Signature
gamma_t = W_t * G_t / pi_(t-1); UC_t = r - pi_tilde_(t-1) + delta_t
| Inputs | Definition | Unit |
|---|---|---|
W_t | Money value of one unit of healthy time, measured by the wage | currency per healthy day |
G_t | Increase in healthy time from one more unit of health capital | healthy days per year per unit of health capital |
pi_(t-1) | Cost of producing one more unit of gross investment in period t-1, from medical care and own time | currency per unit of health capital |
r | Market rate of interest on financial assets | proportion per year |
pi_tilde_(t-1) | Percentage rate of change in the marginal cost of gross investment between periods t-1 and t, the capital gains term | proportion per year |
delta_t | Share of the health stock that depreciates in period t | proportion per year |
gamma_t | Marginal monetary rate of return on an investment in health | proportion per year |
|---|---|---|
UC_t | User cost of health capital in terms of the price of gross investment: interest, less capital gains, plus depreciation | proportion per year |
Function
Health capital accumulation and investment function
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.
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Implementations
Excel
Marginal efficiency and opportunity cost of health capital
With the inputs in Wage, MargProdHealth, MargCostInvest, InterestRate, MargCostGrowth and DeprRate, the first formula returns the marginal efficiency of health capital and the second its opportunity cost.
=Wage*MargProdHealth/MargCostInvest; =InterestRate-MargCostGrowth+DeprRate
Assumptions
Pure investment model
The marginal utility of healthy time is set to zero, so the psychic return drops out of Grossman's general condition. In the pure consumption version the monetary return is zero instead, and some predictions differ.
Constant returns in producing gross investment
Gross investment is produced with constant returns to scale and input prices that do not depend on the quantity produced, so the marginal cost pi_(t-1) equals the average cost. The condition also omits the small product of delta_t and pi_tilde_(t-1), which Grossman assumes is close to zero.
Interior solution with positive gross investment
Gross investment is positive in period t, so the first-order condition holds with equality. G_t falls as the stock rises, which makes the marginal efficiency schedule slope downward.
Worked examples
Optimal stock at a younger age
A worker values a healthy day at 150 pounds, one more unit of health capital adds 0.4 healthy days a year and gross investment costs 750 pounds a unit, so gamma_t is 0.08. With interest of 3%, a 1% rise in the marginal cost of investment and depreciation of 6%, the opportunity cost is also 0.08, so the stock is at its optimum. The figures are illustrative.
W_t = 150; G_t = 0.4; pi_(t-1) = 750; r = 0.03; pi_tilde_(t-1) = 0.01; delta_t = 0.06; gamma_t = 0.08; UC_t = 0.08
Optimal stock after depreciation rises with age
When depreciation rises to 9%, the opportunity cost rises to 0.11. The condition is restored only at a lower stock, where the marginal product of health capital has risen to 0.55 healthy days a year. This is the model's prediction that the optimal stock falls with age.
W_t = 150; G_t = 0.55; pi_(t-1) = 750; r = 0.03; pi_tilde_(t-1) = 0.01; delta_t = 0.09; gamma_t = 0.11; UC_t = 0.11
Common errors
Omitting depreciation from the cost of capital
Equating the return on health to the interest rate alone, 0.03 in the first example, sets the cost of holding health capital far too low. Depreciation is part of its user cost and is the channel through which age lowers the optimal stock.
Using the price of medical care as the marginal cost of investment
pi_(t-1) is the marginal cost of producing health investment from medical care and own time. It depends on the wage and on education as well as on the price of care.
Sources
Grossman's optimality condition in the pure investment model
Grossman M. The human capital model of the demand for health. NBER Working Paper 7078. Cambridge, MA: National Bureau of Economic Research; 1999. Published as: The human capital model. In: Culyer AJ, Newhouse JP, editors. Handbook of Health Economics. Vol 1A. Amsterdam: Elsevier; 2000. p. 347-408. Equation (2-11) (value of the marginal product of health capital equal to its user cost), footnote 6 (the product of delta_t and pi_tilde_(t-1) assumed close to zero), equation (2-24) (gamma_t plus a_t equals r minus pi_tilde_(t-1) plus delta_t) and equation (3-1) (pure investment model: gamma_t = W_t G_t / pi_(t-1) = r minus pi_tilde_(t-1) plus delta_t).
Original pure investment model of the demand for health
Grossman M. On the concept of health capital and the demand for health. Journal of Political Economy. 1972;80(2):223-255. Marginal efficiency of health capital and the equilibrium condition equating it to the cost of capital in the pure investment model.
Canonical Identity
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