Functions & Formulae

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

Break-even price, cost-neutral price and service volume function

p_star = f(lambda, Delta_E, Delta_C_0, u); p_0 = g(Delta_C_0, u); Q_star = h(F, P, V)

Solves for the value of one input at which a decision criterion is exactly zero, holding every other input at its base-case value. For a new health technology the input is the unit price and the criterion is incremental net monetary benefit at a stated threshold (the break-even price) or incremental cost alone (the cost-neutral price). For a service paid per unit of activity the input is volume and the criterion is the surplus of income over fixed and variable costs (the break-even volume). The records follow the notation of the Break-Even Analysis article. NICE's manual (PMG36) calls the general calculation threshold analysis and its result a switching value; the switching value of a non-linear input by interpolation is HE-FM-OWSA-002, and the payer's maximum price per patient in price negotiation is HE-FM-BARG-002.

  • Break-even and cost-neutral unit prices of a health technology

    p_star = (lambda * Delta_E - Delta_C_0) / u; p_0 = -Delta_C_0 / u

    Splits the incremental cost per patient into the acquisition cost of the technology, u units at unit price p, and every other cost difference, Delta_C_0, which is negative when the technology avoids more cost than it adds. Setting incremental net monetary benefit, lambda times Delta_E less (u times p plus Delta_C_0), to zero gives the break-even price, the highest unit price at which the technology is cost-effective at threshold lambda. Setting the incremental cost alone to zero gives the cost-neutral price. The gap between them, lambda times Delta_E divided by u, is the money value of the health gain per unit, so for a technology that improves health the break-even price lies above the cost-neutral price. With u equal to 1 the break-even price is the payer's maximum price per patient of HE-FM-BARG-002. Solving the same condition for the effect gives Delta_C divided by lambda, the health displaced of HE-FM-CET-002.

  • Break-even activity volume and margin of safety for a health service paid per unit

    Q_star = F / (P - V); MS = (Q_e - Q_star) / Q_e

    Applies the cost-volume form of break-even analysis from management accounting to a service paid a fixed price or tariff per unit of activity. Each unit contributes the price less the variable cost, so the break-even volume is the fixed cost for the period divided by the contribution per unit, and the margin of safety is the share by which expected activity could fall before the service makes a loss. When fixed costs step up with capacity, as when a second team is needed, the calculation is repeated for each block of capacity.