Functions & Formulae

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

Healthcare production function mapping staff, capital, consumables and technology to provider activity

f(L, K, M, T) = Q

Maps a provider's labour L, capital K, drugs and consumables M and technology and organisation T to the maximum activity Q it can produce in a period, such as consultations or case-mix-adjusted episodes. It is the provider counterpart of the household function HE-FN-HPF-001, in which health itself is the output. The quantities read off the function are held in live records and are cited here, not restated. The Cobb-Douglas form with exponents summing to one is HE-FM-HPF-001 with A = exp(rho_H E), alpha = 1-K and beta = K, K there being the own time share, and the marginal products, each exponent times output over the input, and their ratio, the marginal rate of technical substitution, take the form of those in HE-FM-HPF-002, which computes them only for exponents summing to one; the same form holds for any exponents (derived). Marginal cost as the price of an input over its marginal product, MC = w / MP_L when labour is the only variable input, is Grossman's condition held in HE-SR-HPF-003 and tested in HE-VT-HPF-006, and average variable cost w / AP_L is the unit labour cost w / a_h of HE-FM-BAUM-001. A discrete marginal product from a production schedule is a change in output over a change in input, the same difference-quotient form as HE-FM-ICER-001, and an average product is a partial productivity ratio such as the bed turnover rate of HE-FM-BTO-001. Distance from an estimated frontier is measured by HE-FM-EFF-001 to HE-FM-EFF-003. Average cost, cost functions and the marginal cost of a change in scale belong to the Average Cost, Cost Function and Marginal Analysis pages. This package adds the degree of returns to scale of a Cobb-Douglas function whose output elasticities are estimated freely (HE-FM-HCPF-001). Notation follows the Healthcare Production Function article.

  • Returns to scale of a Cobb-Douglas hospital production function with free output elasticities

    Q = A * L^alpha * K^beta; RTS = alpha + beta; Q_s = A * (s * L)^alpha * (s * K)^beta; g_s = Q_s / Q

    Gives a provider's activity from labour and capital in the Cobb-Douglas form that hospital studies estimate, with the output elasticities alpha and beta left free, and returns the degree of returns to scale RTS, their sum, with the output Q_s and the ratio g_s when every input is multiplied by s. Sielska and Nojszewska define economies of scale as the sum of the output elasticities, the percentage change in output when all inputs rise by 1%, and g_s equals s to the power RTS for any s (derived), so returns to scale are decreasing when RTS is below 1 and increasing when it is above 1. With alpha + beta = 1 the function is the constant-returns HE-FM-HPF-001, under which g_s always equals s. The marginal products alpha Q / L and beta Q / K have the form of those in HE-FM-HPF-002, each exponent times output over the input, a form that holds at any RTS (derived), although HE-FM-HPF-002 computes them only for exponents summing to one. RTS is a property of the estimated function and is the same at every size, unlike scale efficiency HE-FM-EFF-003, a ratio of frontier scores for one unit.