Functions & Formulae

Each applied formula has its own function page, with a signature, implementations, and tests.

Phelps welfare loss and optimal coinsurance function for medical insurance

z(C, r, sigma, eps, mu) = (WL, C_star)

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.

  • Phelps second-order expected welfare loss at a coinsurance rate

    RP = 0.5 * r * sigma^2 * C^2; MH = 0.5 * (1 - C)^2 * eps * mu; WL = RP + MH

    Adds two losses that move in opposite directions as cover changes. The first is the risk premium of the spending the patient still bears, one half of r times the variance of that spending, which is sigma squared times C squared when insurance pays the share 1 minus C of every bill. The second is the welfare-loss triangle of insurance-induced care, one half of (1 minus C) squared times eps times mu, which is HE-FM-MH-002 with the demand slope written as eps mu over the square of the full price (derived, for a straight-line demand curve with elasticity eps at the full price). Lowering C cuts the first term and raises the second, both quadratically, so the expected loss has a single minimum at the rate given by HE-FM-MHHI-002. Phelps prints the sum as his equation (4), with the elasticity entered as a negative number and subtracted.

  • Phelps optimal coinsurance rate from risk aversion, spending variance and price elasticity

    C_star = eps * mu / (eps * mu + r * sigma^2)

    Gives the coinsurance rate that minimises the expected welfare loss of HE-FM-MHHI-001, found by setting its derivative with respect to C to zero. The rate is eps mu over eps mu plus r sigma squared, which Phelps puts in words as the moral hazard loss over the sum of the moral hazard loss and the risk premium. It rises with the price elasticity and falls with risk aversion and with the variance of spending, so a large, barely price-sensitive risk such as hospital care calls for near-complete cover and small, predictable, price-sensitive spending for a high rate. Phelps also gives it in relative terms with the coefficient of variation and the share of income at risk (his equation (7)), the same rule divided through by mu (derived).