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  • The High-Frequency Trading Arms Race: Frequent Batch Auctions as a Market Design Response — Supplementary Data
    First published online July 23 2015 doi 10 1093 qje qjv027 Quarterly Journal of Economics November 1 2015 vol 130 no 4 1547 1621 Abstract Free Full Text HTML Free Full Text PDF Supplementary Data Search this journal Advanced Current Issue November 2015 130 4 Alert me to new issues The Journal About the journal Rights permissions We are mobile find out more Journals Career Network Click here to contact the Editorial Office Editorial Office Trina Ott Assistant Editor 1805 Cambridge Street Cambridge MA 02138 617 496 3293 qje admin editorialexpress com Published on behalf of President and Fellows of Harvard University Impact Factor 6 654 5 Yr impact factor 9 794 Editors Pol Antràs Robert J Barro Lawrence F Katz Andrei Shleifer View full editorial board Assistant Editor Trina Ott Alerting Services Email table of contents Email Advance Access CiteTrack XML RSS feed For Authors Services for authors Instructions to authors Submit now Self archiving policy for authors P56qQ0myhZIZ9qtHtIIeI0jcYDo8lVt6 true Looking for your next opportunity Looking for jobs Corporate Services What we offer Advertising sales Reprints Supplements Most Most Read The Impact of Jury Race in Criminal Trials The High Frequency Trading Arms Race Frequent Batch Auctions as a Market Design Response Where is the land of Opportunity The Geography of Intergenerational Mobility in the United States The Employment Effects of Credit Market Disruptions Firm level Evidence from the 2008 9 Financial Crisis The Real Costs of Credit Access Evidence from the Payday Lending Market View all Most Read articles Most Cited The Market for Lemons Quality Uncertainty and the Market Mechanism Job Market Signaling How Much Should We Trust Differences In Differences Estimates A Theory of Fairness Competition and Cooperation A Behavioral Model of Rational Choice View all Most Cited articles Online ISSN 1531 4650 Print ISSN 0033

    Original URL path: https://qje.oxfordjournals.org/content/130/4/1547/suppl/DC1 (2016-02-18)
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  • Behavioral Hazard in Health Insurance
    For Permissions please email journals permissions oup com Previous Next Article Table of Contents This Article The Quarterly Journal of Economics 2015 130 4 1623 1667 doi 10 1093 qje qjv029 First published online July 15 2015 Abstract Free Full Text HTML Free Full Text PDF Free Supplementary Data All Versions of this Article qjv029v1 130 4 1623 most recent Classifications Article Services Article metrics Alert me when cited Alert me if corrected Find similar articles Similar articles in Web of Science Add to my archive Download citation Request Permissions Citing Articles Load citing article information Citing articles via CrossRef Citing articles via Scopus Citing articles via Web of Science Google Scholar Articles by Baicker K Articles by Schwartzstein J Search for related content Related Content D03 Behavioral Economics Underlying Principles I12 Health Production I13 Health Insurance Public and Private I30 General I38 Government Policy Provision and Effects of Welfare Programs Load related web page information Share Email this article CiteULike Delicious Facebook Google Mendeley Twitter What s this Search this journal Advanced Current Issue November 2015 130 4 Alert me to new issues The Journal About the journal Rights permissions We are mobile find out more Journals Career Network Click here to contact the Editorial Office Editorial Office Trina Ott Assistant Editor 1805 Cambridge Street Cambridge MA 02138 617 496 3293 qje admin editorialexpress com Published on behalf of President and Fellows of Harvard University Impact Factor 6 654 5 Yr impact factor 9 794 Editors Pol Antràs Robert J Barro Lawrence F Katz Andrei Shleifer View full editorial board Assistant Editor Trina Ott Alerting Services Email table of contents Email Advance Access CiteTrack XML RSS feed For Authors Services for authors Instructions to authors Submit now Self archiving policy for authors P56qQ0myhZIZ9qtHtIIeI0jcYDo8lVt6 true Looking for your next opportunity

    Original URL path: https://qje.oxfordjournals.org/content/130/4/1623.abstract (2016-02-18)
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  • Behavioral Hazard in Health Insurance
    point reduction in mortality generates a value of 3 000 This 3 000 improvement substantially exceeds the standard model s prediction of 26 50 suggesting large negative behavioral hazard Applying the traditional moral hazard calculus in this situation would imply that people place an unrealistically low valuation on their life and health 23 For welfare calculations the theoretical analysis highlights the need to use an estimate of the marginal private health benefit in the presence of behavioral hazard As a rough back of the envelope calculation the 3 000 improvement in mortality minus the 106 increase in spending generates a surplus of 2 894 per person a gross return of 28 per dollar spent The presence of behavioral hazard thus reverses how we interpret the demand response to eliminating copayments moral hazard implies a welfare loss while behavioral hazard implies a gain that is over 30 times larger 24 Previous Section Next Section IV Implications for Optimal Copays We have seen that behavioral hazard can influence whether changing copays from existing levels is good policy This section describes some features of the optimal insurance plan when behavioral hazard is taken into account Consider again equation 5 which gives us the welfare impact of a marginal copay increase Setting this equal to 0 yields a candidate for the optimal copay To limit the number of cases we focus attention on the standard situation where some but not all sick people are treated at the optimum an optimal copay p B satisfies mml math display inline mml mrow mml msup mml mi M mml mi mml mo mml mo mml msup mml mo stretchy false mml mo mml msup mml mi p mml mi mml mi B mml mi mml msup mml mo stretchy false mml mo mml mo mml mo mml mn 0 mml mn mml mrow mml math M pB 0 and mml math display inline mml mrow mml mi M mml mi mml mo stretchy false mml mo mml msup mml mi p mml mi mml mi B mml mi mml msup mml mo stretchy false mml mo mml mo mml mo mml mn 0 mml mn mml mrow mml math M pB 0 This is true under our assumptions for example when people are not too risk averse that is when mml math display inline mml mrow mml mo mml mo mml mstyle mml mfrac mml mrow mml msup mml mi U mml mi mml mrow mml mo mml mo mml mo mml mo mml mrow mml msup mml mrow mml mrow mml msup mml mi U mml mi mml mo mml mo mml msup mml mrow mml mfrac mml mstyle mml mrow mml math U U is sufficiently small over the relevant range of C For presentational simplicity we also focus on the situation where the optimal copay is unique Defining mml math display inline mml mrow mml msup mml mi p mml mi mml mrow mml mi min mml mi mml mrow mml msup mml mo mml mo mml mi inf mml mi mml mrow mml mo stretchy true mml mo mml mrow mml mi p mml mi mml mo mml mo mml mi M mml mi mml mo stretchy false mml mo mml mi p mml mi mml mo stretchy false mml mo mml mo mml mo mml mi q mml mi mml mrow mml mo stretchy true mml mo mml mrow mml mrow mml math pmin inf p M p q to equal the lowest copay where not every sick person demands treatment and mml math display inline mml mrow mml msup mml mi p mml mi mml mrow mml mi max mml mi mml mrow mml msup mml mo mml mo mml mi sup mml mi mml mrow mml mo stretchy true mml mo mml mrow mml mi p mml mi mml mo mml mo mml mi M mml mi mml mo stretchy false mml mo mml mi p mml mi mml mo stretchy false mml mo mml mo mml mo mml mn 0 mml mn mml mrow mml mo stretchy true mml mo mml mrow mml mrow mml math pmax sup p M p 0 to equal the highest copay where some sick person demands treatment we assume the following Assumption 1 The optimal copay is unique and satisfies mml math display inline mml mrow mml msup mml mi p mml mi mml mi B mml mi mml msup mml mo mml mo mml mo stretchy false mml mo mml msup mml mi p mml mi mml mrow mml mtext min mml mtext mml mrow mml msup mml mo mml mo mml msup mml mi p mml mi mml mrow mml mtext max mml mtext mml mrow mml msup mml mo stretchy false mml mo mml mrow mml math pB pmin pmax Proposition 3 Assuming mml math display inline mml mrow mml msup mml mi p mml mi mml mi B mml mi mml msup mml mo mml mo mml mn 0 mml mn mml mrow mml math pB 0 the optimal copay satisfies mml math display block mml mrow mml mfrac mml mrow mml mi c mml mi mml mo mml mo mml msup mml mi p mml mi mml mi B mml mi mml msup mml mrow mml mrow mml msup mml mi p mml mi mml mi B mml mi mml msup mml mrow mml mfrac mml mo mml mo mml mfrac mml mi I mml mi mml mi η mml mi mml mfrac mml mo mml mo mml mfrac mml mrow mml msup mml mi ε mml mi mml mrow mml mtext avg mml mtext mml mrow mml msup mml mrow mml mrow mml msup mml mi p mml mi mml mi B mml mi mml msup mml mrow mml mfrac mml mo mml mo mml mrow mml math c pBpB Iη εavgpB 6 where mml math display inline mml mrow mml mi η mml mi mml mo mml mo mml mo mml mo mml mstyle mml mfrac mml mrow mml msup mml mi M mml mi mml mo mml mo mml msup mml mo stretchy false mml mo mml mi p mml mi mml mo stretchy false mml mo mml mi p mml mi mml mrow mml mrow mml mi M mml mi mml mo stretchy false mml mo mml mi p mml mi mml mo stretchy false mml mo mml mrow mml mfrac mml mstyle mml mrow mml math η M p pM p equals the elasticity of demand for treatment I the insurance value and mml math display inline mml mrow mml msup mml mi ε mml mi mml mrow mml mtext avg mml mtext mml mrow mml msup mml mrow mml math εavg the average size of marginal behavioral hazard all evaluated at p B Proposition 3 expresses the optimal copay in terms of reduced form elasticities as well as the degree of behavioral hazard and the curvature of the utility function It says that fixing insurance value and the cost of treatment the optimal copay is increasing in the demand elasticity and the degree to which behavioral hazard is positive This simple formula illustrates a number of ways behavioral hazard fundamentally changes how we think about optimal copays 1 Optimal Copays Can Substantially Deviate from Cost Even When Coverage Generates Little or No Insurance Value A simple implication of equation 6 is that health insurance can provide more than financial protection it can also improve health care efficiency Even when individuals are risk neutral and there is no value to financial insurance I 0 equation 6 indicates that the optimal copay can differ from cost to provide insurees with incentives for more efficient utilization decisions In fact when consumers are risk neutral the extent of behavioral hazard at the margin fully determines the optimal copay In this case the optimal copay formula reduces to mml math display inline mml mrow mml msup mml mi p mml mi mml mi B mml mi mml msup mml mo mml mo mml mi c mml mi mml mo mml mo mml msup mml mi ε mml mi mml mrow mml mtext avg mml mtext mml mrow mml msup mml mo stretchy false mml mo mml msup mml mi p mml mi mml mi B mml mi mml msup mml mo stretchy false mml mo mml mrow mml math pB c εavg pB the optimal copay acts like a Pigouvian tax to induce marginal insurees to fully internalize their internality Unlike in the standard model there is no clear incentive insurance trade off 2 Optimal Copays Can Be Extreme It can be optimal to fully cover treatments that are ineffective for some insurees or not to cover treatments that benefit insurees A related implication is that optimal copays can be more extreme than in a model with only moral hazard Absent behavioral hazard the optimal copay lies strictly between the value that provides full insurance i e the value that makes I p 0 and cost when insurees are risk averse and demand is elastic Intuitively without behavioral hazard slightly raising the copay from the amount that provides full insurance has only a second order cost through reducing insurance value but a first order benefit through controlling moral hazard slightly reducing the copay from cost has a second order cost through inducing moral hazard but a first order benefit through increasing insurance value In the standard model it cannot be optimal to deny coverage of treatments that benefit some risk averse individuals and it cannot be optimal to fully cover or subsidize treatments when people are price sensitive at the full coverage copay Behavioral hazard alters these prescriptions When behavioral hazard is sufficiently positive the optimal copay can be above cost even when the individual is risk averse it can be good to let insurers discriminate against certain treatments as suggested by Panel B of Figure III When behavioral hazard is sufficiently negative the optimal copay can be below the level that provides full financial protection even if demand is price sensitive at this copay paying people to get treated can be optimal as illustrated in Panel A of Figure III In this spirit some insurers have begun to experiment with paying patients to take their medications Belluck 2010 Volpp et al 2009 3 Optimal Copays Depend on Health Value Not Just Demand Elasticities Optimal copays likely vary more across treatments than in a model with only moral hazard The standard model says that fixing insurance value copays should be higher the larger the cost and elasticity of demand Zeckhauser 1970 as can be seen from plugging mml math display inline mml mrow mml msup mml mi ε mml mi mml mrow mml mtext avg mml mtext mml mrow mml msup mml mo mml mo mml mn 0 mml mn mml mrow mml math εavg 0 into equation 6 That model suggests for example that copays should be lower for emergency care where demand is less elastic than for regular doctor s office visits where it is presumably more price sensitive However it also leads to some counterintuitive prescriptions it suggests that copays should be similar across broad categories of drugs with similar price elasticities even if they have very different efficacies Behavioral hazard alters these prescriptions as well To see this make the approximation mml math display inline mml mrow mml mi ε mml mi mml mo stretchy false mml mo mml mi s mml mi mml mo mml mo mml mi θ mml mi mml mo stretchy false mml mo mml mo mml mo mml mi ε mml mi mml mo mml mo mml mo stretchy false mml mo mml mi s mml mi mml mo mml mo mml mi θ mml mi mml mo stretchy false mml mo mml mrow mml mo mml mo mml mrow mml mo mml mo mml mrow mml mo mml mo mml mrow mml mo stretchy false mml mo mml mi s mml mi mml mo mml mo mml mi θ mml mi mml mo stretchy false mml mo mml mrow mml math ε s θ ε s θ s θ and plug mml math display inline mml mrow mml msup mml mi ε mml mi mml mrow mml mtext avg mml mtext mml mrow mml msup mml mo stretchy false mml mo mml mi p mml mi mml mo stretchy false mml mo mml mo mml mo mml mi p mml mi mml mo mml mo mml mstyle mml mfrac mml mrow mml mi H mml mi mml mo mml mo mml mo stretchy false mml mo mml mi p mml mi mml mo stretchy false mml mo mml mrow mml mrow mml msup mml mi M mml mi mml mo mml mo mml msup mml mo stretchy false mml mo mml mi p mml mi mml mo stretchy false mml mo mml mrow mml mfrac mml mstyle mml mrow mml math εavg p p H p M p Proposition 2 establishes that the second approximation follows from the first into equation 6 yielding mml math display block mml mrow mml mfrac mml mrow mml mi c mml mi mml mo mml mo mml msup mml mi p mml mi mml mi B mml mi mml msup mml mrow mml mrow mml msup mml mi p mml mi mml mi B mml mi mml msup mml mrow mml mfrac mml mo mml mo mml mfrac mml mi I mml mi mml mi η mml mi mml mfrac mml mo mml mo mml mrow mml mo stretchy true mml mo mml mrow mml mfrac mml mrow mml mi H mml mi mml mo mml mo mml mo stretchy false mml mo mml msup mml mi p mml mi mml mi B mml mi mml msup mml mo stretchy false mml mo mml mrow mml mrow mml msup mml mi p mml mi mml mi B mml mi mml msup mml mi M mml mi mml mo mml mo mml mo stretchy false mml mo mml msup mml mi p mml mi mml mi B mml mi mml msup mml mo stretchy false mml mo mml mrow mml mfrac mml mo mml mo mml mn 1 mml mn mml mrow mml mo stretchy true mml mo mml mrow mml mo mml mo mml mrow mml math c pBpB Iη H pB pBM pB 1 7 From equation 7 all else equal copays should be decreasing in the net return to the last private dollar spent on treatment mml math display inline mml mrow mml mstyle mml mfrac mml mrow mml mrow mml mo stretchy true mml mo mml mrow mml mi H mml mi mml mo mml mo mml mo stretchy false mml mo mml mi p mml mi mml mo stretchy false mml mo mml mrow mml mo stretchy true mml mo mml mrow mml mrow mml mrow mml mi p mml mi mml mrow mml mo stretchy true mml mo mml mrow mml mi M mml mi mml mo mml mo mml mo stretchy false mml mo mml mi p mml mi mml mo stretchy false mml mo mml mrow mml mo stretchy true mml mo mml mrow mml mrow mml mfrac mml mstyle mml mo mml mo mml mn 1 mml mn mml mrow mml math H p p M p 1 so the value of treatment now enters into the determination of the optimal copay insofar as it influences mml math display inline mml mrow mml mi H mml mi mml mo mml mo mml mo stretchy false mml mo mml mi p mml mi mml mo stretchy false mml mo mml mrow mml math H p For a given demand response to copays copays should be lower when this demand response has greater adverse effects on health This connects to value based insurance design proposals Chernew Rosen and Fendrick 2007 where all else equal cost sharing should be lower for higher value care While the marginal rather than the average value of care appears in equation 7 knowledge of the average health value of care can provide a useful signal about the marginal health value Consider a case where the demand curve slopes down only because of behavioral hazard mml math display inline mml mrow mml mi V mml mi mml mi a mml mi mml mi r mml mi mml mo stretchy false mml mo mml mi ε mml mi mml mo stretchy false mml mo mml mo mml mo mml mn 0 mml mn mml mrow mml math Var ε 0 but Var b 0 Then the marginal individual at any copay where demand is price sensitive must have a marginal health value equal to the average value b which also can be expressed as mml math display inline mml mrow mml mstyle mml mfrac mml mrow mml mi H mml mi mml mo stretchy false mml mo mml msup mml mi p mml mi mml mrow mml mi min mml mi mml mo mml mo mml mrow mml msup mml mo stretchy false mml mo mml mo mml mo mml mi H mml mi mml mo stretchy false mml mo mml msup mml mi p mml mi mml mrow mml mi max mml mi mml mo mml mo mml mrow mml msup mml mo stretchy false mml mo mml mrow mml mrow mml mi M mml mi mml mo stretchy false mml mo mml msup mml mi p mml mi mml mrow mml mi min mml mi mml mo mml mo mml mrow mml msup mml mo stretchy false mml mo mml mo mml mo mml mi M mml mi mml mo stretchy false mml mo mml msup mml mi p mml mi mml mrow mml mi max mml mi mml mo mml mo mml mrow mml msup mml mo stretchy false mml mo mml mrow mml mfrac mml mstyle mml mrow mml math H pmin H pmax M pmin M pmax Recall that mml math display inline mml mrow mml msup mml mi p mml mi mml mrow mml mi min mml mi mml mo mml mo mml mrow mml msup mml mrow mml math pmin equals the lowest copay where some of the sick do not demand treatment and mml math display inline mml mrow mml msup mml mi p mml mi mml mrow mml mi max mml mi mml mo mml mo mml mrow mml msup mml mrow mml math pmax equals the largest copay where some people still demand treatment Generalizing this example to allow for heterogeneity in private benefits in addition to heterogeneity in behavioral hazard yields the following result Proposition 4 Assume U is linear mml math display inline mml mrow mml msup mml mi M mml mi mml mo mml mo mml msup mml mo stretchy false mml mo mml mi c mml mi mml mo stretchy false mml mo mml mo mml mo mml mn 0 mml mn mml mrow mml math M c 0 and the distribution mml math display inline mml mrow mml mi Q mml mi mml mo stretchy false mml mo mml mi s mml mi mml mo mml mo mml mi θ mml mi mml mo mml mo mml mi γ mml mi mml mo stretchy false mml mo mml mrow mml math Q s θ γ is such that mml math display inline mml mrow mml mi b mml mi mml mo stretchy false mml mo mml mi s mml mi mml mo mml mo mml mi γ mml mi mml mo stretchy false mml mo mml mrow mml math b s γ and mml math display inline mml mrow mml mi ε mml mi mml mo stretchy false mml mo mml mi s mml mi mml mo mml mo mml mi θ mml mi mml mo stretchy false mml mo mml mrow mml math ε s θ are independently distributed according to symmetric and quasi concave densities with mml math display inline mml mrow mml mi V mml mi mml mi a mml mi mml mi r mml mi mml mo stretchy false mml mo mml mi ε mml mi mml mo stretchy false mml mo mml mo mml mo mml mn 0 mml mn mml mrow mml math Var ε 0 mml math display inline mml mrow mml msup mml mi p mml mi mml mi B mml mi mml msup mml mo mml mo mml mi c mml mi mml mrow mml math pB c if mml math display inline mml mrow mml mfrac mml mrow mml mi H mml mi mml mo stretchy false mml mo mml msup mml mi p mml mi mml mrow mml mi min mml mi mml mo mml mo mml mrow mml msup mml mo stretchy false mml mo mml mo mml mo mml mi H mml mi mml mo stretchy false mml mo mml msup mml mi p mml mi mml mrow mml mi max mml mi mml mo mml mo mml mrow mml msup mml mo stretchy false mml mo mml mrow mml mrow mml mi M mml mi mml mo stretchy false mml mo mml msup mml mi p mml mi mml mrow mml mi min mml mi mml mo mml mo mml mrow mml msup mml mo stretchy false mml mo mml mo mml mo mml mi M mml mi mml mo stretchy false mml mo mml msup mml mi p mml mi mml mrow mml mi max mml mi mml mo mml mo mml mrow mml msup mml mo stretchy false mml mo mml mrow mml mfrac mml mo mml mo mml mi c mml mi mml mrow mml math H pmin H pmax M pmin M pmax c and mml math display inline mml mrow mml mi mathvariant double struck E mml mi mml mo stretchy false mml mo mml mi ε mml mi mml mo stretchy false mml mo mml mo mml mo mml mn 0 mml mn mml mrow mml math E ε 0 mml math display inline mml mrow mml msup mml mi p mml mi mml mi B mml mi mml msup mml mo mml mo mml mi c mml mi mml mrow mml math pB c if mml math display inline mml mrow mml mfrac mml mrow mml mi H mml mi mml mo stretchy false mml mo mml msup mml mi p mml mi mml mrow mml mi min mml mi mml mo mml mo mml mrow mml msup mml mo stretchy false mml mo mml mo mml mo mml mi H mml mi mml mo stretchy false mml mo mml msup mml mi p mml mi mml mrow mml mi max mml mi mml mo mml mo mml mrow mml msup mml mo stretchy false mml mo mml mrow mml mrow mml mi M mml mi mml mo stretchy false mml mo mml msup mml mi p mml mi mml mrow mml mi min mml mi mml mo mml mo mml mrow mml msup mml mo stretchy false mml mo mml mo mml mo mml mi M mml mi mml mo stretchy false mml mo mml msup mml mi p mml mi mml mrow mml mi max mml mi mml mo mml mo mml mrow mml msup mml mo stretchy false mml mo mml mrow mml mfrac mml mo mml mo mml mi c mml mi mml mrow mml math H pmin H pmax M pmin M pmax c and mml math display inline mml mrow mml mi mathvariant double struck E mml mi mml mo stretchy false mml mo mml mi ε mml mi mml mo stretchy false mml mo mml mo mml mo mml mn 0 mml mn mml mrow mml math E ε 0 mml math display inline mml mrow mml msup mml mi p mml mi mml mi B mml mi mml msup mml mo mml mo mml mi c mml mi mml mrow mml math pB c if mml math display inline mml mrow mml mfrac mml mrow mml mi H mml mi mml mo stretchy false mml mo mml msup mml mi p mml mi mml mrow mml mi min mml mi mml mo mml mo mml mrow mml msup mml mo stretchy false mml mo mml mo mml mo mml mi H mml mi mml mo stretchy false mml mo mml msup mml mi p mml mi mml mrow mml mi max mml mi mml mo mml mo mml mrow mml msup mml mo stretchy false mml mo mml mrow mml mrow mml mi M mml mi mml mo stretchy false mml mo mml msup mml mi p mml mi mml mrow mml mi min mml mi mml mo mml mo mml mrow mml msup mml mo stretchy false mml mo mml mo mml mo mml mi M mml mi mml mo stretchy false mml mo mml msup mml mi p mml mi mml mrow mml mi max mml mi mml mo mml mo mml mrow mml msup mml mo stretchy false mml mo mml mrow mml mfrac mml mo mml mo mml mi c mml mi mml mrow mml math H pmin H pmax M pmin M pmax c and mml math display inline mml mrow mml mi mathvariant double struck E mml mi mml mo stretchy false mml mo mml mi ε mml mi mml mo stretchy false mml mo mml mo mml mo mml mn 0 mml mn mml mrow mml math E ε 0 This shows that with behavioral hazard the average value of care provides a useful signal for the optimal copay So long as there is some variability in behavioral hazard across people and behavioral hazard does not systematically push people to privately overuse high value treatments or privately underuse low value treatments then the optimal copay is above cost whenever the treatment is not socially beneficial on average and is below cost whenever the treatment is socially beneficial on average Take the case where mml math display inline mml mrow mml mi mathvariant double struck E mml mi mml mo stretchy false mml mo mml mi ε mml mi mml mo stretchy false mml mo mml mo mml mo mml mn 0 mml mn mml mrow mml math E ε 0 The average value of care signals the expected direction of behavioral hazard at the margin since as is familiar from standard signal extraction arguments the marginal patient s expected valuation lies between the copay his revealed valuation if there is no behavioral hazard and the unconditional average valuation his valuation if being marginal was independent of true valuation 25 The marginal degree of behavioral hazard is then negative at copays below the expected value of treatment and positive at copays above the expected value of treatment Returning to the example where Var b 0 the marginal degree of behavioral hazard satisfies mml math display inline mml mrow mml mi b mml mi mml mo mml mo mml mi ε mml mi mml mo mml mo mml mi p mml mi mml mo mml mo mml mi ε mml mi mml mo mml mo mml mi p mml mi mml mo mml mo mml mi b mml mi mml mrow mml math b ε p ε p b which clearly is negative if and only if the copay is below the expected value of treatment These results suggest that optimal copays should depend on the value of treatment in addition to the demand response For example we might expect that we should have high copays for procedures that are not recommended but sought by the patient nonetheless and low copays in situations where people have asymptomatic chronic diseases for which there are effective drug regimens While advocated by some health researchers for example Chernew Rosen and Fendrick 2007 such differential cost sharing is uncommon in practice we return to some possible reasons in Section VI 26 Previous Section Next Section V The Pitfalls of Ignoring Behavioral Hazard Behavioral hazard modifies the central insights of the standard model The goal of this section is to give a sense of how important it is to take behavioral hazard into account how wrong would the analyst be if he ignored behavioral hazard While the optimal copay p B satisfies mml math display inline mml mrow mml msup mml mover accent true mml mi W mml mi mml mo mml mo mml mover mml mo mml mo mml msup mml mo stretchy false mml mo mml msup mml mi p mml mi mml mi B mml mi mml msup mml mo stretchy false mml mo mml mo mml mo mml mn 0 mml mn mml mrow mml math W pB 0 where mml math display inline mml mrow mml msup mml mover accent true mml mi W mml mi mml mo mml mo mml mover mml mo mml mo mml msup mml mrow mml math W is defined in equation 5 a candidate for the neoclassical optimal copay p N satisfies the following condition Definition 1 p N is a candidate for the neoclassical optimal copay when mml math display block mml mrow mml mstyle mml mfrac mml mrow mml mo mml mo mml msup mml mover accent true mml mi W mml mi mml mo mml mo mml mover mml mi N mml mi mml msup mml mo stretchy false mml mo mml msup mml mi p mml mi mml mi N mml mi mml msup mml mo stretchy false mml mo mml mrow mml mrow mml mo mml mo mml mi p mml mi mml mrow mml mfrac mml mstyle mml mo mml mo mml mo mml mo mml msup mml mi M mml mi mml mo mml mo mml msup mml mo stretchy false mml mo mml mi p mml mi mml mo stretchy false mml mo mml mo mml mo mml mo stretchy false mml mo mml mi c mml mi mml mo mml mo mml mi p mml mi mml mo stretchy false mml mo mml mo mml mo mml mi I mml mi mml mo stretchy false mml mo mml mi p mml mi mml mo stretchy false mml mo mml mo mml mo mml mi M mml mi mml mo stretchy false mml mo mml mi p mml mi mml mo stretchy false mml mo mml mo mml mo mml mn 0 mml mn mml mrow mml math W N pN p M p c p I p M p 0 and i mml math display inline mml mrow mml mstyle mml mfrac mml mrow mml mo mml mo mml msup mml mover accent true mml mi W mml mi mml mo mml mo mml mover mml mi N mml mi mml msup mml mrow mml mrow mml mo mml mo mml mi p mml mi mml mrow mml mfrac mml mstyle mml mo mml mo mml mn 0 mml mn mml mrow mml math W N p 0 in a left neighborhood of p N ii mml math display inline mml mrow mml mstyle mml mfrac mml mrow mml mo mml mo mml msup mml mover accent true mml mi W mml mi mml mo mml mo mml mover mml mi N mml mi mml msup mml mrow mml mrow mml mo mml mo mml mi p mml mi mml mrow mml mfrac mml mstyle mml mo mml mo mml mn 0 mml mn mml mrow mml math W N p 0 in a right neighborhood of p N and iii at least one of the inequalities in i or ii is strict for some p in the relevant neighborhoods In other words p N is a copay that an analyst applying the standard model to estimates of the demand and insurance value schedules mml math display inline mml mrow mml mo stretchy false mml mo mml mi M mml mi mml mo stretchy false mml mo mml mo mml mo mml mo stretchy false mml mo mml mo mml mo mml mi I mml mi mml mo stretchy false mml mo mml mo mml mo mml mo stretchy false mml mo mml mo stretchy false mml mo mml mrow mml math M I thinks could be optimal The neoclassical optimal and true optimal copays will clearly coincide when mml math display inline mml mrow mml mi ε mml mi mml mo stretchy false mml mo mml mi s mml mi mml mo mml mo mml mi θ mml mi mml mo stretchy false mml mo mml mo mml mo mml mn 0 mml mn mml mrow mml mo mml mo mml mrow mml mo mml mo mml mrow mml mo mml mo mml mrow mml mi s mml mi mml mo mml mo mml mi θ mml mi mml mrow mml math ε s θ 0 s θ The direction of the deviation between these copays is also intuitive As established in Online Appendix A there is a welfare benefit to raising the copay from the neoclassical optimum whenever behavioral hazard is on average positive for people at the margin and there is a welfare benefit to reducing the copay from the neoclassical optimum whenever behavioral hazard is on average negative for people at the margin 27 Less obvious the deviation between the neoclassical optimal and true optimal copays can be huge Proposition 5 Suppose U is strictly concave mml math display inline mml mrow mml mi ε mml mi mml mo stretchy false mml mo mml mi s mml mi mml mo mml mo mml mi θ mml mi mml mo stretchy false mml mo mml mo mml mo mml mover accent true mml mi ε mml mi mml mo mml mo mml mover mml mo mml mo mml mi mathvariant double struck R mml mi mml mrow mml math ε s θ ε R and mml math display inline mml mrow mml mi b mml mi mml mo stretchy false mml mo mml mi s mml mi mml mo mml mo mml mi γ mml mi mml mo stretchy false mml mo mml mo mml mo mml mi s mml mi mml mrow mml mo mml mo mml mrow mml mo mml mo mml mrow mml mo mml mo mml mrow mml mo stretchy false mml mo mml mi s mml mi mml mo mml mo mml mi γ mml mi mml mo mml mo mml mi θ mml mi mml mo stretchy false mml mo mml mrow mml math b s γ s s γ θ If mml math display inline mml mover accent true mml mi ε mml mi mml mo mml mo mml mover mml math ε is sufficiently large then the neoclassical analyst believes mml math display inline mml mrow mml msup mml mi p mml mi mml mi N mml mi mml msup mml mo mml mo mml mn 0 mml mn mml mrow mml math pN 0 is a candidate for the optimal copay but the optimal copay in fact satisfies mml math display inline mml mrow mml msup mml mi p mml mi mml mi B mml mi mml msup mml mo mml mo mml mi c mml mi mml mrow mml math pB c If mml math display inline mml mover accent true mml mi ε mml mi mml mo mml mo mml mover mml math ε is sufficiently low then the neoclassical analyst believes mml math display inline mml mrow mml msup mml mi p mml mi mml mi N mml mi mml msup mml mo mml mo mml mi c mml mi mml mrow mml math pN c is a candidate for the optimal copay but the optimal copay in fact satisfies mml math display inline mml mrow mml msup mml mi p mml mi mml mi B mml mi mml msup mml mo mml mo mml mn 0 mml mn mml mrow mml math pB 0 When behavioral hazard is extreme the neoclassical optimal copay is exactly wrong the situations in which the neoclassical analyst believes that copays should be really low are precisely

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