Functions & Formulae

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

Health production function mapping medical care, own time and schooling to gross investment in health and its least cost

g(M, TH, E, P, W, C, K, rho_H) = (I, pi)

Maps a person's medical care, own time and stock of schooling to gross investment in health, Grossman's I = I(M, TH; E), and, given the price of care, the wage and an outlay or a target, to the least-cost input mix and the unit cost of gross investment. It uses the Cobb-Douglas member of the class that Grossman fitted, with the intercept suppressed, and treats schooling as an efficiency shift. The output I is the gross investment term of the live stock update HE-FM-DFH-001, which carries the stock into the next period, and the unit cost of HE-FM-HPF-003 is the marginal cost of gross investment pi_(t-1) that the pure investment condition HE-FM-DFH-002 takes as an input, since constant returns and fixed input prices make marginal and average cost equal. The package keeps the production side only, with the Cobb-Douglas gross investment in HE-FM-HPF-001, the least-cost split of an outlay in HE-FM-HPF-002, the unit cost and outlay for a target in HE-FM-HPF-003, and the outlay that holds the stock on a target path in HE-CF-HPF-001. A mix that costs more than the least-cost mix for its output is measured by the cost efficiency ratio of HE-FM-EFF-002. The production of care by providers belongs to the healthcare production function concept and is not drafted here. Notation follows the Health Production Function article.

  • Cobb-Douglas gross investment in health from medical care, own time and schooling

    I = exp(rho_H * E) * M^(1 - K) * TH^K

    Gross investment is the efficiency factor exp(rho_H E) times medical care raised to the power 1-K times own time raised to the power K, the exponential form of Grossman's estimating equation ln I = rho_H E + (1-K) ln M + K ln TH. The exponents sum to one, so doubling both inputs doubles I, and 1-K is the elasticity of gross investment with respect to medical care. Schooling enters as a factor-neutral efficiency shift, so one more year multiplies I by exp(rho_H) for the same inputs, a rise of about 3.05% when rho_H is 0.03. The result I is the gross investment of HE-FM-DFH-001.

  • Least-cost split of an outlay between medical care and own time for Cobb-Douglas gross investment

    M = (1 - K) * C / P; TH = K * C / W; I = exp(rho_H * E) * M^(1 - K) * TH^K; MP_M = (1 - K) * I / M; MP_TH = K * I / TH; R_MP = MP_M / MP_TH

    Spends the share 1-K of the outlay on medical care and the share K on own time, the cost shares that Grossman gives for the Cobb-Douglas form of HE-FM-HPF-001, and returns the gross investment the outlay buys. The quantities follow from Grossman's least-cost condition (derived here), under which the ratio of the marginal product of care, (1-K) I / M, to that of time, K I / TH, equals the ratio of the price of care to the wage. The chain therefore also returns both marginal products and their ratio, so the condition can be checked against P / W. With the outlay fixed, a higher wage buys fewer hours and leaves the units of care unchanged, so the mix shifts towards care.

  • Unit cost of gross investment in health and the outlay for a target gross investment with schooling

    pi = exp(-rho_H * E) * (P / (1 - K))^(1 - K) * (W / K)^K; C = pi * I

    Returns the cost of one unit of gross investment at the least-cost mix and the outlay needed for a target gross investment. For the Cobb-Douglas form of HE-FM-HPF-001 the unit cost is exp(-rho_H E) times (P / (1-K)) to the power 1-K times (W / K) to the power K, derived here by putting the least-cost split of HE-FM-HPF-002 into the production function. Because of constant returns and fixed prices it is both the average and the marginal cost of gross investment, the pi_(t-1) that HE-FM-DFH-002 takes as an input. One more year of schooling divides the unit cost by exp(rho_H), the exact form of Grossman's result that marginal cost falls by rho_H percent per unit of E, and a 1% rise in the wage raises it by about K percent.