Signature
H_(t+1) = H_t - delta_t * H_t + I_t
| Inputs | Definition | Unit |
|---|---|---|
H_t | Health capital at the beginning of period t | units of health capital |
delta_t | Share of the health stock that depreciates during period t, greater than 0 and less than 1 and rising with age after some point | proportion per period |
I_t | Health capital produced during period t from medical care, own time and other inputs | units of health capital per period |
H_(t+1) | Health capital carried into period t+1 | units of health capital |
|---|
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
Next-period health stock
With the current stock in HealthStock, the depreciation rate in DeprRate and gross investment in GrossInvestment, Excel returns the stock carried into the next period.
=HealthStock-DeprRate*HealthStock+GrossInvestment
Assumptions
Depreciation rate exogenous but age-dependent
In Grossman's model delta_t is given to the individual and varies with age; it is not chosen. Death occurs when the stock falls to a minimum level.
Health stock and investment in common units
H_t and I_t are measured in the same units, which are analytical rather than observable. The equation describes the transition of the stock, not whether the investment is optimal.
Worked examples
Investment that raises the stock
Using the article's illustrative values, a stock of 100 units with 8% depreciation and 12 units of gross investment rises to 104 units.
H_t = 100; delta_t = 0.08; I_t = 12; H_(t+1) = 104
Investment below depreciation
With only 5 units of gross investment the stock falls to 97 units, because depreciation of 8 units exceeds investment.
H_t = 100; delta_t = 0.08; I_t = 5; H_(t+1) = 97
Higher depreciation later in life
If the depreciation rate rises to 12%, the same 12 units of investment only maintain the stock at 100 units.
H_t = 100; delta_t = 0.12; I_t = 12; H_(t+1) = 100
Common errors
Reading a rising stock as optimal investment
The equation shows how the stock moves for given investment. Whether 12 units is the right investment depends on its cost and on the value of the healthy time it produces, which the optimality condition addresses.
Treating medical care as the health stock
Medical care is one input to gross investment I_t, not the stock itself. Higher use of care alongside a lower stock, as when depreciation rises with age, is consistent with the model.
Sources
Grossman's net investment identity
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 (2-2): H_(t+1) minus H_t = I_t minus delta_t H_t, with delta_t the exogenous, age-dependent rate of depreciation (0 < delta_t < 1); equation (2-3) for the household production of gross investment from market goods and own time.
Original statement of the health capital model
Grossman M. On the concept of health capital and the demand for health. Journal of Political Economy. 1972;80(2):223-255. Health as a durable capital stock that produces healthy time; net investment equal to gross investment minus depreciation.
Canonical Identity
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