Health Production Function

Health Production Function in Grossman's Model of Health Produced from Medical Care, Time and Education

In health economics, the health production function treats health as something people produce, with medical care as only one of the inputs. Michael Grossman built it into his 1972 model of the demand for health, set out in a Journal of Political Economy paper and a National Bureau of Economic Research (NBER) monograph. In it, people combine market goods, chiefly medical care, with their own time to add to a stock of health that wears down with age, and education makes them more efficient producers. Its economic foundation is Gary Becker's household production theory, which Grossman applied to health. The idea matters because it explains why the demand for medical care is derived from the demand for health, why spending more on care need not buy proportionally more health, and why sicker people use more care.

From household production to the production of health

Traditional demand theory treats market goods as the objects that give satisfaction. Becker's household production model, which Grossman adopted, holds instead that people produce basic commodities, the things that enter the utility function, from market goods and their own time, in the same way that a firm turns inputs into output. Health is one such commodity, and Grossman described his work as motivated by the difference between health as an output and medical care as one of a number of inputs into its production.

Because goods and services are inputs into the production of commodities, the demand for them is a derived demand for a factor of production. The demand for medical care and other health inputs therefore derives from the basic demand for health.

Grossman's gross investment function and the stock of health

Grossman treats health as a durable stock, a form of health capital. People inherit an initial stock that depreciates with age and can be increased by investment, and death occurs when the stock falls below a minimum level. The stock changes from one period to the next as follows.

$$H_{t+1} - H_t = I_t - \delta_t H_t$$

where $H_t$ is the stock of health at age $t$, $I_t$ is gross investment in health and $\delta_t$ is the rate of depreciation during period $t$, which the model takes as exogenous but allows to vary with age. The health production function is the rule that generates gross investment, shown below.

$$I_t = I_t(M_t, TH_t; E)$$

where $M_t$ is the market goods input, which Grossman mostly identifies with medical care, $TH_t$ is the person's own time spent producing health, and $E$ is the stock of knowledge or human capital other than health, measured in practice by years of formal schooling. The semicolon marks $E$ as predetermined, not chosen in the period. Grossman assumes the function is homogeneous of degree one in medical care and own time, so doubling both doubles gross investment.

The stock yields healthy time, which is why people demand it; the demand side is covered on the Grossman model, demand for health and derived demand for healthcare pages.

Medical care, own time, lifestyle and education as inputs

Grossman's 1972 monograph treats medical care as the most important market good in the function because the other inputs are hard to measure, while noting that housing, diet, recreation, cigarette smoking and alcohol consumption also influence health. His later restatement lists medical care use, diet, exercise, cigarette smoking and alcohol consumption as health inputs, and notes that smoking and excessive alcohol use have negative marginal products in producing health. People buy them as inputs into other commodities, such as smoking pleasure, so a lifestyle input can lower health and still be chosen.

Own time is a separate input from sick time. The time a person spends visiting a doctor for a periodic check-up is part of the time input in the health production function, not time lost to illness. Because time is valued at the wage, the cost of producing health depends on earnings as well as on the price of care.

Education enters as an efficiency shifter. Grossman defines efficiency as the amount of health obtained from given health inputs and, following Michael and Becker, treats knowledge capital as raising household efficiency as technology raises the efficiency of firms. In his main case an extra year of schooling raises the marginal products of medical care and own time by the same percentage, so the more educated need fewer inputs for a given gross investment. He separates this productive efficiency from allocative efficiency, in which schooling improves knowledge of the true health effects of inputs, for example of smoking or diet, and so leads to a better input mix. These are productive efficiency and allocative efficiency inside the household.

Least-cost input choices and the predictions they drive

Given the prices of medical care and time, a person producing a chosen amount of gross investment minimises its cost. The choice of input mix is therefore a standard cost-minimisation problem. Grossman's condition is that the ratio of the marginal products of medical care and time equals the ratio of their prices, as shown below.

$$\frac{\partial I_t / \partial M_t}{\partial I_t / \partial TH_t} = \frac{P_t}{W_t}$$

where $P_t$ is the price of medical care and $W_t$ is the wage rate, which is the price of own time. Because the function has constant returns in these two inputs and their prices do not depend on how much is used, the average cost of gross investment is constant and equal to its marginal cost.

For estimation Grossman used a Cobb-Douglas form, with intercepts suppressed, written as follows.

$$\ln I_t = \rho_H E + (1-K)\ln M_t + K \ln TH_t$$

where $\rho_H$ is the percentage increase in gross investment from a one-unit increase in $E$, $K$ is the share of own time in the total cost of gross investment, and $(1-K)$ is the share of medical care, which is also the elasticity of gross investment with respect to medical care. In this form the elasticity of substitution between medical care and time equals one.

Under conditions set out in the full Grossman model, rising depreciation leads older people to hold less health but buy more gross investment and medical care, which is why sick time and medical care move together over the life cycle.

Estimating health production functions from data

Fitting the production function directly is difficult. An ordinary least squares regression of health on medical care is biased because the unobserved depreciation rate sits in the error term. People with high depreciation demand more medical care and have less health, so the medical care coefficient is biased downward, and unmeasured differences in efficiency cause the same bias. Grossman argues that the same kind of bias explains the negative association between health and medical care that, as he reports, Zweifel and Breyer used to criticise the model. Two-stage least squares, a form of instrumental variable analysis with wealth, wage rates and the price of care as instruments for medical care, can give consistent estimates, but only if the instruments are valid and strong, which is doubtful when several inputs matter and some are unmeasured.

Grossman credits Auster, Leveson and Sarachek with first emphasising these biases. Their 1969 study, which treated medical services as an intermediate product in the production of health, applied two-stage least squares to age and sex adjusted death rates across US states in 1960. A 1% increase in the quantity of medical services was associated with a reduction in mortality of about 0.1%, environmental factors led by income and education were the more important determinants of interstate variation in death rates, and the effect of education on mortality was about double that of medical care.

Randomising insurance rather than observing who uses care avoids the selection problem. In the RAND Health Insurance Experiment, 3,958 people aged 14 to 61 were randomly assigned to insurance plans for three or five years, and those required to pay part of their bills made about one-third fewer physician visits and were admitted to hospital about one-third less often than those with free care. In production terms, the extra care that free care induced added little measurable health for the average participant, the confidence intervals ruling out all but a minimal effect, and detectable gains were confined to vision and blood pressure in particular groups. Because the contrast is with cost sharing, the result concerns care at the margin, not care as a whole.

Worked example of least-cost inputs, schooling and depreciation

The figures are illustrative and not drawn from any study. A person produces gross investment with $I = M^{0.5} TH^{0.5}$, so $K = 0.5$, facing a price of 40 GBP per unit of medical care and a wage of 10 GBP per hour, with 800 GBP to spend in the period. Health is measured in abstract stock units, not quality-adjusted life years.

1. Least-cost input mix. With both exponents equal to 0.5, the least-cost mix spends half the budget on each input, 400 GBP each, which buys 10 units of care and 40 hours of time, as shown below.

$$I = (10 \times 40)^{0.5} = 400^{0.5} = 20$$

where 20 is gross investment in stock units. The marginal product of care is $MP_M = 0.5 \times 20 / 10 = 1$ and that of time is $MP_{TH} = 0.5 \times 20 / 40 = 0.25$, a ratio of 4 that equals the price ratio $P / W = 40 / 10 = 4$, so the least-cost condition holds. Each unit of gross investment costs $C / I = 800 / 20 = 40$ GBP, where $C$ is the outlay. Spending 600 GBP on 15 units of care and 200 GBP on 20 hours instead gives $(15 \times 20)^{0.5} = 300^{0.5} = 17.32$, about 13% less for the same outlay.

2. Schooling as efficiency. If one more year of schooling raises $\rho_H E$ by 0.03, the same inputs give $I = 1.030455 \times 20 = 20.61$, where 1.030455 is $e^{0.03}$. Holding gross investment at 20 would instead cost $C = 800 / 1.030455 = 776.36$ GBP, about 24 GBP less.

3. Depreciation and age. The person starts the period with a stock of 100 units and a depreciation rate of 0.15, so the stock next period is given below.

$$H_{t+1} = 0.85 \times 100 + 20 = 105$$

where 0.85 is one minus the depreciation rate and 20 is gross investment. An older person with the same stock, a depreciation rate of 0.25 and the same 800 GBP would end with $H_{t+1} = 0.75 \times 100 + 20 = 95$. Suppose the older person chooses to hold the stock at 100, consistent with the model's prediction of a lower stock but more gross investment when depreciation is higher. That needs gross investment of 25, costing $C = 25 \times 40 = 1{,}000$ GBP, which buys 12.5 units of care and 50 hours of time, since $(12.5 \times 50)^{0.5} = 625^{0.5} = 25$.

4. Why a simple comparison misleads. The younger person uses 10 units of care and ends with 105; the older person uses 12.5 units and ends with 100. More care goes with less health, yet for each person an extra unit of care adds to gross investment. The unrecorded difference in depreciation drives both, which is the downward bias in ordinary least squares described above.

Health production function versus healthcare production function

The two functions have different outputs. The healthcare production function describes how providers turn staff, capital and supplies into care such as consultations or admissions, while the health production function takes that care as one input, alongside own time, lifestyle and education, and asks how much health results. Auster, Leveson and Sarachek linked the two by nesting a production function of medical services, built from physicians, paramedical personnel, medical capital and prescription drugs, inside their production function of health.

Using the concept in economic evaluation and policy

The health production function frames health policy as a choice among inputs such as medical care, education, or changes in diet, housing or smoking. Auster, Leveson and Sarachek used their 1960 estimates to suggest that the returns to spending on education were far greater than those to spending on medical care.

Grossman points out that such a comparison assumes the schooling and health correlation reflects only the effect of schooling on health. It could also reflect better health leading to more schooling, or third variables such as ability, parental characteristics or, in Fuchs's hypothesis, time preference; if so, more spending on schooling would not improve health.

An economic evaluation of a single intervention estimates something narrower, namely the marginal product of one input, in one population, measured in an outcome such as quality-adjusted life years, compared with the opportunity cost of the resources used. The value of extra care depends on the existing level of inputs, the person's depreciation rate and their efficiency as a producer, so the same service can have different marginal products in different groups, as the RAND results showed. A fitted production function does not by itself value health or settle questions of equity.

Limitations and common misreadings

Most errors come from reading observed associations as production relationships. The points below are common misreadings.

• Reading medical care as health. Care is an input. Spending, visits or admissions measure input use, not health output.
• Reading cross-sectional coefficients as causal. Unobserved depreciation and efficiency bias the estimated effect of care downward.
• Treating the schooling and health correlation as an efficiency effect. Reverse causation and third variables may explain it.
• Taking the functional form literally. Constant returns in medical care and time, a Cobb-Douglas form and exogenous depreciation are simplifying assumptions. Ehrlich and Chuma, as quoted by Grossman, argued that constant returns leave optimal investment indeterminate, a criticism Grossman disputes.
• Assuming medical care is the main input. Grossman notes evidence that medical care may be an unimportant determinant of health in developed countries; the model keeps its structure if another purchased input dominates.
• Confusing the two production functions. Evidence that a provider produces care efficiently says nothing on its own about whether that care produces health.

Sources

• Auster R, Leveson I, Sarachek D. The production of health, an exploratory study. Journal of Human Resources. 1969;4(4):411-436. doi:10.2307/145166. Reprinted in: Fuchs VR, ed. Essays in the Economics of Health and Medical Care. New York: National Bureau of Economic Research; 1972:135-158.
• Brook RH, Ware JE, Rogers WH, Keeler EB, Davies AR, Donald CA, et al. Does free care improve adults' health? Results from a randomized controlled trial. New England Journal of Medicine. 1983;309(23):1426-1434. doi:10.1056/NEJM198312083092305
• Brook RH, Ware JE, Rogers WH, Keeler EB, Davies AR, Sherbourne CD, et al. The Effect of Coinsurance on the Health of Adults: Results from the RAND Health Insurance Experiment. R-3055-HHS. Santa Monica, CA: RAND Corporation; 1984. doi:10.7249/R3055
• Grossman M. The Demand for Health: A Theoretical and Empirical Investigation. New York: National Bureau of Economic Research; 1972. Chapter 1.
• Grossman M. The human capital model. In: Culyer AJ, Newhouse JP, eds. Handbook of Health Economics. Amsterdam: Elsevier; 2000:347-408. doi:10.1016/S1574-0064(00)80166-3. Version read: The human capital model of the demand for health, NBER Working Paper 7078; 1999. https://www.nber.org/papers/w7078