Functions & Formulae

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

Health capital accumulation and investment function

H_(t+1) = f(H_t, delta_t, I_t)

In the Grossman model health is a durable capital stock that produces healthy time. The stock carried into the next period is the current stock less depreciation plus gross investment, which the individual produces with medical care and own time. The optimal stock in each period equates the marginal return on health capital with its user cost, the real rate of interest plus the rate of depreciation.

  • Health capital stock carried into the next period

    H_(t+1) = H_t - delta_t * H_t + I_t

    Grossman's equation (2-2) sets net investment, the change in the stock, equal to gross investment minus depreciation. Rearranged, the stock at the start of the next period is the current stock, less the share delta_t that depreciates during the period, plus gross investment. Holding the stock constant requires gross investment equal to delta_t × H_t.

  • Pure investment condition for the optimal health stock

    gamma_t = W_t * G_t / pi_(t-1); UC_t = r - pi_tilde_(t-1) + delta_t

    In the pure investment version of the Grossman model, health yields no direct utility and is valued only for the healthy time it adds to earnings. The marginal monetary return on health capital, gamma_t, is the wage multiplied by the marginal product of health capital in healthy time, divided by the marginal cost of gross investment. At the optimal stock it equals the opportunity cost of health capital, the real-own rate of interest r minus pi_tilde_(t-1) plus the rate of depreciation. The formula returns both sides so that the condition can be checked.