Article #1 and #5 FOR SMART-GURU ONLY

On the Concept of Health Capital and the Demand for Health

Michael Grossman National Bureau of Economic Research

The aim of this study is to construct a model of the demand for the commodity ''good health." The central proposition of the model is that health can be viewed as a durable capital stock that produces an output of healthy time. It is assumed that individuals inherit an initial stock of health that depreciates with age and can be increased by investment. In this framework, the "shadow price" of health depends on many other variables besides the price of medical care. It is shown that the shadow price rises with age if the rate of depreciation on the stock of health rises over the life cycle and falls with education if more educated people are more efficient producers of health. Of particular importance is the conclusion that, under certain conditions, an increase in the shadow price may simultaneously reduce the quantity of health demanded and increase the quantity of medical care demanded.

I. Introduction

During the past two decades, the notion that individuals invest in them- selves has become widely accepted in economics. At a conceptual level, increases in a person's stock of knowledge or human capital are assumed to raise his productivity in the market sector of the economy, where he produces money earnings, and in the nonmarket or household sector, where he produces commodities that enter his utility function. To realize

This paper is hased on part of my Columbia University Ph.D. dissertation, "The Demand for Health: A Theoretical and Empirical Investigation," which will be pub- lished by the National Bureau of Economic Research. My research at the Bureau was supported by the Commonwealth Fund and the National Center for Health Services Research and Development (PHS research grant 2 P 01 HS 00451-04). Most of this paper was written while I was at the University of Chicago's Center for Health Ad- ministration Studies, with research support from the National Center for Health Ser- vices Research and Development (PHS grant HS 00080). A preliminary version of this paper was presented at the Second World Congress of the Econometric Society. I am grateful to Gary S. Becker, V. K. Chetty, Victor R. Fuchs, Gilbert R. Ghez, Robert T. Michael, and Jacob Mincer for their helpful comments on earlier drafts.

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potential gains in productivity^ individuals have an incentive to invest in formal schooling or in on-the-job training. The costs of these invest- ments include direct outlays on market goods and the opportunity cost of the time that must be withdrawn from competing uses. This frame- work has been used by Becker (1967) and by Ben-Porath (1967) to develop models that determine the optimal quantity of investment in human capital at any age. In addition, these models show how the optimal quantity varies over the life cycle of an individual and among individuals of the same age.

Although several writers have suggested that health can be viewed as one form of human capital (Mushkin 1962, pp. 129-49; Becker 1964, pp. 33-36; Fuchs 1966, pp. 90-91), no one has constructed a model of the demand for health capital itself. If increases in the stock of health simply increased wage rates, such a task would not be necessary, for one could simply apply Becker's and Ben-Porath's models to study the decision to invest in health. This paper argues, however, that health capital differs from other forms of human capital. In particular, it argues that a person's stock of knowledge affects his market and nonmarket productivity, while his stock of health determines the total amount of time he can spend producing money earnings and commodities. The fundamental difference between the two types of capital is the basic justification for the model of the demand for health that is presented in the paper.

A second justification for the model is that most students of medical economics have long realized that what consumers demand when they purchase medical services are not these services per se but, rather, '̂ good health." Given that the basic demand is for good health, it seems logical to study the demand for medical care by first constructing a model of the demand for health itself. Since, however, traditional demand theory assumes that goods and services purchased in the market enter consumers' utihty functions, economists have emphasized the demand for medical care at the expense of the demand for health. Fortunately, a new approach to consumer behavior draws a sharp distinction between fundamental ob- jects of choice—called ^'commodities'^-—and market goods (Becker 1965; Lancaster 1966; Muth 1966; Michael 1969; Becker and Michael 1970; Ghez 1970). Thus, it serves as the point of departure for my health model. In this approach, consumers produce commodities with inputs of market goods and their own time. For example, they use traveling time and transportation services to produce visits; part of their Sundays and church services to produce ''peace of mind"; and their own time, books, and teachers' services to produce additions to knowledge. Since goods and services are inputs into the production of commodities, the demand for these goods and services is a derived demand.

Within the new framework for examining consumer behavior, it is assumed that individuals inherit an initial stock of health that depreciates

CONCEPT OF HEALTH CAPITAL 225

over time—at an increasing rate, at least after some stage in the life cycle—and can be increased by investment. Death occurs when the stock falls below a certain level, and one of the novel features of the model is that individuals **choose" their length of life. Gross investments in health capital are produced by household production functions whose direct inputs include the own time of the consumer and market goods such as medical care, diet, exercise, recreation, and housing. The produc- tion function also depends on certain '^environmental variables," the most important of which is the level of education of the producer, that influence the efficiency of the production process.

It should be realized that in this model the level of health of an indi- vidual is not exogenous but depends, at least in part, on the resources allocated to its production. Health is demanded by consumers for two reasons. As a consumption commodity, it directly enters their preference functions, or, put differently, sick days are a source of disutility. As an investment commodity, it determines the total amount of time available for market and nonmarket activities. In other words, an increase in the stock of health reduces the time lost from these activities, and the mone- tary value of this reduction is an index of the return to an investment in health.

Since the most fundamental law in economics is the law of the down- ward-sloping demand curve, the quantity of health demanded should be negatively correlated with its shadow price. The analysis in this paper stresses that the shadow price of health depends on many other variables besides the price of medical care. Shifts in these variables alter the optimal amount of health and also alter the derived demand for gross investment, measured, say, by medical expenditures. It is shown that the shadow price rises with age if the rate of depreciation on the stock of health rises over the life cycle and falls with education if more educated people are more efficient producers of health. Of particular importance is the conclusion that, under cwtain conditions, an increase in the shadow price may simultaneously reduce the quantity of health demanded and increase the quantity of medical care demanded.

n . A Stock Approach to the Demand for Health

A, The Model

Let the intertemporal utility function of a typical consumer be

U = U(<f>ono, . . . , <f>nHn, ZQJ , . . , Zn), ( 1 )

where Ho is the inherited stock of health. Hi is the stock of health in the iih time period, <t>i is the service flow per unit stock, hi = </>ifl"̂ is total consumption of "health services," and Z^ is total consumption of another

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commodity in the ith period.^ Note that, whereas in the usual inter- temporal utility function «, the length of life as of the planning date, is fixed, here it is an endogenous variable. In particular, death takes place when Hi = Hmin- Therefore, length of life depends on the quantities of Hi that maximize utility subject to certain production and resource con- straints that are now outlined.

By definition, net investment in the stock of health equals gross invest- ment minus depreciation:

Hi^^~Hi^h-h,H,, (2)

where Ii is gross investment and 8̂ is the rate of depreciation during the fth period. The rates of depreciation are assumed to be exogenous, but they may vary with the age of the individual.^ Consumers produce gross investments in health and the other commodities in the utility function according to a set of household production functions:

(3)

In these equations, Mi is medical care, Xi is the goods input in the pro- duction of the commodity Z ,̂ THi and Ti are time inputs, and Ei is the stock of human capital.^ It is assumed that a shift in human capital changes the efficiency of the production process in the nonmarket sector of the economy, just as a shift in technology changes the efficiency of the production process in the market sector. The implications of this treat- ment of human capital are explored in Section IV.

It is also assumed that all production functions are homogeneous of degree 1 in the goods and time inputs. Therefore, the gross investment production function can be written as

(4)

where ti ^= THi/Mi. It follows that the marginal products of time and medical care in the production of gross investment in health are

1 The commodity Z,- may be viewed as an aggregate of all commodities besides health that enter the utility function in period i. For the convenience of the reader, a glossary of symbols may be found in Appendix B.

2 In a more complicated version of the model, the rate of depreciation might be a negative function of the stock of health. The analysis is considerably simplified by treating this rate as exogenous, and the conclusions reached would tend to hold even if it were endogenous.

3 In general, medical care is not the only market good in the gross investment func- tion, for inputs such as housing, diet, recreation, cigarette smoking, and alcohol con- sumption influence one*s level of health. Since these inputs also produce other commodities in the utility function, joint production occurs in the household. For an analysis of this phenomenon, see Grossman (1970, chap. 6). To emphasize the key aspects of my health model, I treat medical care as the most important market good in the gross investment function in the present paper.

CONCEPT OF HEALTH CAPITAL 2 2 7

(5)

From the point of view of the individual, both market goods and own time are scarce resources. The goods budget constraint equates the present value of outlays on goods to the present value of earnings income over the life cycle plus initial assets (discounted property income):^

W,TW, = 2. ^ + ^0- (6)

Here Pi and Vi are the prices of Mi and Xi^ W^ is the wage rate, TWi is hours of workj ^o is discounted property income, and r is the interest rate. The time constraint requires that Q, the total amount of time available in any period, must be exhausted by all possible uses:

TW, + TL, + TH, + T,= Q, (7)

where TLi is time lost from market and nonmarket activities due to illness or injury.

Equation (7) modifies the time budget constraint in Becker's time model (Becker 1965). If sick time were not added to market and non- market time, total time would not be exhausted by all possible uses. My model assumes that TL, is inversely related to the stock of health; that is, 'dTLi/'^Hi < 0. If Q were measured in days (Q = 365 days if the year is the relevant period) and if <f>i were defined as the flow of healthy days per unit of Hi, h-, would equal the total number of healthy days in a given year.^ Then one could write

TL, = Q-h. (8)

It is important to draw a sharp distinction between sick time and the time input in the gross investment function. As an illustration of this difference, the time a consumer allocates to visiting his doctor for periodic checkups is obviously not sick time. More formally, if the rate of de- preciation were held constant, an increase in TH.i would increase U and Hij^i and would reduce TL^^i. Thus, TH, and TZ^+i would be negatively correlated.®

^ The sums throughout this study are taken from f r= 0 to n. 5 If the stock of health yielded other services besides healthy days, 0̂ would be a

vector of service flows. This study emphasizes the service flow of healthy days because this flow can be measured empirically.

6 For a discussion of conditions that would produce a positive correlation between and TL^_^_^, see Section III.

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By substituting for TWi from equation (7) into equation ( 6 ) , one obtains the single '*full wealth'' constraint:

+ V,X, + W,{TL, + TH, +

(9)

According to equation ( 9 ) , full wealth equals initial assets plus the present value of the earnings an individual would obtain if he spent all of his time at work. Part of this wealth is spent on market goods, part of it is spent on nonmarket production time, and part of it is lost due to illness. The equilibrium quantities of Hi and Zi can now be found by maximizing the utility function given by equation (1) subject to the constraints given by equations ( 2 ) , ( 3 ) , and (9).'' Since the inherited stock of health and the rates of depreciation are given, the optimal quantities of gross investment determine the optimal quantities of health capital.

B. Equilibrium Conditions

First-order optimality conditions for gross investment in period i — 1

Uhn + . .. + (1 - 8,) ... (1 - K_y) -Y-Gn, (10)

11

The new symbols in these equations are: Vhi ^ "dUfdh; =: the marginal utility of healthy days; ^ = the marginal utihty of wealth; Gi-^'dhi/

i) = the marginal product of the stock of health in the production of healthy days; and :itt_i = the marginal cost of gross investment in health in period i — 1.

"̂ In addition, the constraint is imposed that H^ ^ ^miir 8 Note that an increase in gross investment in period i — 1 increases the stock of

health in all future periods. These increases are equal to

dHi _ _

For a derivation of equation (10), see Part A of the Mathematical Appendix.

CONCEPT OF HEALTH CAPITAL 2 2 9

Equation (10) simply states that the present value of the marginal cost of gross investment in period i — 1 must equal the present value of marginal benefits. Discounted marginal benefits at age / equal

I J

where Gi is the marginal product of health capital—the increase in the number of healthy days caused by a one-unit increase in the stock of health. Two monetary magnitudes are necessary to convert this marginal product into value terms, because consumers desire health for two reasons. The discounted wage rate measures the monetary value of a one-unit in- crease in the total amount of time available for market and nonmarket activities, and the term Uh.i/X measures the discounted monetary equiva- lent of the increase in utihty due to a one-unit increase in healthy time. Thus, the sum of these two terms measures the discounted marginal value to consumers of the output produced by health capital.

While equation (10) determines the optimal amount of gross invest- ment in period i— 1, equation (11) shows the condition for minimizing the cost of producing a given quantity of gross investment. Total cost is minimized when the increase in gross investment from spending an additional dollar on medical care equals the increase in gross investment from spending an addidonal dollar on time. Since the gross investment production function is homogeneous of degree 1 and since the prices of medical care and time are independent of the level of these inputs, the average cost of gross investment is constant and equal to the marginal cost.

To examine the forces that affect the demand for health and gross in- vestment, it is useful to convert equation (10) into a slightly different form. If gross investment in period i is positive, then

. _ i+id^i ( i 4 i ) i + 2 i + 2 / • * l \ ^ I 1 ' / . • • \ . * t O I ' " *

(l+r)

Ov _i -I ) ( 1 O« 1 ) vV«( r-.

— . (12)

From (10) and ( 1 2 ) ,

( l - l

( l + ; - ) i - i " (14-ry + ~ I ~ " ^

Therefore,

(13) )

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where JCi_i is the percentage rate of change in marginal cost between period i— 1 and period i.^ Equation (13) implies that the undiscounted value of the marginal product of the optimal stock of health capital at any moment in time must equal the supply price of capital, 7ti^i{r — jCi_i + 6 0 . The latter contains interest, depreciation, and capital gains components and may be interpreted as the rental price or user cost of health capital.

Condition (13) fully determines the demand for capital goods that can be bought and sold in a perfect market. In such a market, if firms or households acquire one unit of stock in period i— 1 at price Jti_i, they can sell (1 — hi) units at price Tti at the end of period i. Consequently, jt£_i(r — Jti_i + 8,) measures the cost of holding one unit of capital for one period. The transaction just described allows individuals to raise their capital in period / alone by one unit and is clearly feasible for stocks like automobiles, houses, refrigerators, and producer durables. It suggests that one can define a set of single-period flow equilibria for stocks that last for many periods.

In my model, the stock of health capital cannot be sold in the capital market, just as the stock of knowledge cannot be sold. This means that gross investment must be nonnegative. Although sales of health capital are ruled out, provided gross investment is positive, there exists a used cost of capital that in equilibrium must equal the value of the marginal product of the stock.̂ *̂ An intuitive interpretation of this result is that exchanges over time in the stock of health by an indi\adual substitute for exchanges in the capital market. Suppose a consumer desires to increase his stock of health by one unit in period i. Then he must increase gross investment in period i — 1 by one unit. If he simultaneously reduces gross investment in period i hy (1 — 8̂ ) units, then he has engaged in a transaction that raises Hi and Hi alone by one unit. Put differently, he has essentially rented one unit of capital from himself for one period. The magnitude of the reduction in It is smaller the greater the rate of depreciation, and its dollar value is larger the greater the rate of increase in marginal cost over time. Thus, the depreciation and capital gains components are as relevant to the user cost of health as they are to the user cost of any other durable. Of course, the interest component of user cost is easy to interpret, for if one desires to increase his stock of health rather than his stock of some other asset by one unit in a given period, rjt^_i measures the interest payment he

'^Equation (13) assumes S/5t-_j ^ 0 . 10 For similar conclusions with regard to nonsalable physical capital and with regard

to a nonsalable stock of ^'goodwill" produced by advertising, see Arrow (1968) and Nerlove and Arrow (1962).

11 In a continuous time model, the user cost of health capital can be derived in one step. If continuous time is employed, the term b^Ki_^ does not appear in the user cost formula. The right-hand side of (13) becomes Jt^(r — jt̂ + 6^), where K, is the in-

CONCEPT OF HEALTH CAPITAL 2 3 1

A slightly different form of equation (13) emerges if both sides are divided by the marginal cost of gross investment:

^ i, (13')

Here Yi = {WiGi)/Ki_i is the marginal monetary rate of return on an investment in health and

Uh,

is the psychic rate of return. In equilibrium, the total rate of return on an investment in health must equal the user cost of health capital in terms of the price of gross investment. The latter variable is defined as the sum of the real-own rate of interest and the rate of depreciation.

C. The Pure Investment Model

It is clear that the number of sick days and the number of healthy days are complements; their sum equals the constant length of the period. From equation (8), the marginal utility of sick time is —Uhi, Thus, by putting healthy days in the utility function, one implicitly assumes that sick days yield disutility. If healthy days did not enter the utility function directly, the marginal monetary rate of return on an investment in health would equal the cost of health capital, and health would be solely an in- vestment commodity.^- In formalizing the model, I have been reluctant to treat health as pure investment because many observers believe the demand for it has both investment and consumption aspects (see, for example, Mushkin 1962, p. 131; Fuchs 1966, p. 86). But to simplify the remainder of the theoretical analysis and to contrast health capital with other forms of human capital, the consumption aspects of demand are ignored from now on.̂ ^

If the marginal utility of healthy days or the marginal disutility of sick days were equal to zero, condition (13') for the optimal amount of health capital in period i would reduce to

i. ( 1 4 )

stantaneous percentage rate of change of marginal cost at age i. For a proof, see Part B of the Mathematical Appendix.

12 To avoid confusion, a note on terminology is in order. If health were entirely an investment commodity, it would yield monetary, but not utility, returns. Regardless of whether health is investment, consumption, or a mixture of the two, one can speak of a gross investment function since the commodity in question is a durable.

13 Elsewhere, I have used a pure consumption model to interpret the set of phenom- ena that are analyzed in Sections III and IV. In the pure consumption model, the marginal monetary rate of return on an investment in health is set equal to zero (see Grossman 1970, chap. 3).

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Equation (14) can be derived explicitly by excluding health from the utility function and by redefining the full wealth constraint as^^

' = 0̂ + (15)

Maximization of R' with respect to gross investment in periods i — 1 and i yields

+ (1 -f

(16)

+ ...+ (1 —

( 1 - 8 0 . . . (1 - K-

(17)

These two equations imply that (14) must hold. Figure 1 illustrates the determinations of the optimal stock of health

capital at any age i. The demand curve MEC shows the relationship between the stock of health and the rate of return on an investment or the marginal efficiency of health capital, yi- The supply curve S shows the relationship between the stock of health and the cost of capital, r — Si_i - j - 6i. Since the cost of capital is independent of the stock, the supply curve is infinitely elastic. Provided the MEC schedule slopes downward,

F I G . 1

Since the gross investment production function is homogeneous of the first degree,

CONCEPT OF HEALTH CAPITAL 233

365

FIG. 2

the equilibrium stock is given by Hi^j where the supply and demand curves intersect.

In the model, the wage rate and the marginal cost of gross investment do not depend on the stock of health. Therefore, the MEC schedule would be negatively inclined if and only if G ,̂ the marginal product of health capital, were diminishing. Since the output produced by health capital has a finite upper limit of 365 healthy days, it seems reasonable to assume diminishing marginal productivity. Figure 2 shows a plausible relation- ship between the stock of health and the number of healthy days. This relationship may be called the "production function of healthy days." The slope of the curve in the figure at any point gives the marginal product of health capital. The number of healthy days equals zero at the death stock ^min, so that Q = TLi ^ 3 6 5 is an alternative definition of death. Beyond ^min, healthy time increases at a decreasing rate and eventually approaches its upper asymptote of 365 days as the stock becomes large.

In Sections I I I and IV, equation (14) and figure 1 are used to trace out the lifetime path of health capital and gross investment, to explore the effects of variations in depreciation rates, and to examine the impact of changes in the marginal cost of gross investment. Before I turn to these matters, some comments on the general properties of the model are in order. It should be realized that equation (14) breaks down whenever desired gross investment equals zero. In this situation, the present value of the marginal cost of gross investment would exceed the present value of marginal benefits for all positive quantities of gross investment, and equations (16) and (17) would be replaced by inequalities.^^ The re- mainder of the discussion rules out zero gross investment by assumption, but the conclusions reached would have to be modified if this were not the case. One justification for this assumption is that it is observed em- pirically that most individuals make positive outlays on medical care throughout their life cycles.

h~i = h =15 Formally, ŷ ̂ ^ — Jtf_i +

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Some persons have argued that, since gross investment in health cannot be nonnegative, equilibrium condition (14) should be derived by using the optimal control techniques developed by Pontryagin and others. Arrow (1968) employs these techniques to analyze a firm's demand for non- salable physical capital. Since, however, gross investment in health is rarely equal to zero in the real world, the methods I use—discrete time maximization in the text and the calculus of variations in the Mathe- matical Appendix—are quite adequate. Some advantages of my methods are that they are simple, easy to interpret, and familiar to most econo- mists. In addition, they generate essentially the same equilibrium condition as the Pontryagin method. Both Arrow and I conclude that, if desired gross investment were positive, then the marginal efficiency of nonsalable capital would equal the cost of capital. On the other hand, given zero gross investment, the cost of capital would exceed its marginal efficiency.

The monetary returns to an investment in health differ from the returns to investments in education, on-the-job training, and other forms of human capital, since the latter investments raise wage rates.^^ Of course, the amount of health capital might infiuence the wage rate, but it necessarily influences the time lost from all activities due to illness or injury. To emphasize the novelty of my approach, I assume that health is not a determinant of the wage rate. Put differently, a person's stock of knowl- edge affects his market and nonmarket productivity, while his stock of health determines the total amount of time he can spend producing money earnings and commodities. Since both market time and nonmarket time are relevant, even individuals who are not in the labor force have an incentive to invest in their health. For such individuals, the marginal product of health capital would be converted into a dollar equivalent by multiplying by the monetary value of the marginal utility of time.

Since there are constant returns to scale in the production of gross investment and since input prices are given, the marginal cost of gross investment and its percentage rate of change over the life cycle are exogenous variables. In other words, these two variables are independent of the rate of investment and the stock of health. This implies that con- sumers reach their desired stock of capital immediately. It also implies that the stock rather than gross investment is the basic decision variable in the model. By this I mean that consumers respond to changes in the cost of capital by altering the marginal product of health capital and not the marginal cost of gross investment. Therefore, even though equation (14) is not independent of equations (16) and (17), it can be used to determine the optimal path of health capital and, by implication, the optimal path of gross investment.^'''

difference is emphasized by Mushkin (1962, pp. 132-33). 1*̂ This statement is subject to the modification that the optimal path of capital must

always imply nonnegative gross investment.

CONCEPT OF HEALTH CAPITAL 235

Indeed, the major differences between my health model and the human capital models of Becker (1967) and Ben-Porath (1967) are the assump- tions made about the behavior of the marginal product of capital and the marginal cost of gross investment. Both Becker and Ben-Porath assume that any one person owns only a small amount of the total stock of human capital in the economy. Therefore, the marginal product of his stock is constant. To rule out solutions in which the desired stock of capital is either zero or infinite, they postulate that the marginal cost of producing gross additions to the stock is positively related to the rate of gross investment. Since marginal cost rises, the desired stock of human capital is not reached immediately. Moreover, since the marginal product of capital is constant, gross investment is the basic decision variable in these models.^^ In my model, on the other hand, the marginal product of health capital falls because the output produced by this capital has a finite upper limit. Consequently, it is not necessary to introduce the assumption of rising marginal cost in order to determine the optimal stock.

To illustrate how the implications of the health and human capital models differ, suppose the rate of depreciation on either the stock of health or human capital rises. This upsets the equality between the cost of capital and its marginal efficiency. To restore this equality in the health model, the marginal product of health capital must rise, which would occur only if the stock of capital declines. To restore this equality in the human capital model, marginal cost must fall, which is possible only if gross investment declines.^^

HL Life Cycle Variations in Depreciation Rates

Equation (14) enables one to study the behavior of the demand for health and gross investment over the life cycle. To simplify the analysis, it is assumed that the wage rate, the stock of knowledge, the marginal cost of gross investment, and the marginal productivity of health capital are independent of age. These assumptions are not as restrictive as they may seem. To be sure, wage rates and human capital are undoubtedly cor- related with age, but the effects of shifts in these variables are treated in Section IV. Therefore, the results obtained in this section may be viewed as partial effects. That is, they show the impact of a pure increase in age on the demand for health, with all other variables held constant.

IS For a complete discussion of these points, see Becker (1967, pp. 5-12) and Ben- Porath (1967, pp. 353-61). For models of the demand for physical capital by firms in which the marginal cost of investment and the amount of investment are positively correlated, see, for example, Eisner and Strotz (1963) and Gould (1968).

1^ Section I I I demonstrates that an increase in the rate of depreciation on health capital might cause gross investment to increase.

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Since marginal cost does not depend on age, Jti_i — 0 and equation (14) reduces to

K (18)

It is apparent from equation (18) that, if the rate of depreciation were independent of age, a single quantity of H would satisfy the equality between the marginal rate of return and the cost of health capital. Con- sequently, there would be no net investment or disinvestment after the initial period. One could not, in general, compare ^ 0 and H^ because accumulation in the initial period would depend on the discrepency be- tween the inherited stock and the stock desired in period 1. This dis- crepency in turn would be related to variations in H^y and other variables across individuals. But, given zero costs of adjusting to the desired level immediately, H would be constant after period 1. Under the stated condition of a constant depreciation rate, individuals would choose an infinite life if they choose to hve beyond period 1. In other words, if Hx > ^min? then Hi would always exceed the death stock.-^

To permit the demand for health to vary with age, suppose the rate of depreciation depends on age. In general, any time path of hi is possible. For example, the rate of depreciation might be negatively correlated with age during the early stages of the life cycle. Again, the time path might be nonmonotonic, so that hi rises during some periods and falls during others. Despite the existence of a wide variety of possible time paths, it is ex- tremely plausible to assume that 6̂ is positively correlated with age after some point in the life cycle. This correlation can be inferred because, as an individual ages, his physical strength and memory capacity deteriorate. Surely, a rise in the rate of depreciation on his stock of health is merely one manifestation of the biological process of aging. Therefore, the anal- ysis focuses on the effects of an increase in the rate of depreciation with age.

Since a rise in 8̂ causes the supply curve of health capital to shift up- ward, it would reduce the quantity of health capital demanded over the life cycle. Graphically, an increase in the cost of capital from r -\-hi to r-|-8.i_|_i in figure 3 reduces the optimal stock from Hi to i^i+i. The greater the elasticity of the MEC schedule, the greater the decrease in the optimal stock with age. Put differently, the slower the increase in the marginal product of health capital as H falls, the greater the decrease in the optimal stock.

Differentiation of equation (18) with respect to age quantifies the percentage rate of decrease in the stock of health over the life cycle:

(19)

-0 The possibility that death can occur in period 1 is ruled out from now on.

CONCEPT OF HEALTH CAPITAL 237

r + 5

F I G . 3

In this equation, the tilde notation denotes a percentage time derivative {Hi^= (dHi/di) (I/Hi), etc.), and the new symbols are: 5̂ = 6i/ 8 := the share of depreciation in the cost of health capital and

d \nHi — —d e.,- = —

the elasticity of the MEC schedule (In stands for natural logarithm).^^ Equation (19) indicates that the absolute value of the percentage de- crease in H is positively related to the elasticity of the MEC schedule, the share of depreciation in the cost of health capital, and the percentage rate of increase in the rate of depreciation. If ê and 8,- were constant, the curve relating In Hi to age would be concave unless r = 0, s i ^ ^

(20) di

The absolute value of Hi increases over the life cycle because depre- ciation's share in the cost of capital rises with age.

21 From equation (18), ln(r + 6 )̂ =\nW + ]D.G^ — In JT. Therefore,

8.- 8a

H, or

-2 Differentiation of (19) with respect to age yields

or

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If 8; grows continuously with age after some point in the life cycle, persons would choose to live a finite life. Since H declines over the life cycle, it would eventually fall to i?min, the death stock. When the cost of health capital is r + 8n in figure 3, Hn — Hmin, and death occurs. At death, no time is available for market and nonmarket activities, since healthy time equals zero. Therefore, the monetary equivalent of sick time in period n would completely exhaust potential full earnings, Wn^- More- over, consumption of the commodity Zn would equal zero, since no time would be available for its production if total time equals sick time.̂ -̂ Be- cause individuals could not produce commodities, total utility would be driven to zero at death.-^

Having characterized the optimal path of H,, one can proceed to ex- amine the behavior of gross investment. Gross investment's life cycle profile would not, in general, simply mirror that of health capital. In other words, even though health capital falls over the life cycle, gross investment might increase, remain constant, or decrease. This follows because a rise in the rate of depreciation not only reduces the amount of health capital demanded by consumers but also reduces the amount of capital supplied to them by a given amount of gross investment. If the change in supply exceeded the change in demand, individuals would have an incentive to close this gap by increasing gross investment. On the other hand, if the change in supply were less than the change in demand, gross investment would tend to fall over the life cycle.

To predict the effect of an increase in hi with age on gross investment, note that the net investment can be approximated by HiHir'' Since gross investment equals net investment plus depreciation.

(21)

Differentiation of equation (21) with respect to age yields

+ hiHi + Hu + 8A

rii + Oi

Suppose hi and ê were constant. Then from (19) and ( 2 0 ) , the expres- sion for Ii would simplify to

23 The above s t a t e m e n t assumes t h a t Z- cannot be produced with X^ alone. This would be true if, say, the production function were Cobb-Douglas.

24 Utility equal zero when H = ^nijn provided the death time utiUty function is such t h a t U{0) = 0 .

25 T h a t is,

TJ TJ TJ ' U TJ

dt Hi

The use of this approximation essentially allows one to ignore the one-period lag be- tween a change in gross investment and a change in the stock of health.

CONCEPT OF HEALTH CAPITAL 239

7 8(1 — ̂ ê) (hi — f

Since health capital cannot be sold, gross investment cannot be nega- tive. Therefore, hi ^ —Hi.̂ ^ That is, if the stock of health falls over the life cycle, the absolute value of the percentage rate of net disinvestment cannot exceed the rate of depreciation. Provided gross investment does not equal zero, the term 8̂ — SiSb in equation (22) must exceed zero. It follows that a sufficient condition for gross investment to be positively correlated with the depreciation rate is e < l/si. Thus, Ii would definitely be positive at every point if 8 < 1.

The important conclusion is reached that, if the elasticity of the MEC schedule were less than 1, gross investment and the depreciation rate would be positively correlated over the life cycle, while gross investment and the stock of health would be negatively correlated. Phrased differ- ently, given a relatively inelastic demand curve for health, individuals would desire to offset part of the reduction in health capital caused by an increase in the rate of depreciation by increasing their gross invest- ments. In fact, the relationship between the stock of health and the num- ber of healthy days suggests that 8 is smaller than 1. A general equation for the healthy-days production function illustrated by figure 2 is

hi = 365-BHr^, (23)

where B and C are positive constants. The corresponding MEC schedule

~ (C + 1) In i7^ + In Ŵ — In Jt. (24)

The elasticity of this schedule is given by

din Hi 1 e = = < 1,

a i n y . ( 1 + C ) ^

since C > 0. Observe that with the depreciation rate held constant, increases in

gross investment would increase the stock of health and the number of healthy days. But the preceding discussion indicates that, because the

26 Gross investment is nonnegative as long as /,• = H- (H- ~\- 6̂ -) ^ 0, or 8- ^ — H.. ^'^ If (23) were the production function, the marginal product of health capital would

be

or

Since In YJ = In Ĝ + In W — In Jt, one uses the equation for In G^ to obtain (24)

240 JOURNAL OF POLITICAL ECONOMY

depreciation rate rises with age, it is not unlikely that unhealthy (old) people will make larger gross investments than healthy (young) people. This means that sick time, TL ,̂ will be positively correlated with M, and THij the medical care and own time inputs in the gross investment func- tion, over the life cycle.^^ In this sense, at least part of TLi or TH, may be termed "recuperation time/^

Unlike other models of the demand for medical care, my model does not assert that **need" or illness^ measured by the level of the rate of depreciation, will definitely be positively correlated with utihzation of medical services. Instead, it derives this correlation from the magnitude of the elasticity of the MEC schedule and indicates that the relationship between the stock of health and the number of healthy days will tend to create a positive correlation. If e is less than 1, medical care and "need'' will definitely be positively correlated. Moreover, the smaller the value of e, the greater the explanatory power of "need" relative to that of the other variables in the demand curve for medical care.

It should be realized that the power of this model of life cycle behav- ior is that it can treat the biological process of aging in terms of con- ventional economic analysis. Biological factors associated with aging raise the price of health capital and cause individuals to substitute away from future health until death is "chosen." It can be concluded that here, as elsewhere in economics, people reject a prospect^—the prospect of longer life in this case—because it is too costly to achieve. In particular, only if the elasticity of the MEC schedule were zero would individuals fully compensate for the increase in 8̂ and, therefore, maintain a constant stock of health.

Market and Nonmarket Efficiency

Persons who face the same cost of health capital would demand the same amount of health only if the determinants of the rate of return on an investment were held constant. Changes in the value of the marginal product of health capital and the marginal cost of gross investment shift the MEC schedule and, therefore, alter the quantity of health demanded even if the supply curve of capital does not change. I now identify the variables that determine the level of the MEC schedule and examine the effects of shifts in these variables on the demand for health and medical care. In particular, I consider the effects of variations in market effi- ciency, measured by the wage rate, and nonmarket efficiency, measured by human capital, on the MEC schedule.

28 Note that the time path of H. or h^ would be nonmonotonic if the time path of b^ were characterized by the occurrence of peaks and troughs. In particular, k. would be relatively low and TH, and M- would be relatively high (if e < 1) when 6- was relatively high; these periods would be associated with relatively severe illness.

CONCEPT OF HEALTH CAPITAL 2 4 I

Before beginning the analysis, two preliminary comments are in order. First, the discussion pertains to uniform shifts in variables that influence the rate of return across persons of the same age. That is, if the variable Xi is one determinant, then

Second, the discussion proceeds under the assumption that the real rate of interest, the rate of depreciation, and the elasticity of the MEC schedule are constant. These two comments imply that an increase in Xi will alter the amount of capital demanded but will not alter its rate of change over the life cycle.^^ Note from equation (21):

d]nl dlnH (25)

dX dX

since the rate of depreciation and the percentage rate of net investment do not depend on X.^^ Equation (25) indicates that percentage changes in health and gross investment for a one-unit change in X are identical. Consequently, the effect of an increase in X on either of these two vari- ables can be treated interchangeably.

A, Wage Effects

Since the value of the marginal product of health capital equals WG, an increase in the wage rate, W, raises the monetary equivalent of the mar- ginal product of a given stock. Put differently, the higher a person's wage rate, the greater the value to him of an increase in healthy time. A con- sumer's wage rate measures his market efficiency or the rate at which he can convert hours of work into money earnings. Hence, it is obviously positively correlated with the benefits of a reduction in the time he loses from the production of money earnings due to illness. Moreover, a high wage rate induces an individual to substitute market goods for his own time in the production of commodities. This substitution continues until in equilibrium the monetary value of the marginal product of consump- tion time equals the wage rate. So the benefits from a reduction in time lost from nonmarket production are also positively correlated with the wage.

2^ Strictly speaking, shifts in X. would definitely have no effects on H^ if and only if X- = 0. Even though a uniform shift in X. implies that there is no correlation be- tween its level and rate of change, H^ might be altered if X^ ^ 0. For a complete dis- cussion of this point, see Grossman (1970, p. 49).

^^ Since the analysis in this section deals with variations in X among individuals of the same age, time subscripts are omitted from now on. Note also that (25), like the expression for /^, ignores the one-period lag between an increase in gross investment and an increase in the stock of health.

242 JOURNAL OF POLITICAL ECONOMY

FIG. 4

If an upward shift in the wage rate had no effect on the marginal cost of gross investment, a 1 percent increase in it would increase the rate of return, y? associated with a fixed stock of capital by 1 percent. In fact, this is not the case because own time is an input in the gross investment function. If K is the fraction of the total cost of gross investment ac- counted for by time, then a 1 percent rise in W would increase marginal cost, X, by K percent. After one nets out the correlation between W and Kj the percentage growth in y would equal 1 — K, which exceeds zero as long as gross investment is not produced entirely by time.

Since the wage rate and the level of the MEC schedule are positively correlated, the demand for health would be positively related to W. Graphically, an upward shift in W from Wi to W^ in figure 4 shifts the MEC schedule from MECi to MEC2 and, with no change in the cost of health capital, increases the optimal stock from H-i to Ho. A formula for the wage elasticity of health capital iŝ ^

(26)

This elasticity is larger the larger the elasticity of the MEC schedule and the larger the share of medical care in total gross investment cost.

Although the wage rate and the demand for health or gross invest-

31 Differentiation of the natural logarithm of (18) with respect to In W yields

dln(r4-h)

dlnW = 0 = 1 +

dlnG dlnH

dlnH din W dlnW

0=1 — K —

CONCEPT OF HEALTH CAPITAL 243

ment are positively related, W has no effect on the amount of gross in- vestment supplied by a given input of medical care. Therefore, the demand for medical care would rise with the wage. If medical care and own time were employed in fixed proportions in the gross investment production function, the wage elasticity of M would equal the wage elasticity of H. On the other hand, given a positive elasticity of substitution, M would increase more rapidly than H. This follows because consumers would have an incentive to substitute medical care for their relatively more ex- pensive own time. A formula for the wage elasticity of medical care is

, (27)

where Op is the elasticity of substitution between M and Th in the pro- duction of gross investment.'^^ The greater the value of 0 ,̂ the greater the difference between the wage elasticities of M and H.

Note that an increase in the price of either medical care or own time raises the marginal or average cost of gross investment. But the effects of changes in these two input prices are not symmetrical. In particular, an upward shift in the price of medical care lowers the MEC schedule and causes the demand for health to decline. This difference arises be- cause the price of time influences the value of the marginal product of health capital while the price of medical care does not.

B. The Role oj Human Capital

Up to now, no systematic allowance has been made for variations in the efficiency of nonmarket production. Yet it is known that firms in the market sector of an economy obtain varying amounts of output from the same vector of direct inputs. These differences have been traced to forces like technology and entrepreneurial capacity, forces that shift production functions or that alter the environment in which firms operate. Reason- ing by analogy, one can say that certain environmental variables influ- ence productivity in the nonmarket sector by altering the marginal products of the direct inputs in household production functions. This study is particularly concerned with environmental variables that can be associated with a particular person—his or her race, sex, stock of human capital, etc. While the analysis that follows could pertain to any environ- mental variable, it is well documented that the more educated are more efficient producers of money earnings. Consequently, it is assumed that shifts in human capital, measured by education, change productivity in

32 For a proof, see Part C of the Mathematical Appendix. The corresponding equa- tion for the wage elasticity of the own time input is

This elasticity is positive only if e > o^^.

244 JOURNAL OF POLITICAL ECONOMY

the household as well as in the market, and the analysis focuses on this environmental variable.

The specific proposition to be examined is that education improves nonmarket productivity. If this were true, then one would have a con- venient way to analyze and quantify what have been termed the non- monetary benefits to an investment in education. The model can, however, treat adverse as well as beneficial effects and suggests empirical tests to discriminate between the two.̂ ^

To determine the effects of education on production, marginal cost, and the demand for health and medical care, recall that the gross invest- ment production function is homogeneous of degree 1 in its two direct inputs—medical care and own time. It follows that the marginal product of £ , the index of human capital, would be

a/ _ ,„ „ , , , ___ a£~ ~ 6£ dE '

where g — tg' is the marginal product of medical care and g' is the mar- ginal product of time.^^ If a circumflex over a variable denotes a per- centage change per unit change in E, the last equation can be rewritten as

dl 1 'Mig-tg')

Equation (28) indicates that the percentage change in gross investment supplied to a consumer by a one-unit change in £ is a weighted average of the percentage changes in the marginal products of M and TH^^

If E increases productivity, then TH > 0. Provided E raises both mar- ginal products by the same percentage, equation (28) would simplify to

rH = g = g'. (29)

33 The model developed here is somewhat similar to the one used by Michael (1969). 34 If / is homogeneous of degree 1 in Af and TH^ then from Euler's theorem

Differentiation of this equation with respect to £, holding M and TH constant, yields the marginal product of human capital.

35 Instead of putting education in the gross investment production function, one could let it affect the rate of depreciation or the marginal productivity of health capital. This approach has not been taken because a general treatment of environmental vari- ables like education must permit these variables to influence all household commodities. Since depreciation rates and stock-flow relationships are relevant only if a particular commodity is durable, a symmetrical development of the role of environmental vari- ables requires that they affect household production functions and not depreciation rates or stock-flow relationships. In a more complicated version of the model, the gross investment function, the rate of depreciation, and the marginal productivity of health capital might all depend on education. But the basic implications of the model would not change.

CONCEPT OF HEALTH CAPITAL 245

In this case, education would have a '^neutral" impact on the marginal products of all factors. The rest of the discussion assumes "factor neu- trality."

Because education raises the marginal product of the direct inputs, it reduces the quantity of these inputs required to produce a given amount of gross investment. Hence, with no change in input prices, an increase in E lowers average or marginal cost. In fact, one easily shows that

JC = —rs = —g = —f, (30)

where K is the percentage change in average or marginal cost.^^ So, if education increases the marginal products of medical care and own time by 3 percent, it would reduce the price of gross investment by 3 percent.

Suppose education does in fact raise productivity so that n and E are negatively correlated. Then, with the wage rate and the marginal product of a given stock of health held constant, an increase in education would raise the marginal efficiency of health capital and shift the MEC schedule to the right.̂ *^ In figure 5, an increase in E from Ei to £2 shifts the MEC curve from MECi to MECo- If the cost of capital were independent of £ , there would be no change in the supply curve, and the more educated would demand a larger optimal stock (compare Hi and H2 in fig. 5).

The percentage increase in the amount of health demanded for a one- unit increase in E is given by^^

A

H — r^e. {61)

Since rji indicates the percentage increase in gross investment supplied by a one-unit increase in E, shifts in this variable would not alter the demand for medical care or own time if r^ equaled H. For example, a person with ten years of formal schooling might demand 3 percent more health than a person with nine years. If the medical care and own time inputs were held constant, the former individual's one extra year of

•̂ ^ For a proof, see Part D of the Mathematical Appendix, where the human capital formulas are developed in more detail.

3*̂ It should be stressed that the model of nonmarket productivity variations pre- sented here examines the partial effect of an increase in education with the wage rate held constant. Although these two variables are surely positively correlated, this corre- lation does not appear to be large enough to prevent one from isolating pure changes in nonmarket productivity at the empirical level. For some evidence on this point, see Grossman (1970, chap. 5) and Michael (1969, chaps. 4 and 5).

38 If p^ and r + 5 are fixed and if G depends only on i/, then

d\n(r A- b) d In G dlnH dlnK ^ = 0 = ,

dE d\nH dE dE

or A

H 0=

E

246 JOURNAL OF POLITICAL ECONOMY

FIG. 5

schooling might supply him with 3 percent more health. Given this con- dition, both persons would demand the same amounts of M and TH. As this example illustrates, any effect of a change in E on the demand for medical care or time reflects a positive or negative difference between H and ^^

M = TH ^ — 1). ( 3 2 )

Equation (32) suggests that, if the elasticity of the MEC schedule were less than unity, the more educated would demand more health but less medical care. Put differently, they would have an incentive to offset part of the increase in health caused by an increase in education by re- ducing their purchases of medical services. Note that if TH were negative and 8 were less than 1, H would be negative and M would be positive. Since education improves market productivity, I have examined the im- plications of the hypothesis that rn is positive. But the model is appli- cable whether r^ is positive or negative and gives empirical predictions in either case.

V. S u m m a r y and Conclusions

The main purpose of this paper has been to construct a model of the demand for the commodity "good health." The central proposition of the model is that health can be viewed as a durable capital stock that pro- duces an output of healthy time. A person determines his optimal stock of health capital at any age by equating the marginal efficiency of this capital to its user cost in terms of the price of gross investment. Graphi- cally, each person has a negatively inclined demand curve for health

^^ The terms M and TH are equal because, by the definition of factor neutrality, E has no effect on the ratio of the marginal product of M to the marginal product of TH.

CONCEPT OF HEALTH CAPITAL 247

capital, which relates the marginal efficiency of capital to the stock, and an infinitely elastic supply curve. The equilibrium stock is determined by the intersection of these two functions. The demand curve slopes down- ward due to diminishing marginal productivity of health capital

Although in recent years there have been a number of extremely in- teresting explorations of the forces associated with health differentials (Adelman 1963; Fuchs 1965; Larmore 1967; Newhouse 1968; Auster, Leveson, and Sarachek 1969), these studies have not developed behav- ioral models that can predict the effects that are in fact observed. Con- sequently, the framework I have developed is important because of its ability to bridge the existing gap between theory and empiricism in the analysis of health differentials. My model explains variations in both health and medical care among persons in terms of variations in supply and demand curves for health capital. This paper has traced upward shifts in the supply curve to increases in the rate of depreciation on the stock of health with age, and it has traced upward shifts in the demand curve to increases in the wage rate and education.

One prediction of the model is that if the rate of depreciation increases with age, at least after some point in the life cycle, then the quantity of health capital demanded would decline over the life cycle. At the same time, provided the elasticity of the marginal efficiency of capital schedule were less than unity, expenditures on medical care would rise with age. A second prediction is that a consumer's demand for health and medical care should be positively correlated with his wage rate. A third prediction is that if education increases the efficiency with which gross investments in health are produced, then the more educated would demand a larger optimal stock of health. On the other hand, given a relatively inelastic demand curve, the correlation between medical outlays and education would be negative. It should be noted that one of the advantages of the model is that it enables one to study the effects of demographic variables like age and education without assuming that these variables are posi- tively or negatively correlated with consumers' ^'tastes" for health. In- stead, these variables enter the analysis through their impact on either the cost of health capital or its marginal efficiency, and one can make strong predictions concerning their effects on health levels or medical care.

It must be admitted that this paper has made a number of simplifjdng assumptions, all of which should be relaxed in future work. A more gen- eral model would treat the depreciation rate as an endogenous variable and would not rule out periods in which the optimal amount of gross investment is zero. Most important of all, it would modify the assump- tion that consumers fully anticipate intertemporal variations in depre- ciation rates and, therefore, know their age of death with certainty. Since in the real world length of life is surely not known with perfect foresight.

2 4 8 JOURNAL OF POLITICAL ECONOMY

it might be postulated that a given consumer faces a probability distribu- tion of depreciation rates in each period. This uncertainty would give persons an incentive to protect themselves against the "losses" associated with higher than average depreciation rates by purchasing various types of insurance and perhaps by holding an "excess" stock of health.'*^ But whatever modifications are made, it would be a mistake to neglect the essential features of the model I have presented in this paper. Any model must recognize that health is a durable capital stock, that health capital differs in important respects from other forms of human capital, and that the demand for medical care must be derived from the more fundamental demand for "good health."

Appendix A

Mathematical Appendix

A. Utility Maximization—Discrete Time

To maximize utility subject to the full wealth and production function con- straintSj form the Lagrangian expression

nil TJ A TJ 7

+ M ^ - 2 — — — J , (Al)

where Ĉ = P^M^ + W-.TH^ and C,i - V^X^ + W^T^. Differentiating L with re- spect to gross investment in period i — 1 and setting the partial derivative equal to zero, one obtains

OH, \-

-i-...-\-Uhn —— — = I

But

. (A2)

4̂ For an attempt to introduce uncertainty into a model that views health as a durable capital stock, see Phelps (in preparation).

CONCEPT OF HEALTH CAPITAL 249

i l i

i) ( n i ) , — 7 7 = Jt.i_i, a n d -——— = —G,. dli^i

Therefore,

( 1 - 6 0 . . . ( l - 8 « - i ) P F «

UhiA.1 ^ Uhn

+ (1 - 80 — y ^ G , + i -f . . . + (1 - 80 . . . (1 - 8 « _ ) I (A3)

B. Utility Maximization—Contimions Time

Let the utility function be

(A4)

where mi is tbe weight attached to utility in period i. Equation (A4') defines an additive utility function, but any monotonic transformation of this function could be employed.'*^ Let all household production functions be homogeneous of degree 1. Then Ci — TtJ.^ Cn^qiZi,^- and full wealth can be written as

/ { + g + ) (A5)

By definition,

(A6)

where ffj is the instantaneous rate of change of capital stock. Substitution of (A6) into (AS) yields

ihiHi + :i,Hi + q^Zi -f W{TL,)di. (A7)

To maximize the utility function, form the Lagrangian

nA + QiZ, + Wi TL,)] di, (A8)

or

L-lR = iQiH,, H,, Zi, i)di, (A9)

where

Q = m (AlO)

Euler's equation for the optimal path of H^ is

41 Strotz (1955-56) has shown, however, that certain restrictions must be placed on the m-. In particular, the initial consumption plan will be fulfilled if and only if w^ =

42 The variable q, equals the marginal cost of Z^.

250 JOURNAL OF POLITICAL ECONOMY

dQ d dQ

di

In the present context,

dQ

di

Consequently,

T which is the continuous time analogue of equation (13)

(All)

(A12)

(A13)

C. Wage Effects

To obtain the wage elasticities of medical care and the time spent producing health, three equations must be partially differentiated with respect to the wage. These equations are the gross investment production function and the two first- order conditions for cost minimization;

, TH-E)= Mg{t; E) = {H + h)H,

W = Jig',

P = n{g-tg').

Since / is linear homogenous in M and TH,

td(g-tg')

dM dTH

dTH dTH

Op = I{[dig-tg')]/dTH]

Therefore, the follo^^^ng relationships hold:

dis-tg') t{g-tg')g'

dM Io p

iS-tg')s' dTH t

is - ts')s' dTH

(A14)

CONCEPT OF HEALTH CAPITAL 251

Carrying out the differentiation, one gets

dTH

dW + ig -

dM

~dW

H{H dn J

'dW~W

dn

dW

0=(g-

( dg' dTH dg' dM j ^ I I

\dTH dW dM dW

dn

'dW

d{g-tg') dTH dig-tg') dM

dTH dW dM dWJ

Using the cost-minimization conditions and (A14) and rearranging terms, one has

dn dTH dM hn Ie

dW dW

dn 1 dTH /Op P

dW t dW

dW W

dM n

dW W 'P) (A15)

dn dTH dM lop \- W tw = 0.

dW dW dW Since (A15) is a system of three equations in three unknowns—dTH/dW,

dM/dWj and d^/dW—Cramer^s rule can be applied to solve for, say, dM/dW:

1

t

dM

— 0

IE + W

W — tw

The determinant in the denominator reduces to terminant in the numerator is

:^P)/THM. The de-

Therefore,

THM

dM THM

dW In WTH

252 JOURNAL OF POLITICAL ECONOMY

In elasticity notation, this becomes

eM^w = (I ~ K)B-}-Koj,. (A16)

Along similar lines, one easily shows that

(A17)

D. The Role of Human Capital

To convert the change in productivity due to a shift in human capital into a change in average or marginal cost, let the percentage changes in the marginal products of medical care and own time for a one-unit change in human capital be given by

d{g — tg') 1 gg — tg't dE g-tg' g-

BE g'

If a shift in human capital were "factor neutral," the percentage changes in these two marginal products would be equal:

— tss 6 ± f y

g — ig or

(A18)

The average cost of gross investment in health is defined as

Jc = {PM +

Given factor neutrality,

dn I g

dE %

This coincides with the percentage change in marginal cost, since

(A19)

and

1 f ss — ts't dE n \ g-tg'

(A20)

Part B of Section IV outlines a derivation of the human capital parameter in the demand curve for medical care but does not give a rigorous proof. Taking the total derivative of E in the gross investment function, one computes this parameter thus:

dl 1 is-tg') THg' = M — M H — TH +dE I I I

Since M — TH and H = I, the last equation can be rewritten as

CONCEPT OF HEALTH CAPITAL 2 5 3

A

Solving for M and noting that H = rjjz, one gets

(A21)

Appendix B

Glossary of Mathematical Terms

n Total length of life i Age HQ Inherited stock of health Hi Stock of health in period i

Death stock Service flow per unit stock or number of healthy days per unit stock Total number of healthy days in period i Consumption of an aggregate commodity in period i Gross investment in health Rate of depreciation Medical care Time input in gross investment function

Xl Goods input in the production of Ẑ T^ Time input in the production of Z^ El Stock of human capital g — tig' Marginal product of medical care in the gross investment

production function g* Marginal product of time Pi Price of medical care Vi Price of Z^ Wi Wage rate AQ Initial assets r Rate of interest TW: Hours of work TLi Sick time Q Constant length of the period R Full wealth Gi Marginal product of health capital Uk^ Marginal utility of healthy days ^ Marginal utility of wealth Tt̂ Marginal cost of gross investment in health TCi Percentage rate of change of marginal cost Ŷ Monetary rate of return on an investment in health or

marginal efficiency of health capital Gi Psychic rate of r e t u m on an investment in health ^ A tilde over a variable denotes a percentage time derivative s^ Share of depreciation in the cost of health capital e Elasticity of the MEC schedule K Fraction of the total cost of gross investment accounted for

by time Op Elasticity of substitution betwen medical care and own time in

the production of gross investment

2 54 JOURNAL OF POLITICAL ECONOMY

eff-^ Elasticity of H with regard to W e^f-w Elasticity of M with regard to W A A circumflex over a variable denotes a percentage change per

unit change in E rjj Percentage change in gross investment for a one unit change

in E Ci Total cost of gross investment in health in period i Cii Total cost of Zi mi Weight attached to total utility in period i qi Marginal cost of Z^

References

Adelman. Irma. "An Econometric Analysis of Population Growth." AE.R. SI rjune 1963): 314-49.

Arrow, Kenneth J. "Optimal Capital Policy with Irreversible Investment." In Value, Capital a?id Growth: Papers in Honour of Sir John Hicks, edited by J. N. Wolfe. Edinburgh: Edinburgh Univ. Press, 1968.

Auster, Richard D.; Leveson, Ir\ang; and Sarachek, Deborah. "The Production of Health: An Exploratory Study." / . Human Resources 4 (Fall 1969): 411-36.

Becker, Gary S. Human Capital. New York: Columbia Univ. Press (for Nat. Bur. Econ. Res.), 1964.

. "A Theory of the Allocation of Time." Eco7i, J. 75 (September 1965): 493-517.

-. Human Capital and the Personal Distribution of Income: An Analytical Approach. W. S. Woytinsky Lecture no. 1. Ann Arbor: Univ. Michigan, 1967.

Becker, Gary S., and Michael, Robert T. "On the Theory of Consumer De- mand." Unpublished paper, 1970.

Ben-Porath, Yoram. "The Production of Human Capital and the Life Cycle of Earnings." J.P.E. 75 (August 1967): 353-67.

Eisner. Robert, and Strotz, Robert H. "Determinants of Business Investment." In Impacts of Monetary Policy. Englewood Cliffs, N J . : Prentice-Hall (for Commission on Money and Credit), 1963.

Fuchs, Victor R. "Some Economic Aspects of Mortality in the United States." Mimeographed. New York: Nat. Bur. Econ. Res., 1965.

. "The Contribution of Health Services to the American Economy." Milbaiik Meinorial Fund Q. 44 (October 1966): 65-102.

Ghez. Gilbert R. "A Theory of Life Cycle Consumption." Ph.D. dissertation, Columbia Univ., 1970.

Gould, John P. "Adjustment Costs in the Theor\' of Investment in the Firm." Rev. Econ. Studies 25 (January 1968): 47-55.

Grossman, Michael. "The Demand for Health: A Theoretical and Empirical Investigation." Ph.D. dissertation, Columbia Univ., 1970.

Lancaster, Kelvin J. "A New Approach to Consumer Theory." J.P.E. 74 (April 1966): 132-57. "

Larmore, Mary Lou. "An Inquiry into an Econometric Production Function for Health in the United States." Ph.D. dissertation, Northwestern Univ., 1967.

Michael, Robert T. "The Effect of Education on Efficiency in Consumption." Ph.D. dissertation, Columbia Univ., 1969; to be published by Nat. Bur. Econ. Res.

CONCEPT OF HEALTH CAPITAL 2 5 5

Mushkin, Selma J. "Health as an Investment." J.P.E. 70, no. 2, suppl (Octo- ber 1962): 129-57.

Muth, Richard. "Household Production and Consumer Demand Functions." Econometrica 34 (July 1966): 699-708.

Nerlove, Marc, and Arrow, Kenneth J. "Optimal Advertising Policy under Dynamic Conditions." Economica 29 (May 1962): 129^2.

Newhouse, Joseph P. "Toward a Rational Allocation of Resources in Medical Care." Ph.D. dissertation, Harvard Univ., 1968.

Phelps, Charles E. "The Demand for Health Insurance: Theory and Empirical Results." Ph.D. dissertation, Univ. Chicago (in preparation).

Strotz, Robert H. "Myopia and Inconsistency in Dynamic Utility Maximiza- tion." Rev. Econ. Studies 23 (1955-56): 165-180.