Unit cost of gross investment in health and the outlay for a target gross investment with schooling

Signature

pi = exp(-rho_H * E) * (P / (1 - K))^(1 - K) * (W / K)^K; C = pi * I

Function

Health production function mapping medical care, own time and schooling to gross investment in health and its least cost

Computational function

• Computational function for the outlay and least-cost inputs that hold the health stock on a target path

  Takes the stock at the start of a period, the depreciation rate, the stock wanted at the start of the next period, the price of care, the wage, the time share and schooling, and returns the gross investment required, its least-cost outlay and the care and own time it buys. The required gross investment is the stock update HE-FM-DFH-001 solved for I and set to zero when the target is below the depreciated stock, since Grossman notes that health capital cannot be sold and gross investment cannot be negative; the outlay uses the unit cost of HE-FM-HPF-003 and the inputs the split of HE-FM-HPF-002. The R and Python code run the same steps over a path of periods from a starting stock, carrying each period's end stock into the next through HE-FM-DFH-001, the target when gross investment is needed and the depreciated stock otherwise, with depreciation that may rise with age, the pattern behind the article's comparison of a younger and an older person.

  Inputs and outputs: H_t is the stock at the start of the period, delta_t the depreciation rate, H_target the stock wanted at the start of the next period, P the price of care, W the wage, K the time share, rho_H the schooling parameter and E the years of schooling above the reference level. I_req is the gross investment required, pi the unit cost, C the outlay, M the units of care and TH the hours of own time. In the code H0 is the stock at the start of the first period, and d and H_target hold one entry per period.

  Assumption: Depreciation is exogenous and applies to the stock at the start of the period, as in HE-FM-DFH-001, prices are fixed within each period, and the target path is set by the user, not derived from the optimality condition HE-FM-DFH-002. The path total is undiscounted at one unit cost; costs of later periods are discounted with HE-FN-DISC-001. The figures are the article's illustrative values.

  Worked example (Older person holding the stock at 100): With a stock of 100, depreciation of 0.25 and a target of 100, the gross investment required is 25, the outlay 1,000 GBP and the mix 12.5 units of care and 50 hours, as in the article. H_t = 100; delta_t = 0.25; H_target = 100; P = 40; W = 10; K = 0.5; rho_H = 0.03; E = 0; I_req = 25; pi = 40; C = 1000; M = 12.5; TH = 50

  Worked example (Younger person raising the stock to 105): With depreciation of 0.15 the article's 20 units of gross investment raise the stock of 100 to 105, so a target of 105 needs 800 GBP, 10 units of care and 40 hours. H_t = 100; delta_t = 0.15; H_target = 105; P = 40; W = 10; K = 0.5; rho_H = 0.03; E = 0; I_req = 20; pi = 40; C = 800; M = 10; TH = 40

  Worked example (Target below the depreciated stock): A target of 80 with depreciation of 0.15 is below the 85 units left after depreciation, so no gross investment is required, the outlay is zero and the stock ends at 85, above the target, a limiting case (computed here for illustration). H_t = 100; delta_t = 0.15; H_target = 80; P = 40; W = 10; K = 0.5; rho_H = 0.03; E = 0; I_req = 0; pi = 40; C = 0; M = 0; TH = 0

  Excel: For one period, =MAX(TargetStock-(1-DeprRate)*HealthStock,0) in RequiredInvest, =UnitCostInvest*RequiredInvest in HoldOutlay, =(1-TimeShare)*HoldOutlay/PriceCare in HoldCareUnits and =TimeShare*HoldOutlay/Wage in HoldTimeHours, with UnitCostInvest from HE-IM-HPF-003. For a path with one row per period in the ranges StartPath, DeprPath and TargetPath, where StartPath holds the starting stock in its first row and in each later row the previous row's end stock, the larger of that row's TargetPath and (1-DeprPath)*StartPath, =UnitCostInvest*SUMPRODUCT((TargetPath-(1-DeprPath)*StartPath)*(TargetPath>(1-DeprPath)*StartPath)) in PathOutlay gives the undiscounted total outlay over the path with each period's required investment floored at zero.

  R: hold_path <- function(H0, d, H_target, P, W, K, rho, E) { n <- length(d); H_target <- rep_len(H_target, n); pi <- exp(-rho*E)*(P/(1-K))^(1-K)*(W/K)^K; H <- numeric(n); I_req <- numeric(n); h <- H0; for (t in seq_len(n)) { H[t] <- h; I_req[t] <- max(H_target[t]-(1-d[t])*h, 0); h <- (1-d[t])*h+I_req[t] }; C <- pi*I_req; data.frame(H = H, I_req = I_req, C = C, M = (1-K)*C/P, TH = K*C/W) } Called as hold_path(100, c(0.15, 0.20, 0.25), 100, 40, 10, 0.5, 0.03, 0) it returns required gross investment of 15, 20 and 25 and outlays of 600, 800 and 1000, 2400 in total. Called as hold_path(100, c(0.15, 0.15, 0.15), c(105, 80, 100), 40, 10, 0.5, 0.03, 0) it carries 105 into the second period, where the depreciated stock of 89.25 is above the target and no investment is needed, and 89.25 into the third, which needs 24.1375 units and 965.50 GBP.

  Python: With import math and from itertools import accumulate (Python 3.8 or later), def hold_path(H0, d, H_target, P, W, K, rho, E): pi = math.exp(-rho*E)*(P/(1-K))**(1-K)*(W/K)**K; H = list(accumulate(zip(d, H_target), lambda h, x: max(x[1], (1-x[0])*h), initial=H0)); req = [max(t-(1-dt)*h, 0) for h, dt, t in zip(H, d, H_target)]; return [(h, i, pi*i, (1-K)*pi*i/P, K*pi*i/W) for h, i in zip(H, req)] Called as hold_path(100, [0.15, 0.20, 0.25], [100, 100, 100], 40, 10, 0.5, 0.03, 0) it returns outlays of 600, 800 and 1000, and with the second R call's inputs it returns the same starting stocks and outlays.

  Test (Older person holding the stock): With HealthStock 100, DeprRate 0.25, TargetStock 100, PriceCare 40, Wage 10, TimeShare 0.5 and SchoolingYears 0, HoldOutlay equals 25 units at 40 GBP, 1,000 GBP. Expected result TRUE. FALSE shows the depreciation rate used in place of one minus the rate, which requires 75 units and gives 3,000, or only the net change in the stock financed, which gives 0. Excel check =ABS(HoldOutlay-25*40)<0.005

  Test (Younger person raising the stock to 105): With HealthStock 100, DeprRate 0.15 and TargetStock 105 and the same prices, HoldOutlay equals the article's 800 GBP. Expected result TRUE. FALSE shows only the net change of 5 units financed, which gives 200, or depreciation charged on the target stock in place of the starting stock, which requires 20.75 units and gives 830. Excel check =ABS(HoldOutlay-800)<0.005

  Common error (Older person kept at the younger person's outlay): Keeping the 800 GBP outlay when depreciation rises to 0.25 buys only 20 units and the stock falls to 95, as in the article; holding it at 100 needs 1,000 GBP, 25% more (computed here for illustration).

  Source: 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 (2-2), net investment equal to gross investment minus depreciation, with exogenous age-dependent depreciation rates; section 2.2, equation (2-10) and the text after it on the least cost of a given gross investment; section 3.1, which states that since health capital cannot be sold, gross investment cannot be negative.

  I_req = max(H_target - (1 - delta_t) * H_t, 0); pi = exp(-rho_H * E) * (P / (1 - K))^(1 - K) * (W / K)^K; C = pi * I_req; M = (1 - K) * C / P; TH = K * C / W

Try this function

Implementations

• Excel

  Unit cost of gross investment and outlay for a target from named cells

  With PriceCare, Wage, TimeShare, SchoolingEffect and SchoolingYears as in HE-IM-HPF-002 and the target in TargetInvest, the first formula fills UnitCostInvest and the second OutlayForTarget.

  =EXP(-SchoolingEffect*SchoolingYears)*(PriceCare/(1-TimeShare))^(1-TimeShare)*(Wage/TimeShare)^TimeShare; =UnitCostInvest*TargetInvest

Assumptions

• Constant returns and fixed prices make the unit cost equal marginal cost

  pi does not depend on I only because the function has constant returns and P and W do not change with the amounts used, the conditions under which Grossman states that average cost is constant and equal to marginal cost, so C is proportional to I. Ehrlich and Chuma, as Grossman reports, introduce rising marginal cost of investment. C would then rise faster than I (derived here).

• Schooling lowers the unit cost of gross investment by the efficiency factor

  With factor-neutral efficiency the least-cost ratio of care to time is the same at every E, and a unit of gross investment needs the inputs divided by exp(rho_H) for each extra year, so pi falls in the same proportion. Grossman's equation (3-10) gives the percentage form, a fall in pi of rho_H per unit of E, and his 1972 example states that education raising both marginal products by 3 percent would reduce the price of gross investment by 3 percent; the exponential factor is the exact form under the log-linear equation (5-3) (derived here).

• Prices of the period in which the gross investment is produced

  P and W are the prices of the period in which the gross investment is made, so pi computed with period t-1 prices is the pi_(t-1) of HE-FM-DFH-002. C is the cost within that period; costs of later periods are discounted with HE-FN-DISC-001.

Worked examples

• Unit cost of 40 GBP in the health production example

  With a price of care of 40 GBP, a wage of 10 GBP, K of 0.5 and schooling at the reference level, the unit cost is 2 times the square root of 40 times 10, 40 GBP, so 20 units of gross investment cost 800 GBP, the article's C / I = 800 / 20 = 40.

  rho_H = 0.03; E = 0; P = 40; W = 10; K = 0.5; I = 20; pi = 40; C = 800

• Outlay for 20 units of gross investment after one more year of schooling

  One more year of schooling at rho_H of 0.03 divides the unit cost by exp(0.03), to 38.8178 GBP, so holding gross investment at 20 costs 776.36 GBP, the article's figure, about 24 GBP less.

  rho_H = 0.03; E = 1; P = 40; W = 10; K = 0.5; I = 20; pi = 38.8178; C = 776.36

• Outlay for 25 units of gross investment for the older person

  The article's older person needs 25 units of gross investment to hold the stock at 100 with depreciation of 0.25 (HE-FM-DFH-001), which at 40 GBP a unit costs 1,000 GBP.

  rho_H = 0.03; E = 0; P = 40; W = 10; K = 0.5; I = 25; pi = 40; C = 1000

• Unit cost of gross investment when own time is a quarter of the cost

  With the illustrative K of 0.25 of HE-EX-HPF-005 the unit cost is 49.6323 GBP, so 20 units cost 992.65 GBP and 800 GBP buys 800 / 49.6323 = 16.12 units, the gross investment of HE-EX-HPF-005 (computed here for illustration).

  rho_H = 0.03; E = 0; P = 40; W = 10; K = 0.25; I = 20; pi = 49.6323; C = 992.65

Common errors

• Price of medical care taken as the unit cost of gross investment

  In the article's example the unit cost equals the price of care, 40 GBP, only because 2 times the square root of 40 times 10 happens to be 40. With a wage of 20 GBP and the same price of care the unit cost is 56.57 GBP, so using 40 GBP would understate it by 29% (computed here for illustration). HE-ER-DFH-004 makes the same point for the pure investment condition.

• Wage left out of the cost of producing health

  Own time is an input, so the unit cost rises with the wage. Grossman states that a 1 percent rise in the wage raises the marginal cost of gross investment by K percent; in the article's example a 10% rise in the wage, to 11 GBP, raises the unit cost by 4.88%, to 41.95 GBP (computed here for illustration).

• Linear reading of the schooling saving over several years

  Reading rho_H E as a percentage cut in cost is close for one year, 776.00 GBP against 776.36 GBP for 20 units, but the gap widens as the change in E grows. For four extra years at 0.03 the outlay is 800 times exp(-0.12), 709.54 GBP, against 704.00 GBP from the linear reading (computed here for illustration).

Sources

• Grossman on the marginal cost of gross investment, the wage and schooling

  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, the text after equation (2-10), which states that with linear homogeneity and prices independent of the level of the inputs the average cost of gross investment is constant and equal to the marginal cost; section 3.2, which states that if K is the fraction of the total cost of gross investment accounted for by time a 1 percent rise in W would increase marginal cost by K percent, and equation (3-10), in which the percentage change in marginal cost per unit change in E equals minus rho_H because a rise in E reduces the inputs required for a given gross investment; section 2.4, which reports that Ehrlich and Chuma introduce rising marginal cost of investment. The closed-form unit cost with the share terms is derived here from equation (5-3) and the least-cost condition.

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• Grossman's 1972 statement that schooling and the wage shift the marginal cost of gross investment

  Grossman M. The Demand for Health: A Theoretical and Empirical Investigation. National Bureau of Economic Research, New York; 1972. Chapter 2, The Shadow Price of Health, pages 11 to 30. Section 3, Market and Nonmarket Efficiency, page 22, which states that a 1 percent rise in W would increase marginal cost by K percent, and page 26, equation (2-17), which states that under factor neutrality education lowers average or marginal cost and that education raising the marginal products of medical care and own time by 3 percent would reduce the price of gross investment by 3 percent.

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Canonical Identity