Moral Hazard in Health Insurance

Moral Hazard in Health Insurance: Arrow, Pauly and the Optimal Coinsurance Rate

Kenneth Arrow identified moral hazard as a limit on medical insurance in 1963, drawing on a long-standing insurance idea: he observed that the cost of care depends on the doctor chosen and that doctor's willingness to use medical services as well as on the illness, and that widespread medical insurance increases the demand for care. Mark Pauly argued in 1968 that, because fully insured people demand care as if it were free while their premiums must still cover its cost, complete cover may not be worth buying, so in an optimal arrangement some uncertain medical expenses should not be insured. Later work turned this into the question of the optimal coinsurance rate. This page is the insurance design companion to the general moral hazard page, which covers the definition, hidden action and the basic welfare triangle. It covers Arrow's and Pauly's arguments, the two forms of moral hazard in a contract, the optimal coinsurance formula with an illustrative calculation, the RAND and Oregon evidence, Nyman's critique and design responses.

Arrow and Pauly: why complete insurance stops being optimal

Arrow's 1963 paper called the welfare case for insurance policies of all sorts overwhelming. In his formal results, if the insurer adds a fixed-percentage loading to the actuarial value, the policy a consumer most prefers pays 100 per cent of costs above a fixed deductible; if the insurer is itself risk averse, the Pareto optimal policy also includes some coinsurance above the deductible.

Moral hazard enters Arrow's paper as a practical limit: he noted that coinsurance had been introduced into major medical policies partly to meet the extra demand, and that insurance removes the incentive for patients and physicians to shop around for better prices for hospital and surgical care.

Pauly's 1968 comment put the demand response at the centre. A person whose insurance covers all costs demands care as though it had a zero price, but when buying the insurance must take account of its positive cost through the actuarially necessary premium, and so may not wish to buy cover at that premium. Pauly described a prisoners' dilemma: each insured person may recognise that excess use raises the premium, but nobody restrains their own use, because its cost is spread over the other insurance holders. He called the extra use rational economic behaviour, not moral perfidy, and concluded that even if everyone is risk averse, some uncertain medical expenses will not and should not be insured in an optimal situation. Einav and Finkelstein state the general result: when use responds to price and underlying health cannot be written into the contract, optimal insurance is incomplete, balancing risk reduction against incentives.

The two papers reach partial insurance for different reasons. In Arrow's formal analysis the deductible follows from the loading and the coinsurance from the insurer's risk aversion; in Pauly's argument partial cover is needed because insurance changes how much care is bought.

Ex ante and ex post moral hazard inside an insurance contract

The two forms differ in what a contract can condition on. Ex ante moral hazard is a change in prevention or health behaviour before illness because insurance covers part of its financial cost. Einav and Finkelstein note that it has received very little empirical attention, since the financial cost of being sick is probably not its most important cost. In the RAND experiment, paying more of their own costs did not change participants' rates of smoking or obesity.

Ex post moral hazard is the response of demand for care to the out-of-pocket price, holding health status fixed, and it is the focus of most of the literature. Einav and Finkelstein point out that the label stretches the hidden-action idea: healthcare use is observed and contractible, but the insurer cannot observe underlying health, so the problem is closer to hidden information, the kind of problem behind adverse selection at enrolment.

Optimal coinsurance: risk protection against induced demand

Phelps credits Zeckhauser's 1970 paper with the first formal model of this trade-off between risk spreading and moral hazard in choosing medical insurance. Phelps, in a 2002 working paper revisiting Zeckhauser's problem, used a second-order approximation that reduces it to two quadratic terms in the coinsurance rate.

$$ WL(C) = \frac{1}{2} r \sigma^{2} C^{2} + \frac{1}{2} (1 - C)^{2} \varepsilon \mu $$

where $WL(C)$ is the expected welfare loss per person per year at coinsurance rate $C$ (the patient's share of each unit of spending, from 0 to 1), $r$ is the coefficient of absolute risk aversion, $\sigma^{2}$ is the variance of annual medical spending without insurance, $\varepsilon$ is the absolute price elasticity of demand and $\mu$ is expected annual spending at the full price. The first term is the risk premium, which falls as $C$ falls; the second is the welfare-loss triangle from induced demand, which grows as cover rises. Setting the derivative with respect to $C$ to zero gives

$$ C^{*} = \frac{\varepsilon \mu}{\varepsilon \mu + r \sigma^{2}} $$

where $C^{*}$ is the optimal coinsurance rate and the other symbols are as above. Phelps states it in words as the moral hazard loss divided by the sum of the moral hazard loss and the risk premium.

The formula is a second-order approximation that ignores the skewness of medical spending, and it assumes straight-line demand with the elasticity measured at the full price, the same elasticity across illnesses, actuarially fair premiums and a single coinsurance rate. In Phelps's base case, using RAND elasticities and spending distributions, the optimal rate was 0.07 for hospital care, 0.96 for well care and close to 0.5 for physician care, with physician and dental rates sensitive to the risk aversion assumed. Einav and Finkelstein describe the usual US schedule of deductible, coinsurance and out-of-pocket maximum as a natural way to make the trade-off, since cover is worth most when spending is high; the cost sharing page explains such schedules.

Worked example: hospital cover and routine outpatient cover

Two services show how far the answer varies with the kind of care. All figures are illustrative, in USD per person per year, with a risk aversion coefficient $r = 0.0001$ per USD, the baseline value Phelps used.

1. Size the two terms. For hospital care the variance is 5,000 × 5,000 = 25,000,000, so the risk term is 0.0001 × 25,000,000 = 2,500, and the induced-demand term is 0.1 × 1,000 = 100. For outpatient visits the variance is 500 × 500 = 250,000, the risk term is 0.0001 × 250,000 = 25 and the induced-demand term is 0.3 × 500 = 150.

2. Find the optimal coinsurance rate. For hospital care, 100 / 2,600 = 0.038, about 4 per cent. For outpatient visits, 150 / 175 = 0.857, about 86 per cent.

3. Compare three designs. For hospital care, full insurance leaves only the induced-demand loss, 0.5 × 100 = 50, no insurance leaves only the risk premium, 0.5 × 2,500 = 1,250, and the optimal rate gives 0.5 × 2,500 × 100 / 2,600 = 48.08. For outpatient visits the three losses are 0.5 × 150 = 75, 0.5 × 25 = 12.5 and 0.5 × 25 × 150 / 175 = 10.71. Substituting $C^{*}$ into $WL(C)$ gives a minimum loss of one half of $r\sigma^{2} \varepsilon \mu / (r\sigma^{2} + \varepsilon \mu)$.

4. Interpret the result. The hospital risk is large and barely price sensitive, so near-complete cover costs little in induced use and removes a large risk premium. Outpatient spending is small and predictable but price sensitive, so a high coinsurance rate does best. The result depends on the elasticities and variances chosen and on counting all induced care as a loss, the assumption Nyman disputed.

What the RAND and Oregon experiments measure

The two large randomised studies answer different questions. Einav and Finkelstein describe the Oregon experiment as comparing insurance with no insurance, whereas the RAND Health Insurance Experiment asked whether, among insured people, the comprehensiveness of cover affects use.

Its fee-for-service arms assigned families at random to free care or to 25, 50 or 95 per cent coinsurance, with out-of-pocket spending capped. Its research brief reports that participants on the 25 per cent plan spent 20 per cent less than those with free care and those on the 95 per cent plan about 30 per cent less, and that the fall came mainly from people deciding not to start care. Aron-Dine, Einav and Finkelstein explain that the price elasticity of about −0.2 attributed to RAND comes from experimental data combined with modelling assumptions. In their example of a move from RAND's 25 per cent plan to a constant 28 per cent plan, three ways of summarising the original plan as one price predict spending falls of 18, 9 and 14 per cent, a twofold difference.

The Oregon Health Insurance Experiment followed a 2008 lottery for places in a Medicaid programme for low-income uninsured adults. Winning the lottery raised the chance of having cover in the first year by about 25 percentage points, and cover brought more use of primary, preventive and hospital care. Scaling the lottery effect up to the effect of being insured, and weighting each type of extra use by its average cost, the authors' rough calculation put the rise in annual spending from insurance at about USD 778, roughly 25 per cent above the implied control mean.

Nyman's access motive: whether induced care is a loss

The welfare-loss term in the formula treats every unit of insurance-induced care as worth less than it costs. Nyman argued in 1999 that health insurance is also bought for access to care that would otherwise be unaffordable: a USD 300,000 procedure is out of reach for a person with USD 50,000 in net worth, but insurance makes it accessible because the annual premium is a fraction of its cost. In a second paper that year he argued that Pauly's analysis overstates the welfare loss because it includes the effect of income on consumption, arising from the transfer from the healthy to the ill, and that the remaining loss can be treated as a transaction cost of insurance. In 2004 he argued that much of what is called moral hazard is efficient.

An illustrative case shows the size of the transfer. If 1 person in 1,000 needs a USD 300,000 procedure in a year, the fair premium for full cover is 300,000 / 1,000 = 300, and the person who falls ill receives a transfer of 300,000 − 300 = 299,700 from the 999 who stayed well. On Nyman's argument, care bought with that transfer is not part of the welfare-loss triangle, so the induced-demand term is too large, and, from the formula, the optimal coinsurance rate too high.

Plan design responses to moral hazard

Insurers and employers use several tools. Each sets the patient's price in a different way.

• Deductibles and coinsurance. These raise the price of each unit of care up to an out-of-pocket maximum. Einav and Finkelstein note that most spending comes from a small share of high-cost people whose spending lies where deductibles and co-payments no longer bind, so provider incentives may matter more for total spending.
• Reference pricing. The plan pays up to a set amount and the patient pays the full difference for a dearer provider. In one employer's laboratory test scheme studied by Robinson, Whaley and Brown, reference pricing was associated with a 31.9 per cent lower average price per test by the third year, relative to members of an insurer without the policy.
• Value-based insurance design. Chernew, Rosen and Fendrick proposed using cost sharing to encourage services whose clinical benefits exceed their cost, an answer to the RAND finding that cost sharing cut highly effective and less effective care in roughly equal proportions.

Using moral hazard estimates in benefit design decisions

Estimates of moral hazard enter health insurance policy analysis as predicted changes in use and spending. Two choices drive the answer.

1. The optimal rate is service-specific. As Phelps's results and the example above show, a single coinsurance rate for all care sits far from the optimum for services whose variances and elasticities differ.
2. The elasticity must match the contract. The −0.2 figure depends on modelling choices and on how a non-linear plan is reduced to one price, so costing a change in benefit design needs an elasticity defined on the same price measure.

Sources

• Aron-Dine A, Einav L, Finkelstein A. The RAND Health Insurance Experiment, three decades later. Journal of Economic Perspectives. 2013;27(1):197-222. https://doi.org/10.1257/jep.27.1.197
• Arrow KJ. Uncertainty and the welfare economics of medical care. American Economic Review. 1963;53(5):941-973.
• Chernew ME, Rosen AB, Fendrick AM. Value-based insurance design. Health Affairs. 2007;26(2):w195-w203. https://doi.org/10.1377/hlthaff.26.2.w195
• Einav L, Finkelstein A. Moral hazard in health insurance: what we know and how we know it. Journal of the European Economic Association. 2018;16(4):957-982. https://doi.org/10.1093/jeea/jvy017
• Finkelstein A, Taubman S, Wright B, et al. The Oregon Health Insurance Experiment: evidence from the first year. Quarterly Journal of Economics. 2012;127(3):1057-1106. https://doi.org/10.1093/qje/qjs020
• Nyman JA. The value of health insurance: the access motive. Journal of Health Economics. 1999;18(2):141-152.
• Nyman JA. The economics of moral hazard revisited. Journal of Health Economics. 1999;18(6):811-824.
• Nyman JA. Is 'moral hazard' inefficient? The policy implications of a new theory. Health Affairs. 2004;23(5):194-199.
• Pauly MV. The economics of moral hazard: comment. American Economic Review. 1968;58(3):531-537.
• Phelps CE. Medical insurance: risk spreading vs. moral hazard revisited. Working paper. Rochester, NY: University of Rochester; 2002.
• RAND Corporation. The Health Insurance Experiment: a classic RAND study speaks to the current health care reform debate. Research Brief RB-9174-HHS. Santa Monica, CA: RAND; 2006.
• Robinson JC, Whaley C, Brown TT. Association of reference pricing for diagnostic laboratory testing with changes in patient choices, prices, and total spending for diagnostic tests. JAMA Internal Medicine. 2016;176(9):1353-1359.
• Zeckhauser R. Medical insurance: a case study of the tradeoff between risk spreading and appropriate incentives. Journal of Economic Theory. 1970;2(1):10-26. https://doi.org/10.1016/0022-0531(70)90010-4