Signature
M = (1 - K) * C / P; TH = K * C / W; I = exp(rho_H * E) * M^(1 - K) * TH^K; MP_M = (1 - K) * I / M; MP_TH = K * I / TH; R_MP = MP_M / MP_TH
| Inputs | Definition | Unit |
|---|---|---|
K | Share of own time in the total cost of gross investment, greater than 0 and less than 1 | proportion |
C | Money spent in the period on medical care and on own time valued at the wage | currency per period |
P | Price of one unit of medical care | currency per unit of medical care |
W | Wage rate, the price of one hour of own time spent producing health | currency per hour |
rho_H | Proportional rise in gross investment from one more year of schooling with the inputs held constant | 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 | Units of medical care in the least-cost mix for the outlay C | units of medical care per period |
|---|---|---|
TH | Hours of own time in the least-cost mix for the outlay C | hours per period |
I | Gross investment in health produced by the least-cost mix | units of health capital per period |
MP_M | Extra gross investment from one more unit of medical care at the mix, (1-K) I / M | units of health capital per unit of medical care |
MP_TH | Extra gross investment from one more hour of own time at the mix, K I / TH | units of health capital per hour |
R_MP | Marginal product of medical care over marginal product of own time, equal to P / W at the least-cost mix | hours of own time per unit of medical care |
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.
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Implementations
Excel
Least-cost split of an outlay between care and own time from named cells
With the outlay in Outlay, the price of care in PriceCare, the wage in Wage, the time share in TimeShare and the schooling inputs in SchoolingEffect and SchoolingYears, the six formulas fill CareUnits, TimeHours, GrossInvestSplit, MargProdCare, MargProdTime and MargProdRatio in turn.
=(1-TimeShare)*Outlay/PriceCare; =TimeShare*Outlay/Wage; =EXP(SchoolingEffect*SchoolingYears)*CareUnits^(1-TimeShare)*TimeHours^TimeShare; =(1-TimeShare)*GrossInvestSplit/CareUnits; =TimeShare*GrossInvestSplit/TimeHours; =MargProdCare/MargProdTime
Assumptions
Input prices fixed and the outlay spent on care and own time
P and W do not depend on how much care or time the person uses, as Grossman assumes, and the whole outlay C goes on medical care and on own time priced at the wage, the two prices in Grossman's condition (2-10). Time lost to illness is not an input and does not enter C.
Interior least-cost mix with both inputs used
Both inputs are bought in positive amounts, which the Cobb-Douglas form ensures for K greater than 0 and less than 1, and the split is the same whether the person fixes the outlay or the gross investment (derived here). The least-cost condition settles how gross investment is produced; how much to produce is set by the optimality condition HE-FM-DFH-002.
Worked examples
Article's 800 GBP split between 10 units of care and 40 hours
With an outlay of 800 GBP, a price of care of 40 GBP, a wage of 10 GBP and K of 0.5, half the outlay buys 10 units of care and half buys 40 hours, giving 20 units of gross investment. The marginal products are 1 and 0.25, a ratio of 4 equal to the price ratio of 40 to 10, as in the article. The figures are illustrative.
C = 800; P = 40; W = 10; K = 0.5; rho_H = 0.03; E = 0; M = 10; TH = 40; I = 20; MP_M = 1; MP_TH = 0.25; R_MP = 4
Time share of 0.25 makes the article's alternative mix least-cost
For an illustrative person whose own time is a quarter of the cost of gross investment, K of 0.25, the same 800 GBP at the same prices buys 15 units of care and 20 hours, the mix that the article shows is not least-cost when K is 0.5. Gross investment is 16.12 units and the ratio of marginal products is again 4 (computed here for illustration).
C = 800; P = 40; W = 10; K = 0.25; rho_H = 0.03; E = 0; M = 15; TH = 20; I = 16.12; MP_M = 0.806; MP_TH = 0.2015; R_MP = 4
Older person's 1,000 GBP split between 12.5 units of care and 50 hours
The article's older person spends 1,000 GBP to hold the stock at 100 with depreciation of 0.25, which buys 12.5 units of care and 50 hours and so 25 units of gross investment, the investment that HE-FM-DFH-001 requires. The figures are illustrative.
C = 1000; P = 40; W = 10; K = 0.5; rho_H = 0.03; E = 0; M = 12.5; TH = 50; I = 25; MP_M = 1; MP_TH = 0.25; R_MP = 4
Common errors
Equal quantities of care and time in place of the least-cost shares
Buying the same number of units of each input, 800 GBP over 50 GBP for 16 units of care and 16 hours in the article's example, gives 16 units of gross investment against 20, 20% less for the same outlay (computed here for illustration). The least-cost rule sets the spending shares to 1-K and K, not the quantities.
Shortfall from a non-least-cost mix of care and time read as wasted inputs
The article's alternative mix of 15 units and 20 hours lies on the production function, so no input is wasted, yet it yields 17.32 units against 20 for the same 800 GBP. In the terms of HE-FM-EFF-002 the minimum cost of 17.32 units is 692.82 GBP, so cost efficiency and input-mix allocative efficiency are both 692.82 / 800 = 0.866 while technical efficiency is 1 (computed here for illustration). The remedy is a change of mix, not fewer inputs.
Least-cost shares swapped when the time share is not one half
With K of 0.25 the least-cost mix spends three quarters of the outlay on care. Swapping the shares buys 5 units of care and 60 hours, which gives 9.31 units of gross investment against 16.12, 42% less (computed here for illustration). With the article's K of 0.5 the swap leaves the result unchanged, so HE-VT-HPF-004 tests the split at another K.
Sources
Grossman's least-cost condition for producing gross investment in health
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 2.2, equation (2-10), which sets the marginal cost of gross investment equal to the price of medical care over the marginal product of medical care and to the wage over the marginal product of own time, and the text after it, which states that this is the condition for minimising the cost of producing a given quantity of gross investment; section 5.1, which states the least-cost equilibrium condition that the ratio of the marginal product of medical care to the marginal product of time must equal the ratio of the price of medical care to the wage rate, and gives 1-K as the share of medical care in the total cost of gross investment; section 3.2, which defines K as the fraction of the total cost accounted for by time. The quantities (1-K) C / P and K C / W are derived here from these statements and equation (5-3).
Grossman's 1972 condition for minimising the cost of gross investment
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 5, which states that equation (1-11) shows the condition for minimising the cost of producing a given quantity of gross investment, met when the change in gross investment from spending an additional dollar on medical care equals the change from spending an additional dollar on time, and that with a function homogeneous of degree one and prices independent of the level of the inputs average cost is constant and equal to marginal cost.
Canonical Identity
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