Signature
RP = 0.5 * r * sigma^2 * C^2; MH = 0.5 * (1 - C)^2 * eps * mu; WL = RP + MH
| Inputs | Definition | Unit |
|---|---|---|
r | Pratt coefficient of absolute risk aversion, minus the ratio of the second to the first derivative of utility, above zero, in the reciprocal of the currency used for spending | per unit of currency |
sigma | Standard deviation of annual medical spending per person without insurance, zero or above; the formula uses its square, the variance | currency per person per year |
C | Share of each unit of spending paid by the patient, the policy's only choice parameter, at least 0 and at most 1 (0 is full insurance and 1 is no insurance) | proportion |
eps | Absolute value of the price elasticity of demand for the insured care, evaluated at the full price, zero or above | dimensionless |
mu | Expected annual medical spending per person when the patient pays the full price, zero or above | currency per person per year |
RP | Second-order risk premium of the share C of medical spending the patient still pays, one half of r times sigma squared times C squared | currency per person per year |
|---|---|---|
MH | Welfare-loss triangle of care induced by the price reduction 1 minus C, the form of HE-FM-MH-002 in spending units | currency per person per year |
WL | Expected welfare loss per insured person per year at coinsurance rate C, the risk premium RP plus the induced-demand loss MH | currency per person per year |
Function
Phelps welfare loss and optimal coinsurance function for medical insurance
Maps a coinsurance rate C, the coefficient of absolute risk aversion r, the standard deviation sigma and mean mu of annual medical spending without insurance, and the absolute price elasticity of demand eps to the expected welfare loss per person of a policy with that rate, and maps the same parameters to the coinsurance rate that minimises the loss. The loss is the second-order approximation of Phelps, the risk premium of the share of spending the patient still bears plus the welfare-loss triangle of insurance-induced care. The triangle is the live HE-FM-MH-002 written in spending units (HE-FM-MH-001 gives it from prices and quantities, and HE-FN-MH-001 is the general ex post moral hazard function), and HE-FM-DWL-003 gives the fall in that loss when the rate rises, from spending at the two rates (with straight-line demand it equals the triangle at the lower rate minus the triangle at the higher rate, derived). The risk premium has no live record, since HE-FM-EV-002 gives the second-order expansion only in the output's own units. The cost-sharing split of a plan with a deductible and an out-of-pocket maximum is HE-FM-AV-002, the RAND arc elasticity behind the usual elasticity estimates is HE-FM-MH-003, and the fair premium in the article's Nyman illustration is HE-FM-ACTA-001 with HE-FM-ACTA-002 at zero loadings. Notation follows the Moral Hazard in Health Insurance article.
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Implementations
Excel
Phelps welfare loss from named coinsurance and spending cells
With the coinsurance rate in PhelpsCoinsRate, the risk aversion coefficient in RiskAversionCoef, the standard deviation and mean of uninsured spending in SpendSD and MeanSpendFullPrice and the absolute elasticity in PriceElasticityAbs, the three formulas return the risk premium in RiskPremiumLoss, the induced-demand loss in MoralHazardLoss and their sum in PhelpsWelfareLoss.
=0.5*RiskAversionCoef*SpendSD^2*PhelpsCoinsRate^2; =0.5*(1-PhelpsCoinsRate)^2*PriceElasticityAbs*MeanSpendFullPrice; =RiskPremiumLoss+MoralHazardLoss
Assumptions
Second-order approximation of the risk premium in the Phelps loss
The risk term keeps only the variance term of a Taylor series expansion of expected utility about mean spending, which Phelps notes replicates Pratt's result that the risk premium is proportional to the variance. Applied to the utility of income net of the retained spending, the risk term is minus the correction of HE-FM-EV-002 (one half of the second derivative of utility times C squared times sigma squared) divided by the first derivative of utility, which converts a loss of expected utility into money (derived). Skewness and kurtosis terms are dropped. Phelps finds that for hospital care, with the skewness and kurtosis of the RAND spending data, the third and fourth order terms are of the same order of magnitude as the variance term, while for the other types of care they are vanishingly small.
Straight-line demand with one elasticity at the full price in the Phelps loss
Demand for the insured care is a straight line over the price range, the elasticity is evaluated at the full-price demand and one elasticity eps applies across illnesses, so that Phelps's expected term E(pm eta) equals minus eps times mu. The general constant-slope form of the triangle and its own assumptions are HE-FM-MH-002 and HE-AS-MH-004.
Fair premium, a single coinsurance rate and no income effect of the premium in the Phelps loss
The coinsurance rate is the only choice parameter of the policy, the loading charged by the insurer is left out, and the income-price interaction of paying the premium in advance is ignored. Phelps reports in his appendix that for all medical care, at a budget share of 0.05, that ignored term is about one fifth of the welfare triangle, and that for components of care other than hospital care the relevant budget share is on the order of 0.01.
Induced care valued by the demand curve in the Phelps loss
The second term counts every unit of insurance-induced care as worth what the demand curve says, so all of it below its cost is a loss. HE-AS-MH-001 records that Nyman's access motive and behavioural hazard reject this reading for part of the extra use, and the article notes that on Nyman's argument the induced-demand term is then too large and the optimal rate from HE-FM-MHHI-002 too high (a smaller second term lowers C_star, derived).
Worked examples
Full insurance of the illustrative hospital risk
The article's illustrative hospital care has expected spending of USD 1,000, a standard deviation of USD 5,000 and an absolute elasticity of 0.1, with r = 0.0001 per USD. With full insurance (C = 0) the risk premium is zero and only the induced-demand loss remains, 0.5 times 100, giving 50 USD per person per year.
C = 0; r = 0.0001; sigma = 5000; eps = 0.1; mu = 1000; RP = 0; MH = 50; WL = 50
No insurance of the illustrative hospital risk
With no insurance (C = 1) there is no induced use and the whole loss is the risk premium, 0.5 times 0.0001 times 25,000,000, giving 1,250 USD per person per year, as in the article. The inputs are illustrative.
C = 1; r = 0.0001; sigma = 5000; eps = 0.1; mu = 1000; RP = 1250; MH = 0; WL = 1250
Illustrative hospital risk at its optimal coinsurance rate
At the optimal rate 100 / 2,600 = 0.0384615 from HE-FM-MHHI-002, the risk premium is 1.85 and the induced-demand loss 46.23, a total of 48.08 USD per person per year, as in the article. This equals one half of 2,500 times 100 over 2,600, the minimum loss the article states (derived by substituting C_star into the formula). The inputs are illustrative.
C = 0.0384615; r = 0.0001; sigma = 5000; eps = 0.1; mu = 1000; RP = 1.85; MH = 46.23; WL = 48.08
Illustrative outpatient risk at its optimal coinsurance rate
The article's illustrative outpatient visits have expected spending and a standard deviation of USD 500 each and an absolute elasticity of 0.3. At the optimal rate 150 / 175 = 0.857143 the risk premium is 9.18 and the induced-demand loss 1.53, a total of 10.71 USD per person per year, below the 75 of full insurance and the 12.5 of no insurance.
C = 0.857143; r = 0.0001; sigma = 500; eps = 0.3; mu = 500; RP = 9.18; MH = 1.53; WL = 10.71
Common errors
Entering the price elasticity with its negative sign in the Phelps loss
Phelps writes the elasticity as a negative number and subtracts its term, while the formula here takes the absolute value. Typing minus 0.1 for the illustrative hospital care turns the full-insurance loss of 50 into minus 50, a gain from induced use that the model cannot produce.
Using the standard deviation of spending in place of its variance
The risk term needs sigma squared. With the illustrative hospital standard deviation of 5,000 used unsquared, the no-insurance risk premium is 0.25 instead of 1,250, and the loss at the optimal rate is 46.23 instead of 48.08, so the value of cover almost disappears.
Reading the second-order risk premium as exact for hospital spending
The risk term ignores skewness. Phelps finds that for hospital care in the RAND data the third and fourth order terms are of the same order of magnitude as the variance term, so the second-order risk premium for heavily skewed hospital spending is an approximation to report with that caveat, even though the optimal rate he computes changes little between the second and fourth order.
Sources
Phelps on the risk premium and moral hazard loss as quadratics in the coinsurance rate
Phelps CE. Medical insurance: risk spreading vs. moral hazard revisited. Working paper. Rochester, NY: University of Rochester; April 2002. Section II, The Model, equation (2c), whose variance term minus r sigma squared over 2 Phelps says replicates Pratt's conclusion, with the statement that insurance paying 1 minus C of any loss reduces the risk to C squared sigma squared; equation (3), the expected moral hazard loss one half of (1 minus C) squared times E(pm eta) with the elasticity evaluated at the full-price demand; and equation (4), the sum of the two losses, with the statement that smaller C decreases the risk losses quadratically and increases the moral hazard losses quadratically. The appendix sizes the ignored income-price term.
Canonical Identity
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