Signature
e_MW = (1 - K) * eps + K; e_MP_scale = -(1 - K) * eps; e_MP_subst = -K; e_MP = e_MP_scale + e_MP_subst; s_ME = rho_H * (eps - 1); e_Mdelta = 1 - eps; M_ratio = exp(e_MW * log(W_1 / W_0) + e_MP * log(P_1 / P_0) + s_ME * (E_1 - E_0) + e_Mdelta * log(delta_1 / delta_0))
| Inputs | Definition | Unit |
|---|---|---|
K | Share of own time in the total cost of gross investment, greater than 0 and less than 1 and constant under the Cobb-Douglas form, so that 1-K is the share of medical care | proportion |
eps | Elasticity of the marginal efficiency of capital schedule, the parameter of the demand for the health stock, a positive constant | dimensionless |
rho_H | Proportional rise in the marginal products of medical care and own time from one more year of schooling | proportion per year of schooling |
W_1 | Wage rate in the second state | currency per hour |
W_0 | Wage rate in the first state | currency per hour |
P_1 | Money price of one unit of medical care in the second state, above 0 | currency per unit of medical care |
P_0 | Money price of one unit of medical care in the first state, above 0 | currency per unit of medical care |
E_1 | Years of formal schooling in the second state | years |
E_0 | Years of formal schooling in the first state; only the difference E_1 minus E_0 matters | years |
delta_1 | Rate of depreciation of the health stock in the second state, above 0 | proportion per year |
delta_0 | Rate of depreciation of the health stock in the first state, above 0 | proportion per year |
e_MW | Percentage change in medical care demanded for a 1% rise in the wage, (1-K) eps + K | dimensionless |
|---|---|---|
e_MP_scale | Part of the price elasticity of care that works through the lower health stock demanded, minus (1-K) eps | dimensionless |
e_MP_subst | Part of the price elasticity of care from substituting own time for care, minus K | dimensionless |
e_MP | Percentage change in medical care demanded for a 1% rise in its price, the sum of the scale and substitution parts | dimensionless |
s_ME | Proportional change in medical care demanded for one more year of schooling, rho_H (eps minus 1) | proportion per year of schooling |
e_Mdelta | Percentage change in medical care demanded for a 1% rise in the depreciation rate, 1 minus eps | dimensionless |
M_ratio | Medical care demanded in the second state divided by care demanded in the first, M_1 / M_0 | ratio |
Function
Derived demand for medical care from the Grossman demand for the health stock
Maps the elasticity of the marginal efficiency of health capital, the time share of the cost of gross investment, the schooling efficiency parameter and a change in the wage, the marginal cost of gross investment or the price of care, schooling and the depreciation rate to the proportional change in the health stock demanded, in gross investment and in medical care demanded, in the pure investment version of the Grossman model with a zero real-own rate of interest. HE-FM-DDHC-001 gives Grossman's demand function for the health stock and HE-FM-DDHC-002 his reduced-form demand function for medical care with its wage, price, schooling and depreciation coefficients. Both build on live records that this package does not restate, the pure investment condition HE-FM-DFH-002, the stock update HE-FM-DFH-001 (gross investment of about delta H when net investment is small), the Cobb-Douglas gross investment of HE-FM-HPF-001, the least-cost care of HE-FM-HPF-002 and the unit cost pi of HE-FM-HPF-003. Because Grossman suppresses the intercepts, both formulas compare two states, such as two ages of one person or two people, as ratios, and for a change in one price alone each ratio has the constant-elasticity form of HE-FM-XPE-003. Notation follows the Derived Demand for Healthcare article.
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Implementations
Excel
Elasticities and ratio of the derived demand for medical care from named cells
With MECElasticity, WageBefore, WageAfter, DeprRateBefore and DeprRateAfter as in HE-IM-DDHC-001, the time share in TimeShareCost, the schooling parameter in SchoolingEffect, the years of schooling in SchoolingBefore and SchoolingAfter and the prices of care in PriceCareBefore and PriceCareAfter, the seven formulas fill CareWageElast, CarePriceScale, CarePriceSubst, CarePriceElast, CareSchoolingEffect, CareDeprElast and CareDemandRatio.
=(1-TimeShareCost)*MECElasticity+TimeShareCost; =-(1-TimeShareCost)*MECElasticity; =-TimeShareCost; =CarePriceScale+CarePriceSubst; =SchoolingEffect*(MECElasticity-1); =1-MECElasticity; =EXP(CareWageElast*LN(WageAfter/WageBefore)+CarePriceElast*LN(PriceCareAfter/PriceCareBefore)+CareSchoolingEffect*(SchoolingAfter-SchoolingBefore)+CareDeprElast*LN(DeprRateAfter/DeprRateBefore))
Assumptions
Cobb-Douglas gross investment with a unit elasticity of substitution in the derived demand for care
Gross investment has the Cobb-Douglas form of HE-FM-HPF-001, so the elasticity of substitution between care and own time is one and K is constant. Grossman states that with an elasticity of substitution other than one the term K in the wage and price elasticities of care would be multiplied by it, and that K would then depend on W and P, so the coefficients would not be constant.
Pure investment model, zero real-own rate and small net disinvestment in the derived demand for care
The formula keeps the conditions of HE-FM-DDHC-001, the pure investment model and a zero real-own rate of interest, and leaves out the term in the rate of net disinvestment, which Grossman states could be ignored if net disinvestment were small relative to the depreciation rate. The depreciation ratio covers both a rise with age and a difference in initial depreciation between people of the same age.
Money price of care and two inputs in the derived demand for care
Care has a money price P and medical care and own time are the only inputs. Grossman shows that where the money price is zero and care is rationed by time, so that its price is the wage times the hours needed, and a third input needs no time, the wage elasticity of care becomes (1-K)(eps minus sigma_MX), with K reinterpreted as the share of care and own time together in the cost of gross investment and sigma_MX the partial elasticity of substitution between care and that input, and can be zero or negative. This formula does not cover that case.
Factor-neutral schooling in the derived demand for care
One more year of schooling raises the marginal products of care and own time by the same rho_H, Grossman's factor-neutrality assumption, so it leaves the least-cost mix unchanged and changes care only through the health stock demanded and the inputs needed per unit of gross investment.
Worked examples
Doubling of depreciation with eps of 0.5 in the derived demand for care
The article's person with eps of 0.5, K of 0.4 and rho_H of 0.06 faces a doubling of depreciation, 0.02 a year before and 0.04 after, with the wage, the price of care (10 and 40 here, illustrative) and schooling unchanged. Care rises by the depreciation elasticity 0.5 times ln 2, 0.347, a ratio of 1.4142, the same 41% as gross investment in HE-EX-DDHC-001. The price elasticity is -0.7, made up of -0.3 through the lower health stock and -0.4 from substitution towards own time, and the schooling effect is -0.03 a year, as in the article.
eps = 0.5; K = 0.4; rho_H = 0.06; W_0 = 10; W_1 = 10; P_0 = 40; P_1 = 40; E_0 = 0; E_1 = 0; delta_0 = 0.02; delta_1 = 0.04; e_MW = 0.7; e_MP_scale = -0.3; e_MP_subst = -0.4; e_MP = -0.7; s_ME = -0.03; e_Mdelta = 0.5; M_ratio = 1.4142
One more year of schooling in the derived demand for care
With eps of 0.5 and rho_H of 0.06 one more year of schooling changes care by rho_H (eps minus 1) = -0.03, the article's fall of 3% in log terms. The exact ratio is exp(-0.03) = 0.9704, a fall of 2.96%, while the health stock demanded rises by rho_H eps = 0.03 (computed here).
eps = 0.5; K = 0.4; rho_H = 0.06; W_0 = 10; W_1 = 10; P_0 = 40; P_1 = 40; E_0 = 0; E_1 = 1; delta_0 = 0.02; delta_1 = 0.02; e_MW = 0.7; e_MP_scale = -0.3; e_MP_subst = -0.4; e_MP = -0.7; s_ME = -0.03; e_Mdelta = 0.5; M_ratio = 0.9704
Doubling of depreciation with eps of 1.5 in the derived demand for care
With the article's counterfactual eps of 1.5 the depreciation elasticity of care is -0.5, so doubling depreciation cuts care to 0.7071, a fall of about 29%, as in the article. The price elasticity is -1.3, of which -0.9 is the scale part, and the schooling effect turns positive at 0.03 a year (computed here).
eps = 1.5; K = 0.4; rho_H = 0.06; W_0 = 10; W_1 = 10; P_0 = 40; P_1 = 40; E_0 = 0; E_1 = 0; delta_0 = 0.02; delta_1 = 0.04; e_MW = 1.3; e_MP_scale = -0.9; e_MP_subst = -0.4; e_MP = -1.3; s_ME = 0.03; e_Mdelta = -0.5; M_ratio = 0.7071
Ten per cent rise in the price of care in the derived demand for care
With eps of 0.5 and K of 0.4 an illustrative rise in the price of care to 44 from 40, with nothing else changed, gives 1.1 to the power -0.7, a ratio of 0.9355, a fall of 6.45% in care demanded (computed here for illustration). This is HE-FM-XPE-003 with own-price elasticity -0.7.
eps = 0.5; K = 0.4; rho_H = 0.06; W_0 = 10; W_1 = 10; P_0 = 40; P_1 = 44; E_0 = 0; E_1 = 0; delta_0 = 0.02; delta_1 = 0.02; e_MW = 0.7; e_MP_scale = -0.3; e_MP_subst = -0.4; e_MP = -0.7; s_ME = -0.03; e_Mdelta = 0.5; M_ratio = 0.9355
Unit MEC elasticity in the derived demand for care
With eps of 1 the schooling and depreciation effects on care vanish and the price elasticity is -1, so a doubling of depreciation together with one more year of schooling leaves care unchanged, a limiting case that checks an implementation (computed here for illustration).
eps = 1; K = 0.4; rho_H = 0.06; W_0 = 10; W_1 = 10; P_0 = 40; P_1 = 40; E_0 = 0; E_1 = 1; delta_0 = 0.02; delta_1 = 0.04; e_MW = 1; e_MP_scale = -0.6; e_MP_subst = -0.4; e_MP = -1; s_ME = 0; e_Mdelta = 0; M_ratio = 1
Common errors
Coefficient on log depreciation read as the percentage change for a doubling
The coefficient 1-eps applies to ln delta, so with eps of 0.5 a doubling of depreciation raises care by exp(0.5 times ln 2), 41.4%, not by 50%, which overstates the rise by about a fifth (computed here for illustration). The article takes the same route, 0.347 and its exponential.
Price elasticity of care taken as the substitution part or the scale part alone
With eps of 0.5 and K of 0.4 the price elasticity of care is -0.7. Using the substitution part alone, -0.4, understates the response by 43%, and using the scale part alone, -0.3, by 57% (computed here for illustration). The scale part is the response of the health stock demanded to the price of care, so a care elasticity estimated with health held fixed leaves it out (derived here).
Negative health coefficient in a conditional demand for care read as evidence against derived demand
Grossman states that when the conditional demand for care is estimated with health treated as exogenous, the initial depreciation rate in the disturbance is positively correlated with care and negatively with health, so the coefficient of health is biased downward and could well be negative. In HE-EX-DDHC-005 the same doubling of depreciation gives 41% more care with 29% less health (HE-EX-DDHC-001).
Sources
Grossman on the reduced-form demand function for medical care and its elasticities
Grossman M. The human capital model. In Culyer AJ, Newhouse JP, editors. Handbook of Health Economics. Vol 1A. Elsevier, Amsterdam; 2000. p. 347-408. Section 5.1, Structure and reduced form, which states that equations (42) to (44) and the least-cost condition that the ratio of the marginal products of care and time equals the ratio of the price of care to the wage generate the reduced form demand curves, equation (46) for medical care with coefficients [(1-K) eps + K] on ln W, minus [(1-K) eps + K] on ln P, rho_H (eps minus 1) on E and 1-eps on the depreciation terms; that the last term could be ignored if net disinvestment were small relative to the rate of depreciation; that the investment model predicts a positive wage and a negative price coefficient for care and, if eps is below one, a negative schooling and a positive age coefficient; and that with an elasticity of substitution other than one the term K in the wage and price elasticities would be multiplied by it. Section 3.2, Market and nonmarket efficiency, equation (32), the wage elasticity of care K sigma_p plus (1-K) eps, equation (33) for care rationed by time with a third input, and equation (36), the schooling effect rho_H (eps minus 1). The ratio form and the split of the price elasticity into scale and substitution parts are derived here from equation (46).
Grossman 1972 on the reduced-form demand curves for health and medical care
Grossman M. The Demand for Health: A Theoretical and Empirical Investigation. NBER Occasional Paper 119. National Bureau of Economic Research, New York; 1972. Chapter IV, An Empirical Formulation of the Model, section 1, Structure and reduced form, pages 40 and 41, equation (4-5), the Cobb-Douglas gross investment function, and equations (4-6) and (4-7), the reduced form demand curves for health and medical care, with the statements that the last term in (4-7) could be ignored if net disinvestment were small relative to the rate of depreciation and that the equations contain no intercepts because all variables are expressed as deviations from their means.
Grossman on the bias in a conditional demand function for medical care
Grossman M. The human capital model. In Culyer AJ, Newhouse JP, editors. Handbook of Health Economics. Vol 1A. Elsevier, Amsterdam; 2000. p. 347-408. Section 6.1, Empirical extensions with cross-sectional data, which states that if the conditional demand function is estimated with health treated as exogenous, the depreciation rate in the initial period is positively correlated with medical care and negatively correlated with health, so that the coefficient of health is biased downward and could well be negative.
Canonical Identity
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