Functions & Formulae

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

Ex post moral hazard welfare function

w(p,c,q_0,q_1) = WL

Maps the resource cost of care, the share of that cost the insured patient pays and the quantities of care used at the full and insured prices to the welfare loss attributed to insurance-induced use, and maps observed spending under two cost-sharing arrangements to a price elasticity of demand. The welfare measures assume that the demand curve measures the patient's marginal benefit, an assumption that the access motive and behavioural hazard challenge.

  • Welfare loss triangle of insurance-induced care

    WL = 0.5 * (p - c * p) * (q_1 - q_0)

    With coinsurance rate c the patient pays c × p per unit of care that costs p to provide and uses q_1 units instead of the q_0 chosen at the full price. Each extra unit costs p but is valued by the patient at between c × p and p, so with a linear demand curve the loss is the triangle whose height is the price reduction p minus c × p and whose base is the extra use. The notation follows the moral hazard article.

  • Extra use and welfare loss from a linear demand curve

    Delta_q = D * (p - c * p); WL = 0.5 * D * (p - c * p)^2

    When demand for care is linear with slope D, a fall in the price the patient faces from p to c × p raises use by D × (p minus c × p). Substituting into the triangle gives a welfare loss of one half of D × (p minus c × p) squared. The loss therefore rises with the square of the price reduction: halving the coinsurance rate from 0.5 to 0.25 more than doubles it.

  • RAND arc elasticity of medical spending

    eta_arc = ((q_2 - q_1) / (q_2 + q_1)) / ((p_2 - p_1) / (p_2 + p_1))

    Divides the change in spending, as a proportion of the average of the two spending levels, by the change in price, as a proportion of the average of the two prices. The RAND investigators used arc elasticities because the free care plan had a price of zero, from which an ordinary percentage change cannot be computed. The widely quoted RAND elasticity of about minus 0.2 comes from arc elasticities of this kind computed from an episode-based model; Aron-Dine and colleagues identify Keeler and Rolph 1988 as its underlying source.