Grossman demand for the health stock as a ratio between two states with the implied gross investment

Signature

H_ratio = (W_1 / W_0)^eps * (pi_1 / pi_0)^(-eps) * (delta_1 / delta_0)^(-eps); I_ratio = (delta_1 / delta_0) * H_ratio

Function

Derived demand for medical care from the Grossman demand for the health stock

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Implementations

• Excel

  Health stock demand ratio and gross investment ratio from named cells

  With the elasticity in MECElasticity, the wages in WageBefore and WageAfter, the marginal costs of gross investment in InvestCostBefore and InvestCostAfter and the depreciation rates in DeprRateBefore and DeprRateAfter, the first formula fills HealthStockRatio and the second GrossInvestRatio. InvestCostAfter can hold UnitCostInvest from HE-IM-HPF-003 computed at the new prices.

  =(WageAfter/WageBefore)^MECElasticity*(InvestCostAfter/InvestCostBefore)^(-MECElasticity)*(DeprRateAfter/DeprRateBefore)^(-MECElasticity); =(DeprRateAfter/DeprRateBefore)*HealthStockRatio

Assumptions

• Pure investment model with a zero real-own rate of interest in the health stock demand

  Healthy time yields no utility directly and the cost of health capital reduces to delta because the real-own rate of interest, the interest rate r minus the rate of change in pi, is zero, the assumption Grossman states for his demand function for the stock of health. With a positive real-own rate the cost of capital is r minus pi_tilde plus delta, as in HE-FM-DFH-002, and Grossman's 1972 equation (4-2) keeps it inside the logarithm, so when r minus pi_tilde is positive a change in delta moves the cost of capital by a smaller proportion than delta itself (derived here).

• Constant elasticity of the marginal efficiency schedule from Grossman's healthy-time function

  Grossman's healthy-time function h = 8,760 minus B H^(-C), with B and C positive constants, gives a log-linear marginal efficiency schedule whose elasticity is eps = 1/(1 + C), below 1 since C is positive, and he argues on this ground that eps is likely to be smaller than one. The formula needs only a constant positive eps. A value of 1 or more cannot come from that healthy-time function and serves as a counterfactual, as in the article's case of eps = 1.5.

• Net investment small relative to depreciation for the gross investment ratio

  I_ratio treats gross investment as delta H, the identity of HE-FM-DFH-001 with the rate of net disinvestment small relative to the depreciation rate, the condition under which Grossman drops the net disinvestment term from his estimating equations. When the stock changes quickly within the period the ratio is an approximation (derived here).

Worked examples

• Doubling of depreciation with eps of 0.5 in the derived demand worked example

  The article's illustrative person has eps of 0.5, an unchanged wage and marginal cost of gross investment (10 and 40 here, illustrative) and a depreciation rate that doubles, 0.02 a year before and 0.04 after. The stock demanded falls to 1 over the square root of 2, about 29% less, while gross investment rises by about 41%, as in the article.

  eps = 0.5; W_0 = 10; W_1 = 10; pi_0 = 40; pi_1 = 40; delta_0 = 0.02; delta_1 = 0.04; H_ratio = 0.7071; I_ratio = 1.4142

• Doubling of depreciation with eps of 1.5 in the derived demand worked example

  With the article's counterfactual eps of 1.5 and the same doubling of depreciation the stock demanded falls to 2 to the power -1.5, 0.3536, and gross investment to 0.7071, so investment falls by about 29%, as in the article.

  eps = 1.5; W_0 = 10; W_1 = 10; pi_0 = 40; pi_1 = 40; delta_0 = 0.02; delta_1 = 0.04; H_ratio = 0.3536; I_ratio = 0.7071

• Unit MEC elasticity leaves gross investment unchanged when depreciation doubles

  With eps of 1 the stock demanded halves when depreciation doubles, so gross investment, the doubled rate times the halved stock, is unchanged, the limiting case between the article's two examples (computed here for illustration).

  eps = 1; W_0 = 10; W_1 = 10; pi_0 = 40; pi_1 = 40; delta_0 = 0.02; delta_1 = 0.04; H_ratio = 0.5; I_ratio = 1

• Ten per cent wage rise with the marginal cost of gross investment taken from the unit cost record

  An illustrative wage rise to 11 from 10, with a price of care of 40, K of 0.5 and schooling at the reference level, raises the unit cost of HE-FM-HPF-003 to 41.9524 from 40. With eps of 0.5 and depreciation unchanged at 0.04 the stock demanded rises by 2.41%, which is 1.1 to the power (1-K) eps = 0.25, Grossman's wage elasticity of health, and gross investment rises in the same proportion (computed here for illustration).

  eps = 0.5; W_0 = 10; W_1 = 11; pi_0 = 40; pi_1 = 41.9524; delta_0 = 0.04; delta_1 = 0.04; H_ratio = 1.0241; I_ratio = 1.0241

Common errors

• Fall in the health stock demanded read as a fall in gross investment and care

  In the article's example the stock demanded falls by about 29% when depreciation doubles, yet gross investment, and with it care, rises by about 41%. Grossman shows that when eps is below one gross investment and the depreciation rate are positively correlated over the life cycle while gross investment and the stock of health are negatively correlated, so less health demanded does not mean less care bought.

• Price of medical care used in place of the marginal cost of gross investment

  pi is the unit cost of HE-FM-HPF-003, not the price of care. With K of 0.5 a 10% rise in the price of care raises pi by 4.88%, so with eps of 0.5 the stock demanded falls by 2.35%, while entering the price ratio as the cost ratio gives a fall of 4.65%, about twice as large (computed here for illustration). HE-ER-DFH-004 makes the same point for the pure investment condition.

• Positive real-own rate of interest left out of the cost of health capital

  The formula assumes a zero real-own rate. If r minus pi_tilde were 0.03, doubling depreciation to 0.04 from 0.02 would raise the cost of capital to 0.07 from 0.05, a factor of 1.4, and with eps of 0.5 the stock demanded would fall to 0.8452 of its level, a fall of 15.5% against the 29.3% that the zero-rate formula gives (computed here for illustration with the cost of capital of HE-FM-DFH-002).

Sources

• Grossman on the healthy-time function and the demand function for the stock of health

  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 3.1, Depreciation rate effects, equation (29), the healthy time function h = 8,760 minus B H^(-C) with B and C positive constants, equation (30), the corresponding MEC schedule ln gamma = ln BC minus (C + 1) ln H plus ln W minus ln pi, and the statement that its elasticity is eps = 1/(1 + C), below one since C is positive, with the conclusion that if eps is below one gross investment and the depreciation rate are positively correlated over the life cycle while gross investment and the stock of health are negatively correlated. Section 3.2, Market and nonmarket efficiency, equation (31), the wage elasticity of health (1-K) eps, and equation (35), the schooling effect rho_H eps. Section 5.1, Structure and reduced form, equation (42), the demand function for the stock of health, obtained by solving equation (30) for ln H, with intercepts suppressed and the real-own rate of interest equal to zero; equation (44), the identity that gross investment equals net investment plus depreciation; equation (45), the reduced form for health. The ratio form and the gross investment ratio are derived here.

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• Grossman 1972 on the stock of health demand function in the pure investment model

  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 39 and 40, which derives from the production function of healthy days, equation (4-1), the constant-elasticity MEC schedule with eps = 1/(1 + C) and gives the stock of health demand function, equation (4-2), with the cost of capital r minus pi_tilde plus delta inside the logarithm, and its form with the real-own rate of interest equal to zero, equation (4-3), ln H = B' plus eps ln W minus eps ln pi minus eps ln delta.

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Canonical Identity