Functions & Formulae

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

Derived demand for medical care from the Grossman demand for the health stock

d(eps, K, rho_H, W, pi, P, E, delta) = (H_ratio, I_ratio, M_ratio)

Maps the elasticity of the marginal efficiency of health capital, the time share of the cost of gross investment, the schooling efficiency parameter and a change in the wage, the marginal cost of gross investment or the price of care, schooling and the depreciation rate to the proportional change in the health stock demanded, in gross investment and in medical care demanded, in the pure investment version of the Grossman model with a zero real-own rate of interest. HE-FM-DDHC-001 gives Grossman's demand function for the health stock and HE-FM-DDHC-002 his reduced-form demand function for medical care with its wage, price, schooling and depreciation coefficients. Both build on live records that this package does not restate, the pure investment condition HE-FM-DFH-002, the stock update HE-FM-DFH-001 (gross investment of about delta H when net investment is small), the Cobb-Douglas gross investment of HE-FM-HPF-001, the least-cost care of HE-FM-HPF-002 and the unit cost pi of HE-FM-HPF-003. Because Grossman suppresses the intercepts, both formulas compare two states, such as two ages of one person or two people, as ratios, and for a change in one price alone each ratio has the constant-elasticity form of HE-FM-XPE-003. Notation follows the Derived Demand for Healthcare article.

  • Grossman demand for the health stock as a ratio between two states with the implied gross investment

    H_ratio = (W_1 / W_0)^eps * (pi_1 / pi_0)^(-eps) * (delta_1 / delta_0)^(-eps); I_ratio = (delta_1 / delta_0) * H_ratio

    Grossman's demand function for the health stock, ln H = eps ln W minus eps ln pi minus eps ln delta with the intercept suppressed, is written as the ratio of the stock demanded after a change to the stock demanded before it, so that the intercept cancels. With pi held fixed a 1% rise in the wage raises the stock demanded by eps percent, while a 1% rise in the marginal cost of gross investment or in the depreciation rate lowers it by eps percent. The wage also raises pi through the own time input, so with pi taken from HE-FM-HPF-003 the net wage elasticity of the stock is (1-K) eps and schooling raises it by rho_H eps per year, Grossman's reduced form for health. The second step applies HE-FM-DFH-001 with net investment small, so gross investment is about delta H and its ratio is the depreciation ratio times H_ratio (derived here), which for a change in delta alone is (delta_1 / delta_0) to the power 1-eps and rises with depreciation when eps is below 1. The function follows from the pure investment condition HE-FM-DFH-002 once the marginal efficiency schedule has the constant elasticity eps.

  • Grossman reduced-form demand for medical care with its wage, price, schooling and depreciation elasticities

    e_MW = (1 - K) * eps + K; e_MP_scale = -(1 - K) * eps; e_MP_subst = -K; e_MP = e_MP_scale + e_MP_subst; s_ME = rho_H * (eps - 1); e_Mdelta = 1 - eps; M_ratio = exp(e_MW * log(W_1 / W_0) + e_MP * log(P_1 / P_0) + s_ME * (E_1 - E_0) + e_Mdelta * log(delta_1 / delta_0))

    Grossman's reduced-form demand function for medical care, ln M = [(1-K) eps + K] ln W minus [(1-K) eps + K] ln P plus rho_H (eps minus 1) E plus (1-eps) ln delta, with intercepts suppressed and the small net disinvestment term left out, is written as the ratio of care demanded after and before a change. It follows from the health stock demand of HE-FM-DDHC-001, gross investment of about delta H from HE-FM-DFH-001, the unit cost pi of HE-FM-HPF-003 and the least-cost care M = (1-K) C / P of HE-FM-HPF-002, and every coefficient contains eps, the parameter of the demand for health. The price elasticity splits into a scale part, minus (1-K) eps, through the smaller health stock demanded when care is dearer, and a substitution part, minus K, towards own time. Schooling changes care by rho_H (eps minus 1) per year and depreciation by the elasticity 1-eps, so with eps below 1 the more educated use less care and people whose health depreciates faster use more. For a change in the price of care alone the ratio is HE-FM-XPE-003 with the own-price elasticity e_MP.

Derived Demand for Healthcare — Functions & Formulae | HealthEconomics.wiki