should be reversed. 5. Pay to improve information or move to value-based insurance? Given the optimality and equivalence properties associated with the fully informed demand or marginal benefit curve, it would seem that an ideal remedy when misinformed patients are not using care in accordance with its marginal benefit would be to provide information to all misinformed consumers to move their beliefs closer to those curves, if the cost of

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doing so is low. Indeed, the availability of web-based information to inform patient demands is a centerpiece of most Health Savings Account/Consumer-Directed health plans, which carry high levels of cost sharing. An obvious advantage of value-based coinsurance is that it may entail lower real resource costs than would communicating information. For example, rather than talk patients into taking their medicine on which they owe a copayment, it might be better to lower the copayment. However, there is more to the argument than this. Paradoxically, more information, even at zero resource cost, may not be desirable when consumers undervalue care. Suppose that patient demand falls somewhat short of the truth as in the “primed” curves. Then, as already noted, coinsurance rates can be lowered and more protection against risk provided. The quantity can be driven to a more efficient level without spending any real resources informing patients. Indeed, even if information had zero cost, it could be better to lower cost sharing rather than provide information. That is, for illnesses or types of care where people tend to underuse because of imperfect information, that imperfect information itself reduces the effects of moral hazard and so should not be changed. It is salutary ignorance. In the region of underuse even at zero coinsurance, the story changes. For a service whose demand is not very price responsive and therefore with financial incentives ineffective, the only real option is information. But for the more price responsive services, cutting coinsurance to zero should be the first line of attack. The case of misinformed overuse is different from any of these cases. In the case of overuse, if information is not very costly, it might actually be better to provide information to shrink demand rather than deter overuse by exposing people to higher coinsurance and more risk. That is, it is wasteful to tell people about underuse as long as demand shortfalls relative to the truth are not so large that they cannot be moved to the optimal level by low or zero cost sharing. (Negative cost sharing also adds risk.) In contrast, it may be worthwhile to counsel patients away from overuse, rather than force them to put more skin in the game. Leave underusers in the dark, but inform overusers. 6. A classic numerical example To illustrate the effect of patient misperception on the optimal coinsurance rate, we present a numerical example based on the extended illustration of the optimal coinsurance rate in Phelps’ textbook, Health Economics (2003). (For a more general set of simulations, see Jack and Sheiner, 1997.) Phelps discusses the impact of insurance coverage with an assumed 20% coinsurance rate for a hospitalization insurance that provides coverage for the demand curves and loss probabilities indicated in Table 1 (p. 362); the unit price of a hospital day is $500. He also assumes that consumers have absolute Arrow–Pratt risk aversion parameter r = −0.0003. He shows the total welfare cost of this level of coverage (assuming linear demand curves), compared to no coverage, and the net risk premium for that level of coverage, assuming that insurance has a 10% loading. The main point he makes is that the total welfare cost, at $80, is less than the net risk premium, of $136, so the person would be better off buying insurance with 20% coinsurance than remaining uninsured. Table 2 shows Phelps’ example in the C = 0.2 line. (While Phelps provides a formula for calculating the optimal coinsurance rate, his formula assumes a constant elasticity – rather than a linear – demand curve, and so is not applied to this problem.) To illustrate the influences that determine the optimal coinsurance rate in a linear demand curve example, we first assume that the demand curves are correct reflections of the marginal benefit from inpatient care, and show what the optimal coinsurance rate is. Then we assume that patient information imperfection leads to a particular shift of the demand curve, and show how the coinsurance rate should be altered to compensate. Table 2 shows the total and marginal welfare cost, and the total and marginal risk premiums, for the insurance policy and demand curves in Phelps’ example. At 100% coinsurance, the marginal welfare cost is greater than the marginal risk premium. As coinsurance is reduced, the former falls and the latter rises, and the two become equal at a coinsurance rate of about 0.46. It turns out that, while insurance with C = 0.2 is better than no insurance, 20% coinsurance is not the optimum level. Now we add imperfect information which shifts backwards the demand curve. Such imperfect information will shift back the marginal welfare cost curve, but it will also shift back the marginal risk premium curve. (In the example, and in most plausible cases, the downward shift in the former is greater than the backward shift in the latter, so the optimal coinsurance rate falls. As noted earlier, it would be possible to think of utility functions where this need not occur, but they would be unusual.) Table 3 shows the comparable risk premia and welfare costs for a demand curve shifted backwards by imperfect information by 0.5 days in both illness events. The welfare cost at any coinsurance rate c* along the new demand curve is now the welfare cost from the original demand curve at c + 0.4. For example, the welfare cost at c* = 0.2 is the same as the welfare cost at c = 0.6, or 20. The marginal welfare cost at c* = 0.2 is now 4.7, not the 9.7 as in the full information case. In this example, it turns out that the imperfectly informed optimal coinsurance rate is between 0.3 and 0.35, substantially less than the perfect information optimal coinsurance rate of 0.46. This optimal rate is also greater than the coinsurance rate which, given imperfect information, would produce the same level of use as at the no-ignorance optimal rate. That is, adding imperfect information does more than just change the optimal coinsurance rate; it also changes the optimal rate of use.
Table 1 Parameter values for a detailed calculation of welfare loss Probability 0.3 0.1 Uninsured demand 3.0 8.0 Insured demand 4.0 9.0

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Table 2 Replication of Phelps example with 10% loading total and marginal welfare cost, total and marginal risk premium, and optimal coinsurance rate C 1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 Total welfare cost 0.0 0.3 1.3 2.8 5.0 7.8 11.3 15.3 20.0 25.3 31.3 37.8 45.0 52.8 61.3 70.3 80.0 90.3 101.3 112.8 125.0 Marginal welfare cost 0 0.3 0.9 1.6 2.2 2.8 3.4 4.1 4.7 5.3 5.9 6.6 7.2 7.8 8.4 9.1 9.7 10.3 10.9 11.6 12.2 Marginal risk premium 0.0 14.0 27.6 40.7 53.2 65.1 76.3 86.8 96.5 105.2 113.1 119.9 125.6 130.2 133.5 135.5 136.1 135.3 132.8 128.7 122.9 Total risk premium 0.0 14.0 13.6 13.1 12.5 11.9 11.2 10.5 9.7 8.8 7.8 6.8 5.7 4.6 3.3 2.0 0.6 −0.9 −2.4 −4.1 −5.8 Net gain 0.0 13.7 26.4 37.9 48.2 57.3 65.1 71.5 76.5 79.9 81.8 82.1 80.6 77.4 72.3 65.2 56.1 45.0 31.6 15.9 −2.1

Net gain is maximized at C = 0.46.

A forward shift in the demand curve because imperfect information leads the patient to over-estimate marginal benefit is symmetric. It leads to a higher coinsurance rate, to dampen the additional overuse. There is, however, a potential problem in simply saying that the employer or insurer should offer the plan with the coinsurance rate that optimally deals with patient information imperfection. That advice may be important to a benevolent regulator or planner designing health insurance. But private insurance design may not fit this case. In the backwards shift example, misinformed underusing consumers who judge insurance relative to their own perceived marginal benefit curve will not find a policy with a coinsurance rate of 0.33 for that illness to be the one they find most attractive. Rather, if they judge coverage using their own anticipated demand curves (and the insurance premiums associated with those curves), they would prefer a coinsurance rate of 0.55. It could even happen (though it does not happen in this example) that misinformed consumers could think themselves better off with no insurance at all than with expensive insurance that tricks them into consuming what will turn out to be the right quantity, but is ex ante perceived to be of less value relative to its cost. The

Table 3 Re-calculation of optimal coinsurance rate, misinformation shifts in demand by −0.5 days marginal and total welfare cost, marginal and total risk premium, and optimal coinsurance rate With 10% loading fee C 1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 Moral hazard welfare cost N/A N/A N/A N/A N/A N/A N/A N/A 0.0 0.3 1.3 2.8 5.0 7.8 11.3 15.3 20.0 25.3 31.3 37.8 45.0 Marginal welfare cost N/A N/A N/A N/A N/A N/A N/A N/A 0.0 0.3 0.9 1.6 2.2 2.8 3.4 4.1 4.7 5.3 5.9 6.6 7.2 Risk premium gain 0.0 11.5 22.7 33.4 43.7 53.5 62.6 71.2 79.0 86.2 92.5 97.9 102.5 106.0 108.5 109.8 109.9 108.8 106.3 102.3 96.9 Marginal risk premium 0.0 11.5 11.2 10.7 10.3 9.7 9.2 8.5 7.9 7.1 6.3 5.5 4.5 3.5 2.5 1.3 0.1 −1.1 −2.5 −3.9 −5.5 Net gain N/A N/A N/A N/A N/A N/A N/A N/A 79.0 85.9 91.2 95.1 97.5 98.2 97.2 94.5 89.9 83.5 75.0 64.5 51.9

Net gain is maximized at C = 0.33.

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employer or insurer could offer only the optimal plan, and refuse to offer plans employees–consumers think they like better, but doing so might be difficult. If consumers could be convinced that their own marginal benefit curves are wrong, they might go along with this paternalism—but they might not be easy to convince or, if convinced, might then change their demand curves back closer to the true ones requiring a higher optimal coinsurance rate and sacrificing risk protection. Whether consumers would perceive insurance with value-based cost sharing as providing them with more value than insurance designed around their actual behavior is uncertain. 7. One final illustration to show that demand responsiveness matters We now make demand more price responsive, setting quantities at 20% coinsurance at 5 and 10, respectively. As might be expected, the optimal coinsurance rate under perfect information by similar calculations (not shown) now is higher, at 0.64, compared to the 0.46 for demand curves with less steep slope. Now we impose the same ignorance-induced backward shift in the demand curve of 0.5 days at every coinsurance rate. The optimal rate now becomes 0.55 (versus 0.33 in the prior less price responsive example), making the point that even with identical magnitudes of consumer misconception or non-compliance, the optimal coinsurance rate depends on the price responsiveness of demand. However, the change in the optimal coinsurance rate generated by the same amount of consumer ignorance is smaller in this more responsive case, because the size of the coinsurance reduction needed to push the quantity of care a given amount back closer to the optimum is smaller. But the optimal coinsurance rate from the consumer’s point of view is now about 0.7, so the divergence between the socially optimal and privately optimal rates grew. 8. A classification and some examples from the literature Table 4 summarizes the possible combinations of underuse (marginal benefit from care greater than marginal cost), overuse (marginal benefit less than marginal cost), and ideal use (marginal benefit approximately equal to marginal cost) that might arise, and how coinsurance should be chosen in each case. (The table is based on the assumption that observed demand curves cannot have positive slopes, so overuse at zero coverage but underuse at zero user price is ruled out). The great bulk of examples of application of value-based cost sharing in the literature are of the type described by line 1, where clinically judged underuse is severe and would exist even at zero cost sharing. As already noted, inefficiency from moral hazard cannot arise in such cases. The ideal level of coinsurance here would be zero, assuming negative prices are infeasible. Some of the cases do represent circumstances in which demand is likely to be relatively unresponsive to changes in user price at the margin (for example, Warfarin), but also cases in which demand is probably highly price responsive (for example, proton pump inhibitors) (see Braithwaite and Rosen, 2007). If there are limits to or costs of raising premiums to provide full coverage for all such “line 1” services, it may be better to focus reduced coinsurance or waiving of deductibles on those services that are more price responsive. That is, cost sharing should vary inversely with price responsiveness. Provision of information in this case could improve efficiency if the information is effective relative to its cost in moving demand curves closer to the true curves. Line 2 describes the rather rare case in which exactly zero coinsurance brings forth the appropriate amount of use. Here optimal coinsurance is zero, but provision of information would be efficiency reducing since use is optimal. Line 3 is the case we have been most concerned about in this article, where there is underuse at full price because patients underestimate marginal benefits but there is still overuse at zero price. Here, as discussed earlier, optimal cost sharing should vary directly with price responsiveness, and provision of information can also be efficiency reducing. We can modify an example suggested by Kent and Hayward (2007) to illustrate this case. They discuss the benefit from an intensive treatment regimen for two patients with diabetes, one of whom is at much higher risk than the other, and suggest that the net marginal benefit from treatment will be higher for the higher risk than for the lower risk. If the high risk patient underestimates the benefit from this treatment, he may decline it if it is not covered by insurance. But if the cost of the treatment is completely covered by insurance, the other patient with low but positive net benefit may inefficiently seek treatment.

Table 4 Relationship of user charges to demand responsiveness as a function of appropriateness of use at 100% and zero coinsurance Line no. 1. 2. 3. 4. 5. 6. Appropriateness of use at 100% coinsurance Underuse Underuse Underuse Ideal use Ideal use Overuse Appropriateness of use at zero coinsurance Underuse Ideal use Overuse Ideal use Overuse Overuse Variation in user charges with demand responsiveness Zero or inverse Zero or inverse Direct Zero Direct Direct

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Lines 4 and 5 are cases in which patients are appropriately informed; line 4 is the case of perfectly inelastic demand where coinsurance should be zero, and line 5 is the classic moral hazard case in which coinsurance should be positive and vary directly with demand responsiveness. In terms of the previous diabetes example, the now well-informed high risk demands the (net beneficial) treatment even if charged the full cost. Up to a point, lowering coinsurance would provide risk reduction benefits, but coinsurance should be positive in order to discourage the well-informed low risk from consuming the treatment. In this case, by definition more patient information will be unnecessary, since the patient already has accurate information. Finally, line 6 is the case of overestimation of marginal benefit. A typical example would be the use of a Cox-2 inhibitor for a patient with no signs of gastric bleeding (Spiegel et al., 2003), done only because the patient requests an advertised drug. Ideal coinsurance still varies directly with demand responsiveness, but now information provides positive efficiency benefits if it can shift demand downward and closer to the true marginal benefit schedule. 9. Extensions, applications, and conclusions Thus far we have implicitly been using a model in which consumers are identical in terms of the expected marginal benefit schedules for health and their levels of risk aversion. If this uniformity is not present (in either dimension), optimal coinsurance levels should differ. To the extent that people attach different values to given health outcomes or different values to avoiding financial risk, they will not prefer coinsurance rates that reflect only evidence-based medicine. Rather, their ideal coinsurance rates will depend on their tastes for risk and the value they place on health outcomes. More generally, the reason for using non-zero coinsurance is to control moral hazard. Moral hazard occurs because the insurer cannot precisely tell what illness (loss) state the person is in. When and if the evidence base for describing the marginal benefit of medical treatment associated with particular patients at different illness levels ever becomes complete, and low cost techniques for determining illness severity materialize, cost sharing might be dispensed with entirely; insurance should pay in full for efficient care and pay nothing for care whose benefit falls short of cost. However, the differences in valuations across people would mean that insurance might cover care for one person and not for another, even if their health benefits are identical. With today’s imperfect medical knowledge, information about demand responsiveness gives positive coinsurance a role to play. Moreover, if patients differ in their knowledge about marginal benefit, value-based design will be a blunt instrument compared to correcting selective information imperfections. Which kind of behavioral factors, demand responsiveness or patient misinformation, is more important will vary across illnesses and types of care. The examples used to argue for value-based coinsurance target the low hanging fruit where there is underuse even at full coverage. As long as either variation in unobserved illness severity or variation in benefit with the quantity of care yield smooth marginal benefit curves, we know there must be some patients and some services where there is underuse or appropriate use with high cost sharing and yet overuse at low cost sharing—so optimal coinsurance must be somewhere in between. In these cases, measurement or estimation of both demand responsiveness and information imperfection will be important. Types of care or illnesses with less responsive patient demand curves and patient underestimation of marginal benefit will be candidates for lower cost sharing, while care or illnesses where demand is highly price responsive and benefits are over-estimated will be ones that should have higher cost sharing. Sometimes it will be better to leave consumers ignorant ex ante in order to make them better off ex post. Even with a better information base for medicine, it will still be complex to design voluntary market insurance which is both optimal and feasible. Acknowledgement This research was supported by a grant from the Merck Foundation. References
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