Signature
I = exp(rho_H * E) * M^(1 - K) * TH^K
| Inputs | Definition | Unit |
|---|---|---|
rho_H | Proportional rise in gross investment from one more year of schooling with the inputs held constant, Grossman's percentage increase in gross investment per unit of E | proportion per year of schooling |
E | Years of formal schooling counted from a reference level at which the efficiency factor is normalised to 1 | years |
M | Quantity of medical care used in producing gross investment in the period | units of medical care per period |
K | Share of own time in the total cost of gross investment, greater than 0 and less than 1, so that 1-K is the cost share and output elasticity of medical care | proportion |
TH | Hours of the person's own time spent producing health in the period, such as time spent at a check-up | hours per period |
I | Gross investment in health produced in the period from medical care and own time | units of health capital per period |
|---|
Function
Health production function mapping medical care, own time and schooling to gross investment in health and its least cost
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.
Try this function
Implementations
Excel
Cobb-Douglas gross investment in health from named cells
With the schooling parameter in SchoolingEffect, the years above the reference level in SchoolingYears, the inputs in MedicalCare and OwnTime and the time share in TimeShare, the formula fills GrossInvestCD.
=EXP(SchoolingEffect*SchoolingYears)*MedicalCare^(1-TimeShare)*OwnTime^TimeShare
Assumptions
Cobb-Douglas class with constant returns in medical care and own time
Grossman assumes the gross investment function is homogeneous of degree one in medical care and own time and, for estimation, a member of the Cobb-Douglas class, so the elasticity of substitution between the two inputs is one and the cost share of medical care, 1-K, is constant. Grossman notes that with an elasticity of substitution other than one K would depend on W and P. The closed forms of HE-FM-HPF-002 and HE-FM-HPF-003 would then not apply (derived here).
Factor-neutral schooling effect and a suppressed intercept in gross investment
One more year of E raises the marginal products of M and TH by the same percentage, Grossman's Hicks or factor-neutrality assumption, so schooling changes the gross investment obtained from given inputs but gives no reason to change the ratio of care to time. Grossman writes the estimating form with its intercept suppressed, left out of the equation as shown, so the level of the efficiency factor is not pinned down. Because health capital is measured in abstract units, the formula absorbs the intercept by counting E from the level at which the factor is 1 (a normalisation made here), so only differences in E between people or years carry meaning. Allocative efficiency, in which schooling improves knowledge of the true effects of inputs and so the choice of inputs, is a separate channel that this formula leaves out.
Medical care as the single purchased health input with no joint production
M stands for the purchased health inputs, as Grossman associates the market goods input in the gross investment function with medical care, and the formula rules out joint production, in which inputs such as cigarette smoking have negative marginal products in health but are bought as inputs into other commodities. Grossman notes that the model keeps its structure if the main purchased input is something other than medical care.
Worked examples
Least-cost inputs of 10 units of care and 40 hours in the health production example
The article's illustrative person uses 10 units of medical care and 40 hours of own time with K of 0.5 and schooling at the reference level, so I is the square root of 400, 20 units of health capital. The figures are illustrative and not drawn from any study.
rho_H = 0.03; E = 0; M = 10; TH = 40; K = 0.5; I = 20
Same outlay spent on 15 units of care and 20 hours in the health production example
Spending the same 800 GBP on 15 units of care and 20 hours, the article's alternative mix, gives the square root of 300, 17.32 units, about 13% less than the least-cost mix of HE-EX-HPF-001.
rho_H = 0.03; E = 0; M = 15; TH = 20; K = 0.5; I = 17.32
One more year of schooling with the same inputs in the health production example
With one more year of schooling at rho_H of 0.03 the same 10 units of care and 40 hours give exp(0.03) times 20, 20.61 units, as in the article.
rho_H = 0.03; E = 1; M = 10; TH = 40; K = 0.5; I = 20.61
Common errors
Total years of schooling used as E with a suppressed intercept
Because the intercept that Grossman's estimating form leaves unstated is absorbed into the reference level, rho_H E is a shift from that level. Entering 12 years of schooling in place of 0 for the article's reference person raises I to 28.67 against 20, 43% too high (computed here for illustration), while the comparison the article draws, one more year against the reference, gives 20.61.
Gross investment in health read as the health stock
I is the flow added to the stock in the period, not the stock. In the article's example 20 units of gross investment take a stock of 100 with depreciation of 0.15 to 105 through HE-FM-DFH-001, so the 3.05% rise in I from one more year of schooling raises next period's stock by 0.61 units, about 0.6% (computed here for illustration).
Cross-sectional regression of health on medical care read as the production function
Fitting the production function by ordinary least squares across people gives a coefficient on medical care that Grossman shows is biased downward, because people with high unobserved depreciation demand more care and hold less health, and unmeasured differences in efficiency bias it the same way. Grossman notes that two-stage least squares with wealth, wage rates and the price of care as instruments can give consistent estimates, provided the instruments are valid and explain enough of the variation in care. A negative association between care and health across people does not by itself show that the elasticity 1-K is zero or negative.
Sources
Grossman's gross investment function and its Cobb-Douglas estimating form
Grossman M. The human capital model of the demand for health. NBER Working Paper 7078. National Bureau of Economic Research, Cambridge, MA; 1999. Published as The human capital model, in Culyer AJ, Newhouse JP, editors. Handbook of Health Economics. Vol 1A. Elsevier, Amsterdam; 2000. p. 347-408. Section 2.1, equation (2-3), the gross investment function It = It(Mt, THt; E) with E exogenous or predetermined, the efficiency effect of knowledge capital following Michael and Becker, linear homogeneity in the market goods and own time inputs and the ruling out of joint production; section 3.2, the factor-neutrality assumption that a one unit increase in E raises the marginal products of M and TH by the same percentage rho_H, the percentage increase in gross investment per unit of E; section 5.1, equation (5-3), the Cobb-Douglas form rho_H E + (1-K) ln Mt + K ln THt with intercepts suppressed, an elasticity of substitution of one, 1-K as the share of medical care in total cost and the elasticity of gross investment with respect to medical care, and the downward bias of ordinary least squares estimates of the production function with two-stage least squares as the consistent alternative.
Grossman's 1972 gross investment function homogeneous of degree one
Grossman M. The Demand for Health: A Theoretical and Empirical Investigation. National Bureau of Economic Research, New York; 1972. Chapter 1, A Stock Approach to the Demand for Health, pages 1 to 10. Page 2, equation (1-3), the household production function of gross investment in medical care, own time and the stock of human capital, the statement that a shift in human capital changes the efficiency of production in the nonmarket sector as technology does in the market sector, and the assumption that all production functions are homogeneous of degree one in the goods and time inputs.
Canonical Identity
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