Functions & Formulae

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

Mixed provider payment and the quantity of care chosen under imperfect agency

R = P + r * c * q; alpha * b(q) = (1 - r) * c; r_star = 1 - alpha

Maps a provider payment rule with a prospective amount per case and a reimbursed share of cost to the provider's revenue and margin and, with the index of agency (the weight a provider puts on a pound of patient benefit relative to a pound of net revenue), to the quantity of care it chooses. Ellis and McGuire showed that the efficient reimbursed share equals one minus the index of agency. Notation follows the Contract Theory article and its blended bed-day example.

  • Revenue and provider margin per case under an Ellis and McGuire mixed payment

    R = P + r * c * q; M = P - (1 - r) * c * q

    A mixed payment pays a fixed prospective amount per case plus a share r of the cost of the care given, so revenue rises by r times c for each extra unit and the provider bears the rest, (1 minus r) times c. Pure case payment is r = 0 and full cost reimbursement is P = 0 with r = 1. The margin is revenue minus cost, written in Ellis and McGuire's equation (12) as P + (r minus 1) c q.

  • Bed-days chosen under a mixed payment with a linear marginal benefit of care

    q = max(0, (b_0 - (1 - r) * c / alpha) / b_1)

    Ellis and McGuire's first-order condition alpha b(q) = (1 minus r) c sets the weighted marginal benefit of care equal to the cost the provider bears. With the marginal benefit falling in a straight line, b(q) = b_0 minus b_1 q, the condition solves for the stay; the max function returns no care when even the first unit is not worth its cost to the provider. The article's patient benefit B(q) = 1200q minus 50q squared gives b_0 = 1,200 and b_1 = 100.

  • Efficient reimbursed share of cost and the budget-matched prospective amount

    q_eff = (b_0 - c) / b_1; r_star = 1 - alpha; P = (1 - r_star) * c * q_eff

    Efficiency requires the patient's marginal benefit to equal the marginal cost of care. Comparing this with the provider's condition (HE-FM-CTH-002) gives Ellis and McGuire's optimal reimbursed share r_star = 1 minus alpha, so the provider's own cost share equals the index of agency. The fixed part is then set so that the payment for an efficient stay equals its cost, the article's choice; Ellis and McGuire note that the prospective amount can be lowered as r rises to keep the same total payment per case.

Contract Theory — Functions & Formulae | HealthEconomics.wiki