Signature
C_star = eps * mu / (eps * mu + r * sigma^2)
| Inputs | Definition | Unit |
|---|---|---|
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 |
r | Pratt coefficient of absolute risk aversion, above zero, in the reciprocal of the currency used for mu and sigma | 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_star | Coinsurance rate that minimises the expected welfare loss of HE-FM-MHHI-001, above 0 and below 1 when r, sigma, eps and mu are above zero | proportion |
|---|
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 optimal coinsurance rate from named cells
With the absolute elasticity in PriceElasticityAbs, mean spending at the full price in MeanSpendFullPrice, the risk aversion coefficient in RiskAversionCoef and the standard deviation of spending in SpendSD, the formula returns the optimal rate in OptimalCoinsRate.
=PriceElasticityAbs*MeanSpendFullPrice/(PriceElasticityAbs*MeanSpendFullPrice+RiskAversionCoef*SpendSD^2)
Assumptions
Interior minimum of the Phelps welfare loss
With r above zero and a downward-sloping demand curve, Phelps shows that the optimal rate lies strictly above 0 and below 1 and that the second-order condition holds. He sets aside cases in which the variance of the risk changes with C, through C altering the probabilities of illnesses or the shapes of the demand curves.
Same approximations as the Phelps welfare loss
The rate inherits HE-AS-MHHI-001 to HE-AS-MHHI-004, a second-order risk premium, straight-line demand with one elasticity at the full price, a fair premium with a single rate, and induced care valued by the demand curve. Phelps reports that the optimal hospital rate is little different at the second and fourth order approximations of risk.
Risk aversion measured in the units of the spending figures
r is per unit of the currency in which mu and sigma are measured. Phelps takes r of about 0.0001 per dollar, corresponding to relative risk aversion of about 2 at an income of USD 20,000, and treats 0.00005 to 0.0002 as the range of most of the literature.
Worked examples
Optimal rate for the illustrative hospital risk
For the article's illustrative hospital care, eps mu is 0.1 times 1,000 = 100 and r sigma squared is 0.0001 times 25,000,000 = 2,500, so the optimal rate is 100 / 2,600 = 0.038, about 4 per cent. The inputs are illustrative.
eps = 0.1; mu = 1000; r = 0.0001; sigma = 5000; C_star = 0.038
Optimal rate for the illustrative outpatient risk
For the article's illustrative outpatient visits, eps mu is 0.3 times 500 = 150 and r sigma squared is 0.0001 times 250,000 = 25, so the optimal rate is 150 / 175 = 0.857, about 86 per cent. The inputs are illustrative.
eps = 0.3; mu = 500; r = 0.0001; sigma = 500; C_star = 0.857
Phelps base case for hospital care
Phelps's Table 1 base case for hospital care, with RAND-based mean spending of 500, elasticity 0.15 and variance 3,150 squared at r = 0.0001, gives an optimal rate of 0.07, the value he reports.
eps = 0.15; mu = 500; r = 0.0001; sigma = 3150; C_star = 0.07
Phelps base case for physician care
Phelps's Table 1 base case for physician care, with mean spending of 500, elasticity 0.3 and variance 1,250 squared at r = 0.0001, gives an optimal rate of 0.49.
eps = 0.3; mu = 500; r = 0.0001; sigma = 1250; C_star = 0.49
Physician care with doubled risk aversion
Doubling r to 0.0002 for the same physician care lowers the optimal rate to 0.32, the value in Phelps's Table 2, which shows the rate for intermediate types of care moving with the risk aversion assumed.
eps = 0.3; mu = 500; r = 0.0002; sigma = 1250; C_star = 0.32
Common errors
Using the standard deviation of spending in place of its variance in the optimal rate
For the illustrative hospital care, r times sigma is 0.5 where r times sigma squared is 2,500, so the rate comes out at 0.995 instead of 0.038 and recommends almost no cover for the risk that most needs it.
Risk aversion and spending in different currency units
r = 0.0001 is per dollar. Entering the illustrative hospital spending in thousands of dollars (mu = 1, sigma = 5) with the same r gives 0.1 / 0.1025 = 0.976 instead of 0.038.
Entering the elasticity as a negative number in the optimal rate
Phelps writes the elasticity as negative and the formula here takes its absolute value. Typing minus 0.1 for the illustrative hospital care gives minus 0.042, and minus 0.3 for the outpatient visits gives 1.2, both outside the range above 0 and below 1 that Phelps shows the optimum must lie in.
Applying one coinsurance rate to all types of care
The optimal rate is service-specific. Charging the outpatient optimum of 150 / 175 = 0.857 on the illustrative hospital care raises its expected loss in HE-FM-MHHI-001 to 919.39 against 48.08 at its own optimum, and charging the hospital optimum of 100 / 2,600 = 0.038 on the outpatient visits raises theirs to 69.36 against 10.71.
Sources
Phelps on the optimal coinsurance rate and its RAND-based values
Phelps CE. Medical insurance: risk spreading vs. moral hazard revisited. Working paper. Rochester, NY: University of Rochester; April 2002. Section II, The Model, equation (5a), the first-order condition, equation (5b), C* = minus E(pm eta) over r sigma squared minus E(pm eta), with the statement that r above 0 and minus eta above 0 make C* lie above 0 and below 1, equation (6), C* as the moral hazard loss over the moral hazard loss plus the risk premium, footnote 4 on the second-order condition, and equation (7), the relative form with the coefficient of variation. Section III, Measuring Welfare Losses and C*, which takes r of about 0.0001, Table 1 (base-case mu, eta, sigma squared and C*, including 0.07 for hospital and 0.49 for physician care) and Table 2 (C* for r of 0.00005, 0.0001, 0.0002 and 0.0005, including 0.32 for physician care at 0.0002).
Canonical Identity
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