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On the oncept of Health apital

and the Demand for Health

Michael rossman

National Bureau of Economic Research

Th e aim of this study is to construct a model o f the demand for the

comm odity good health. Th e central proposition o f the model is that

health can be viewed as a durable capital stock that produces an output

o f healthy tim e. It is assumed that individuals inherit an initial stock

o f health tha t depreciates w ith age and can be increased b y invest men t.

In this framew ork. the shadow price

of

health depends on m any other

variables besides the price o f medical care. It is shown t hat the shadow7

price rises with age i f the rate o f depreciation on the stock o f health

rises over the li fe cycle and falls with education i f more educated people

are more efficient producers o f health. O f particular importance is the

conclusion t ha t, under certain co nditions, an increase in th e shadow,

price m ay simultaneously reduce the quant ity o f health demanded and

increase the quan tity o f medical care dema nded .

I ntroduction

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 tha t enter his u ti l i ty function. T o realize

This paper is based on par t of my Colum bia Univers i ty P h.D . disser ta t ion, The

Dem and for Heal th: A Theoret ical and Empir ical Invest igat ion, which will be pub-

l ished by the National Bureau of Economic Research. M y research a t the R ureau w as

suppor ted by the Commonwea l th Fund and the Na t iona l Cen te r fo r Hea l th

Services

Research and Developm ent (P H s research grant 2 P

01

H S

00451-04 .

Most of this

paper was wri t ten while I was at the Univers i ty of Chicago 's Cen ter for Heal th Ad-

minis tra t ion Studies , with research support f rom the National Center for Heal th Ser-

v ices Resea rch and Deve lopmen t ( P H s g ran t H S

00080 .

A preliminary version of this

paper was presented a t the Second World Congress of the Econ ometr ic Society .

I

a m

gra te fu l to G ary S . Becker ,

1 .

K . Che t ty , Vic to r R . Fuchs , Gi lbe r t R. Ghez , Robe r t

T.

Michael , and Jacob Mincer for their helpful comments on ear l ier draf ts .

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potential gains in productivity, individuals have an incentive to invest

in formal schooling or in on-the-jo b training. T h e 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 quanti ty of investment in

human capital a t any age. In addit ion, these models show how the

optim al qu an tit y varies over the life cycle of an individual an d among

individuals of the same age.

Although several writers have suggested that health can be viewed as

one form of hum an capital (M ush kin 1962 , pp . 129-49; Becker 196 4,

pp. 33-36; Fuchs 1966, pp. 90-91)) no one has constructed a model of the

demand for health capital itself.

f

increases in the stoc k of hea lth sim ply

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 dif fers

from other forms of hum an capital . In p ar t icular , i t argues tha t a person's

stock of knowledge affects his market and nonmarket productivity, while

his stock of health determines the tot al amo un t of tim e he can spend

producing money earnings and commodities. The fundamental difference

between the two ty pes of ca pital is the basic justification for the m odel

of the demand for health that is presented in the paper.

A second justification for the model is that m ost stud ents of medical

economics have long realized that what consumers demand when they

purchase medical services are not these services per se bu t, rathe r, good

health. Given tha t the basic demand is for good hea lth, 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'

utility functions, economists have emphasized the

denland for medical

care at the expense of the demand for health. Fortu nately, a new ap proach

to consumer behavior draws a sharp distinction between fundamental ob-

jects of choice-called commodities -and ma rket goods (Be cke r 19 65 ;

Lancaster 1966 ; Bluth 1 966; Blichael 196 9; Becker and Michael 19 70;

Ghez 19 70 ) . Th us, i t serves as the point of dep arture 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 inpu ts into th e production of com modities, the dem and 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

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CONCEPT O F

HE LTH

C PIT L 5

over time-at an increasing rate, at least after some stage in the life

cycle-and can be increased by investment. Dea th occurs when the stock

falls below a certain level, and one of the novel fe ature s of the m odel

is th at individuals choose their length of life. Gross investme nts in

health capital are produced by household production functions whose

direct inputs include the own time of the consumer and marke t 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 pro duct ion process.

I t should be realized th at in this model the level of h ealth of an ind i-

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

investmen t commodity, it determines the to tal am ount of time available

for market and nonmarket activities. In other words, an increase in the

stock of he alth reduces the time lost from these activ ities, and the mone-

tary value of this reduction is an index of the retur n to a n investment

in health.

Since the m ost fundame ntal law in economics is the law of the dow n-

ward-sloping dema nd curve, the quan tity of he alth demanded should be

negatively correlated with its shadow price. The analysis in this paper

stresses that the shadow price of health depends on m any other variables

besides the price of medical care. Shifts in these variables alter the

optimal amou nt of health and also alter the derived demand for gross

investment, measured, say, by medical expenditures. I t is shown tha t the

shadow price rises with age if the r ate of dep reciation o n the stock of

hea lth rises over the life cycle and falls with education if m ore educated

people are more efficient producers of health . Of particu lar im porta nce is

the conclusion th at, under certain conditions, an increase in t he shadow

price may simultaneously reduce the quantity of health demanded and

increase the quantity of medical care demanded.

11 Stock pproach to the Demand for Health

A . The Model

Let the intertemporal utility function of a typical consumer be

where

H o

is the inherited stock of health,

H i

is the stock of health in the

ith time period,

i

is the service flow per unit stock,

hi ;H,

is total

consum ption of health services, and

i

is tot al consumption of an othe r

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6 JOURN L

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commodity in the i th per iod. Note tha t , whereas in the usual inter-

temporal utility function n , th e length of life as of t he plann ing d ate , is

fixed, here it is an endogenous variable. In particu lar, d eath takes place

when H i H m i . Therefo re, length of life depends on th e quantities of

H i that maximize utility subject to certain production and resource con-

straints that are now outl ined.

y

definition, net investment in the stock of health equals gross invest-

ment minus depreciation:

where is gross investm ent and hi is the rate of depreciation during the

ith period. Th e rates of d epreciation are assumed to be exogenous, bu t

t he y may va ry w i th t he age of t he i n d i v i d~ a l .~onsumers produce gross

investments in health and the other commodities in the utility function

according to a set of household production functions:

In these equations,

M

is medical care, X i is the goods input in the pro-

duction of the comm odity Zi, T H i and Ti are t ime inputs, and is the

s toc k of huma n ~ a p i t a l . ~ t ha t a sh if t i n huma nt is assumed capital

changes the efficiency of the production process in the nonmarket sector

of the econom y, just as a shift in technology changes th e efficiency of the

production process in the market sector. Th e implications of this treat-

ment of human capital are explored in Section IV.

I t is also assumed th at all production functions are hom ogeneous of

degree

1

in the goods and t ime inputs. Therefore, the gross investment

production function can be written as

where t i TH i/M i. I t follows tha t the marginal products of t ime and

medical care in the production of gross investment in health are

The commodi ty

Z

may be viewed as an aggregate of all commodities besides health

that enter the ut i l i ty funct ion in per iod i For the convenience of the reader, a glossary

of symbols may be found

in

.4ppendix

B .

V n a more complicated vers ion of the model , the ra te of deprecia t ion might be a

negative function of the stock of health. The analysis is considerably simplified by

treat ing this ra te as exogenous, and the conclusions reached would tend t o hold even

if it were endogenous.

In general , medical care is not th e only m arket good in the gross investment func-

t ion, for inputs such as housing, die t , recreat ion, c igaret te smoking, and a lcohol con-

sum ption influence one s level of he alth Since these inpu ts also produce other

commodit ies in the ut i l i ty funct ion, jo int product ion occurs in the household. For an

analysis of this phenomenon, see Grossman (1970, chap. 6 ) . To emphasize the key

aspects of my heal th model , I t reat medical care as the most important market good

in the gross investment funct ion in the present paper .

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CONCEPT O F HE LTH C A P I T A L

Fro m the point of view of th e indiv idual, both ma rke t goods and own

time are scarce resources. Th e 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 (discoun ted prop erty income) :

Here P, a nd V , are the prices of and i Wi is the wage rate, TW i is

hou rs of work, A ,, is discounted property income, and is the interest rate .

The t ime constraint requires that

Q , the tota l amou nt of tim e available

in any period, must be exhausted by all possible uses:

where T L , is time lost from market and nonmarket activities due to illness

or injury.

Equa t ion ( 7 ) modifies the time budget constraint in Becker's time

model (Becker

1 9 6 5 ) .

If sick t ime were not added to market and non-

market t ime, total t ime would

not

be exhausted b y al l possible uses. M y

model assumes that

T L ,

is inversely related to the stock of h ealth : th at is,

a T L , / a H , 0 If Q were measured in days Q= 3 6 5 days if the year

is the relevant period) and if 4 were defined as the flow of healthy day s

per unit of H I ,

h ,

would equal the total number of healthy d ays in a

given year.5 Then one could write

I t is importan t to dram a sharp dist inction between sick t ime and the

time inpu t in the gross investment functio n. As an illustratio n of this

difference, the time a consum er allocates to visiting his do ctor for periodic

checkups is obviously not sick time. More form ally, if th e rate of de-

preciation were held constant, an increase in

TH i

would increase I , and

H I ,

nd would reduce

T L L b l .

T h u s ,

T H ,

a nd

T L i + ,

would be negatively

~ o r r e l a t e d . ~

The s u m s t h r o u g h ou t th is s tu d y a re ta k e n f ro m = to n

S I f

the stock of health yielded other services besides health y day s.

d

would be a

vecto r of service flows Th is study emphasizes the service flow of h ealth y da ys because

this flow can be measured empirically.

For a d iscussion of condi t ions that would produce a posi t ive corre la t ion between

TH i

and T t l see Section

111.

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By substi tut ing for l 'Wi from equation

( 7 )

into equation

( 6 ) ,

one

obtain s the single full wealth cons traint:

PiMi ViX, Wi(TLi TH4 Ti)

= Z

WiQ

+A = R

(1 ( 1 + r I i

9 )

According to equatio n (9 ), full wealth equals initial assets plus th e present

value of the earnings a n individual would obt ain if he s pent all of his time

a t work. Pa rt of th is wealth is spent on market goods, pa rt of it is spent

on nonmarket production t ime, and par t of i t is lost due to i l lness. The

equilibr ium quanti t ies of H i and Zi can now be found b y maximizing the

uti l i ty function given by equation ( I ) subject to the constraints given by

equations

Z ) ,

3) ,

a nd ( 9 )

5

Since the inherited stock of health and the

rates of depreciation are given, the optimal quantities of gross investment

determine the optimal quanti t ies of health capital .

B Equilibrium

Conditions

First-ord er optim ality conditions for gross investment in period 1

are:8

+

( I 86)

( 1

(

+

1

?-)

tin-

I )

WnGn

Pi-

wi-

xi-1 =

g ti-1

g'

Th e new symbols in these equations are: Uh, = a U / a h , t he m a rg in al

uti l i ty of healthy da ys ;

=

the marginal utility of wealth; G ,

=

a h , /

a H , = - ( a T L , / a H , ) = the marginal product of the stock of health in

the production of h ealthy day s; a nd xi- ,

=

th e marg inal cost of gross

investment in health in period i I .

In add i t ion , the cons t r a in t i s imposed tha t H

Hllli,,.

Note that an increase in gross investment in per iod 1 increases the stock of

health in all future periods. These increases are equal to

dHi

dH i+

dHn

1,

=

( 1 S i ) ,

. ,-

d l i p l d r i p ,

= ( 1 - 8 ; ) ( l - & + I )

. . .

(1 - t i , , -1) .

For a der ivat ion of equat ion

l o ) ,

see Pa r t of the Mathem atical Appendix.

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CONCEPT

OF HEALTH

CAPITAL 9

Equation

10)

simply states that the present value of the marginal

cost of gross investment in period

1

must equal the present value of

marginal benefits. Discounted marginal benefits at age equal

where G 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

U h i h

measures the discounted monetary equiva-

lent of the increase in utility 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.

LI hile equation 10) determines the optimal amount of gross invest-

ment in period

i

1, equation I I ) shows the condition for minimizing

the cost of producing a given quantit y 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 additional 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

From

10)

and

12) ,

Therefore,

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23 J O U R N A L

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E C O N O M Y

where is the percentage rat e of chan ge in margina l cost between

period

i

1 and period i. quation (1 3) implies th at the undiscounted

value of the marginal prod uct of the optim al stock of he alth ca pital a t

an y moment in time must equal th e supply price of c apital ,

I - C - ~ Y

z-1 a i) . Th e 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 m arket. In such a m arke t, if firms or

households acqu ire one uni t of stock in period 1 a t price

-I,

they

can sell (1 a i units a t price ni at the end of period

i

Consequently,

~ t i - l ( r

Z i 1

ai) measures the cost of holding one unit of capital for

one period. The transaction just described allows individuals to raise their

capital in period

i lone

by one unit and is clearly feasible for stocks like

automo biles, houses, refrigerators, an d producer durables. I t suggests tha t

one can define a set of single-period flow equilibria for stocks that last

for many periods.

In my m odel, the stock of h ealth capital c annot be sold in the cap ital

market, just as the stock of knowledge cannot be sold. This means that

gross investment m ust be nonnegative. Although sales of health c apital a re

ruled ou t, provided gross inve stme nt is positive, there exists a used cost of

capital th at in equilibrium m ust equal the value of the marginal product

of the

stock.1 An intuitive in terpre tation of th is result is tha t exchanges

over time in the stock of health by an individual 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 un it. If he simultaneously reduces gross investment

in period i by (1 a i) units , then he has engaged in a transaction tha t

raises and

lone

by one unit. Put differently, he has essentially

rented one uni t of c apita l from himself for one period. T he m agnit ude of

the reduction in

is smaller the greater th e rate of depreciation, and its

dollar value is larger th e greater the ra te of increase in marginal co st over

time. Thus, the depreciation and capital gains components are as relevant

to the user cost of health a s they are to th e user cost of any oth er durable.

Of course, the intere st compon ent of user cost is easy to in ter pre t, for if

one desires to increase his stock of h ealth rathe r t ha n his stoc k of som e

other asset by one unit in a given period, Y T L ~ I measures the interest

payment he forgoes.ll

quation

13)

assumes t i iZip1

0.

For s imilar conclusions with regard to nonsalable physical capi ta l and with regard

to

a

nonsalable stock of goodwill produced by advertising, see .4rrom7 (1 9 6 8 ) a n d

Nerlove and Arrow ( 1 9 6 2 ) .

In a continuous time model, the user cost of health capital can be derived in one

step. I f cont inuous t ime is employed, the term 6,,?i-l does not appear in the user cost

formula . The r ight-hand s ide of 13) becomes ni r

6i

where j is the in-

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CONCEPT O F HEALTH C A P I T A L

231

A

slightly different form of equation (13) emerges if both sides are

divided by the marginal cost of gross investment:

yi a ;i i (13 )

Here

y

(W iG i) /n ip l is the marginal monetary rate of return on an

investment in health and

is the psychic rate of re turn. I n equilibrium, the tot al rate of return on

an investmen t in health m ust equal the user cost of h ealth c apital in terms

of the price of gross investme nt. The latter variable is defined as the su m

of th e real-own rate of intere st and the ra te of dep recia tion.

C .

T h e Pure I nv e s tme n t Mode l

I t is clear th at the number of sick days and the number of healthy day s

are complem ents; their sum equals the constant length of the period.

From equation

8) ,

the marg inal utility of sick time is

-Uhi .

Thus , by

putting healthy days in the utility function, one implicitly assumes that

sick days yield disut i l i ty .

If

healthy days did not enter the utility function

directly, the marginal monetary rate of return on a n investment in health

would equal the cost of he alth ca pital, and he alth w ould be solely an in-

vestment commodity.lVn formalizing the model, I have been reluctant

to treat health as pure investment because many observers believe the

demand f or it has both investmen t and consumption aspects (see, for

example, Mushkin 1962, p. 131; Fuchs 1966, p. 86 ). But to s implify the

remainder of the theoretical analysis and to contrast health capital with

other forms of hu man capi tal, the consumption aspects of deman d are

ignored fr om now on.13

If the marginal utility of healthy days or the marginal disutility of sick

day s were equal to zero, condition (13 ) for the optima l amoun t of h ealth

cap ital in period would reduce to

stantaneous percentage rate of change of marginal cost at age i For a proof , see Par t

B

of the Mathematical Appendix.

l T O avoid confusion, a note on terminology is in order . I f heal th were ent i re ly an

investment commodity

i t would yie ld monetary, but not ut i l i ty , re turns . Regardless of

whether heal th is investment , consumption, or a mixture of the two, one can speak of

a gross investment j~tnction ince the commodity in quest ion is a durable .

l W l s e w h e r e , I have used a pure consumption model to in terpret the set of phenom-

ena that are analyzed in Sect ions I and

IV.

In the pu re consumpt ion mode l , t he

marginal monetary ra te of re turn on an investment in heal th is set equal to zero (see

Grossman 1970 chap.

3 .

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Equat ion (14) can be derived explicitly by excluding health from the

utility function and by redefining the full wealth constraint as1

Maximization of

R

with respect to gross investment in periods

1

and

yields

These two equations imply that (14) must hold.

Figure

1

illustrates the determ inations of the optimal stock of health

capital at any age

i

The demand curve

ME C

shows the relationship

between the stock of he alth and th e rate of return on an investm ent or the

margina l efficiency of hea lth ca pita l, yl. The supply curve S shows the

relationship between the stock of health and the cost of capital,

z L i

a i Since the cost of c apita l is independ ent of t he stock, the sup ply

curve is infinitely elastic. Provided the

MEC

schedule slopes downward,

1 ince the gross inves tme nt prod uction func tion is homog eneous of the fi rst degree,

P iM ViTHi nili

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CONCEPT O F HEALTH C A P I T A L

the equilibrium stock is given by Hi*, where the supply and demand

curves intersect.

In the model, the wage rate and th e marginal cost of gross investment

do not depend on the stock of h ealth . Therefore, the ME schedule would

be negati vely inclined if a nd o nly if

Gi,

the marginal product of health

capital, were diminishing. Since the output produced by health capital has

a finite upper limit of 365 he althy d ays, it seems reasonable to assume

diminishing marginal productivity. Figure

2

shows a plausible relation-

ship between the stock of h ealth a nd the number of h ealthy days . Th is

relationship may be called the production function of healthy days. Th e

slope of the curve in the figure at any po int gives the marginal prod uct

of health c apital. T he numb er of he althy day s equals zero at the d eath

stock H,,i,,, so that

i

365 is an altern ative definition of d eat h.

Beyond H,i,,, healt hy time increases a t a decreasing rate and eventually

approache s its upper asymp tote of 365 days as the stock becomes large.

In Sections I and IV , equation (1 4) an d figure 1 are used to trace out

the lifetime path of health capital and gross investment, to explore the

effects of va riation s in depreciation rate s, and to exam ine the im pact of

changes in the marginal cost of gross investment. Before I turn to these

ma tters, some comments on the general properties of the model ar e in

order. It should be realized that equation

(14) breaks down whenever

desired gross investment equals zero. In this situation, the present value

of t he marginal co st of gross investm ent would exceed the present value

of m arginal benefits for all positive qu ant itie s of gross inve stme nt, and

equations (16) and (17) would be replaced by inequalities.15 The re-

mainder of the discussion rules out zero gross investment by assumption,

bu t the conclusions reached would hav e 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.

5 Formally Gi i

if

i l 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

(196 8) employs these techniques to ana lyze 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-

matica l Appendix-are qu ite adequ ate. Some advan tages of my meth ods

are that they are simple, easy to interpret , and familiar to most econo-

mists. In ad dit ion, they generate essential ly the same equilibr ium condit ion

as the Pontryagin method. Both Arrow and I conclude t h at , if desired

gross investment were positive, then the marginal efficiency of nonsalable

capital would equal the cost of capital. On the other h and , given zero

gross investment, the cost of capital would exceed its marginal efficiency.

Th e monetary returns to an investment in health differ f rom the returns

to investments in education; on-the- job training, and other forms of hum an

capital, since the latter investments raise wage rates.16 Of course, the

amount of health capital might influence the wage rate, but it necessarily

influences the time lost from all activities due to illness or injury. To

emphasize the novelty of my app roach, I assume that health is not a

determ inant of th e wage rate. Pu t differently, a person s sto ck of know l-

edge affects his market and nonmarket productivity, while his stock of

health determines the total amou nt of time he can spend producing

money earnings and commodities. Since both market t ime 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 inpu t 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 r e a ~ h heir desired stock of capital immediately. I t a lso implies

t h a t th e sto ck r ~ t h e r han 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 ) i s not independent of equa t ions (16 ) and (1 7) , i t can be used to

determine the optimal pa th of h ealth capital and , by implication, the

optimal path of gross investment.17

lCThis difference is emphasized by Mushkin

(1962 ,

pp.

132-33 .

l

This statement is subject to the modification th at th e optimal path of capital must

always imply nonnegative gross investment.

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CONCEPT O F HEALTH C PIT L 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 abo ut the behavior of the marginal product of capital and the

marginal cost of gross investment. Both Becker and Ben-Porath assume

th at an y one person owns only a small amou nt of the tota l stock of

human capital in the economy. Therefore, the marginal product of his

stock is cons tant. T o rule ou t 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. hloreover, since the marginal product

of c apital is cons tan t, gross investment is the basic decision variable in

these models. ls In my model, on the other hand, the marginal product of

health capital fal ls 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.

T o i l lustrate how the implicat ions of the health an d hum an 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 produ ct of health capital m ust rise, which would

occur only if t he stock of capital declines. T o restore this equality in the

human capital model, marginal cost must fall, which is possible only if

gross investment declines.'"

111

Life ycle Variations

in

Depreciation Rates

Equation 1 4 ) enables one to stu dy th e behavior of th e 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 hu man capital are undoubtedly cor-

related with age , bu t th e effects of shifts in these variables are treated in

Section I\ ' . Therefo re, the results obtained in th is section may be viewed

as partial effects. T ha t is, they show the im pact of a p ure increase in age

on the demand for health, with all other variables held constant.

l :

Fo r a complete d iscussion of these points , see Becker (1967, pp . 5-12) an d Ben-

Po r a th (1 9 67 , p p . 3 5 3 -6 1 ) . To r mo d e ls o i th e d em an d io r p h y s ical c ap i ta l b y f i rms in

wh ich th e marg in a l co s t of in v es tm en t an d th e amo u n t o i in v es tmen t a re p o s i tiv e ly

co r re la ted , s ee , fo r e san ~p le .E i s n er a n d S t r o t z ( 1 9 6 3 ) a n d G o u l d ( 1 9 6 8 ) .

l 'Section

I

d e m o n s t r a t c s t h a t a n i n cl e a se i n t h e r a t e o i d ~ p r e c i a t i o n o n h e a l t h

cap i ta l m ig h t cau se g ro ss in v es tmen t to in c rea se.

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236 JOURN L

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P O L I T I C A L ECONOMY

Since marginal cost does not depend on age, and equat ion

(14) reduces to

I t is app aren t from equat ion (1 8) t ha t , if the ra te of depreciat ion were

indep ende nt of age , a single qu ant ity of H would satisfy the equality

between the marginal rate of retu rn and the cost of hea lth cap ital. Con-

sequently, there would be no net investment or dis investment after the

ini tial period. One could not , in general , compare H o and H I because

accumulation in the initial period would depend on the discrepency be-

tween the inheri ted s tock a nd the s tock desired in period 1. Th is dis-

crepency in turn would be related to gariations in Ho and other variables

across individuals. B ut , given zero costs of adjus ting to the desired level

immediately, H would be constant af ter period 1. Cnd er the s ta ted

condition of a con stant de preciation ra te, individuals would choose an

infinite life if the y choose to live beyond period 1. I n othe r w ords, if

H I

>

H,,,, ,, , then H , would always exceed th e de ath stock. O

T o permi t the demand for health to vary w i th age , suppose the ra te of

depreciation d epends on age. In general, any tim e path of 6, is possible.

For example, the rate of depreciation might be negatively correlated with

age during the early s tages of the l i fe cycle. Again, the t ime pa th might be

nonmonotonic, so tha t 6, r ises during some periods and fal ls during others .

Despite the existence of a wide variety of possible time paths, it is ex-

tremely plausible to assume that 6, is positively correlated with age after

some point in the life cycle. This correlation can be inferred because, as

an individual ages, his physical s t rength and memory capaci ty deteriorate.

Surely, a rise in the rate of depreciation on his stock of h ealth is merely

one manifestation of the biological process of aging. The refor e, the ana l-

ysis focuses on the effects of an increa se in the r ate of depreciation with

age.

Since a rise in 6, causes the supply curve of health capital to shift up-

ward, i t would reduce the quant i ty of heal th capi tal demanded over the

life cycle. Graph ically, an increase in the cost of c apital from 6, to

6,+, in figure 3 reduces the op t imal stock f rom H , to H,+1 . Th e

greater the elasticity of th e MEC schedule, the greater the decrease in

the opt imal s tock with age. Put different ly, the s lower the increase in the

marginal product of heal th capi tal as H falls , the greater th e decrease in

the opt imal s tock.

Different iat ion of equat ion (1 8) with respect to age quantifies the

percentage rate of de crease in the stock of h ealth over the life cycle:

H ,

-s, F

6,.

19)

O

T he possibi li ty th at d eath can occur in per iod 1

is

ru led ou t f rom now on

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CONCEPT

O F HEALTH C A P I T A L

In this equat ion, the t i lde notat ion denotes a percentage t ime derivat ive

gi

(d H , / d i ) ( l / H i ) , e t c . ) , an d t h e new s y m bo ls a r e : si f i i r 6i

the share of depreciation in the cost of health capital and

the elasticity of the

MEC

s ch ed ul e ( In s ta n d s f or n a tu r a l l ~ g a r i t h r n ) . ~ ~

Equat ion

19) indicates tha t the ab solute value of the percentage de-

crease in H is positively related to the e lasticity of the MEC schedule,

the share of deprec iation in the cost of hea lth cap ital, and th e percentage

rate of increase in the rate of depreciation. If

ei

and were constant , the

curve relating In to age would be concave unless

r

0, sincez2

s i ( l s , )

ai)

0 .

( 2 0 )

The absolute value of gi increases over the life cycle because depre-

ciation's share in the cost of capital rises with age.

F r o m e q u a t i o n It?), In r b L In TY In G, In x There fo re ,

ai 6, dln i -

Y

6, dln H,

Hi ,

Differentia t ion of ( 1 9 ) with respect to age yie lds

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238 JOURNAL

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If 6,

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 eventua lly fall to H,,,. the de ath stock. When th e cost of

heal th capi tal is

6,

in figure 3 , H , = H and death occurs . At

death, no t ime is avai lable for market and nonmarket act ivi t ies , s ince

healthy tim e equals zero Ther efore, the mon etary equ ivalent of sick time

in period n would completely exhaust potential full earnings, W Q More-

over, consumption of the commodity 2 would equal zero, since no time

would be ava ilable for its produ ction if tota l time eq uals sick time.2 3 Be-

cause individuals could not produce commodities, total utility would be

driven to zero at death. '

Having characterized the opt imal path of H, , one can proceed to ex-

amine the behavior of gross investment. Gross investment's life cycle

profile would no t , in general, s imply m irror tha t of heal th capi tal . 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 ra te of d eprec iation not only reduce s the am oun t of

heal th capi tal d e m a n d e d by consumers but also reduces the amount of

capi tal suppl ied 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 th e

other han d, if the change in supply were less than the change in de man d,

gross investment would tend to fall over the life cycle.

T o predict th e effect of an increase in 6, with age on gross investment,

note th at the n et investment can be approximated by H, H, .2s Since gross

investment equals net investment plus depreciat ion,

Differentiation of equa tion (2 1) with respect to age yields

Suppose

6,

an d ~ were cons tan t . The n f rom (19 ) an d (2 0) , the expres-

sion for would simplify to

The above statement assumes that i cannot be produced with

X i

alone. This

would be true if, say. the production function were Cobb-Douglas .

- Utility equal zero when

H

= Hlllillprovided the death time utility function is

such that

U 0 )

=

0

That is,

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

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CONCEPT O F HEALTH C A P I T A L =39

.d d

-

b

1- S,E)

b ,

- s , E ~ )

, E ~ ~

I ,

,

6, - s,e6

Since heal th capi tal cannot be sold, gross investment cannot be nega-

tive. Therefore, 6,

-g .26 T h a t is, if t he stock of he alth falls over the

life cycle, the abso lute value of t he percentage rate of ne t disinvestmen t

cannot exceed the rate of depreciation. Provided gross investment does

-

not equal zero, the term

6, -

s , ~ 6n equation (22 ) mus t exceed zero . I t

follows that a sufficient condition for gross investment to be positively

correlated with the depreciation rate is

E -

H I .

7 If 2

1 w e r e

the p roduc t ion func t ion , the marg ina l p roduc t o f hea l th cap i ta l wou ld

be

ci BCH; - ,

or

I n G i 1 n B C

-

( C + l ) l n H i .

Since In y i n G i n W In x one u ses the equ a t ion fo r In Gi t o o b t a i n (2 4 )

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24

J O U R N A L O F P O L I T I C A L E C O N O M Y

depreciat ion rate rises wi t h age, i t i s not u nl ikely th at unh eal thy (o ld )

people ui l l ma ke larger gross in vestm ents than heal th y (y ou ng ) people.

Th is means that s ick t im e, T L i , wi ll be posi tively correlated wit h M , and

T H i, the medical care and o wn t ime inputs in the gross investme nt fu nc-

t ion , over th e l i f e cyc le .28 In th i s sense , a t l east part o f T L , or T H , ma y

be termed recuperation time.

Unlike other models o f the demand for medical care, m y model does

not

assert

that need or i l lness , measured b y the level o f the rate o f

depreciation, will definit ely be positively correlated wi th utilization o f

medical services . Instead, i t derives this correlat ion from the magnitude

o f the elast icity o f the ME C schedule and indicates that the relat ionship

between the s tock o f heal th and th e number o f heal thy day s wil l tend

to create a posi tive correlat ion. I f

e

is less than

1 ,

medical care and

'he ed wil l def initely be posi tively correlated. RI ~r eo ve r, he smaller t he

va lue o f E the greater the explanatory power of need relative to that

o f the other variables i n the dem and curve for medical care.

I t should be realized t ha t the power o f this model o f l i fe cycle beha v-

ior is that it can treat the biological process of aging in terms of con-

ventional economic analysis. Biological factors associated wi th aging raise

the price o f heal th capi tal and cause individuals to subst i tute aw ay fro m

futu re heal th unt i l death is chosen. I t can be concluded tha t here, as

elsewhere in econ omic s, people reject a prospect-the prospect o f longer

l i fe in this case-because i t i s too cost ly to achieve. In part icular, on ly i f

the elast ici ty o f the M E C schedule were zero would individuals ful ly

compensate for the increase in

6

and , therefore, maintain a constant

s tock o f hea l th .

IV

Market and Nonmarket fficiency

Persons who face the same cost of health capital would dem and the same

amount o f hea l th on ly i f the de terminants o f the ra te o f re turn on an

investment were held constant . Changes in the value of the marginal

product o f heal th capital and the marginal cost of gross inv estme nt s hi f t

the M E C schedule and , there fore , a lt er the qua nt i t y o f hea l th demanded

even i f the supply curve o f capi ta l does no t change. I no w id en t i f y the

variables that determine the level o f the M E C schedule and examine the

ef fe cts o f shi f ts in these variables on the demand for heal th and medical

care. In part icular, I consider the e f fec ts o f variat ions in m arket ef f i -

ciency, measured b y the wage rate, and nonmarket ef f icien cy, measured

b y h u m a n ca p it a l, o n t h e M E C s ched ul e.

2 R S ~ t ehat the t ime path oi

H i

or i would be nonmonotonic if the time path oi

6i were characterized by the occurrence of peaks and troughs. In particular, i would

be relatively low and

T H i

an d would be relatively high if E < 1 when

i

was

relatively high; these periods would be associated with relatively severe illness.

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CONCEPT O F HEALTH C A P I T A L 24

Before beginning the ana lysis , two prel iminary comments a re in order.

Firs t , the discussion pertains to uniform shifts in variables that influence

the ra te of re turn across persons of th e same age. T h a t is, if the variable

X , is one determinan t , then

d In X i

= 1,

all i

d l n X i - l

Second, the discussion proceeds under the assumption that the real rate

of intere st, th e rate of dep recia tion, and th e elasticity of the MEC

schedule are constant . These two comments imply that an increase in Xi

wil l al ter the amount of capi tal demanded but wil l not al ter i ts rate of

change over the l i fe cycle.29 No te from equ at ion ( 2 1 ) :

since the rate of depreciation and the percentage rate of net investment

do not depend on

X 3 0

E q u a t i o n ( 2 5 ) indicates that percentage changes

in health and gross illvestment for a one-unit change in X are ident ical .

Consequently, the effect of an increase in

X

on either of these two vari-

ables can be t reated interchangeably.

A .

Wage Effec ts

Since the value of the marginal product of health capital equals WG, a n

increase in the wage rate, W, raises the monetary equivalent of the mar-

ginal product of a given stock . P u t differently, the higher a person's wage

rate, the greater the value to him of an increase in heal thy t ime.

con-

sumer's wage rate measures his market efficiency or the rate at which he

can convert ho urs of w ork into mone y earnings. Henc e, it is obviously

positively c orrelated w ith the benefits of a reductior. in the tim e he loses

from the production of m oney earning s due to illness. Moreover, a high

wage rate induces an individual to subst i tute 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 pioduct ion are also posi t ively correlated with the

wage.

'St r ictly speaking, shif ts in iwould def ini tely have no effects i f an d onlyon

if

Xi = 0

Even though a uni form $if t i n i implies that-there is no correlation be-

tween i ts level and rate of change,

imight

be altered if

Xi 0

F o r

a

complete dis-

cussion of this point , see Gros sm in 1970, p . 49 .

"Since the analys i s i n t h i s sec t ion deal s wi th var i a t ions in X among indiv idual s of

the same age , t ime subscr ip t s a r e omi t t ed f rom now on. Note a l so tha t 2 5 ) , l i ke the

expression for I ; ignores the one-per iod lag between an increase in gross investment

and an increase in the stock of heal th.

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JOURNAL OF POLITICAL ECONOMY

If an upward shift in the wage rate had no effect on the marginal cost

of gross inves tmen t a

1 percent increase in it would increase the rate of

re turn

y

associated with a fixed stock of capital by 1 percent . In fac t

this is not the case because own time is an input in the gross investment

funct ion.

If

K is the fraction of the total cost of gross investment ac-

counted for by time then a percent rise in

W

would increase marginal

cost n b y K percent . After one nets out the correlat ion between

W

and

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 rat e and the level of the

MEC

schedule are positively

correlated the deman d for hea lth would be positively related to

W .

Graphical ly an upward shift in

W

from

W 1

to

W 2

in figure 4 shifts the

MEC

schedule from

MEC l

to

MEC2

and with no change in the ccst of

heal th capi tal increases the opt imal s tock from

H I

to

H z

A formula for

the wage elastic ity of health cap ital is31

Th is elasticity is larger th e 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 heal th or gross invest-

Differentiation oi the natural logarithm of

18)

with respcct to In W yields

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C O N C E P T O F H E L T H C P I T L 43

ment ar e positively re lated ,

W

has no effect on the amount of gross in-

vestment supplied by a given inpu t of medical care. Therefore, the d emand

for medical care would rise with the wage.

If

medical care and own t ime

were employed in fixed proportions in the gross investment production

fun ctio n, 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 incen tive to sub stitu te medical care for their relatively more ex-

pensive own time. formula for the wage elasticity of medical care is

where o, is the elasticity of sub stitution betw een

M

and

h

in the pro-

duction of gross investment." T h e grea ter the value of o,, the greate r

the difference between the wage elasticities of M and

H

No te th at an increase in the price of e ither 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 part icular,

an upward shift in the price of medical care lowers the

M E C schedule

and causes the demand for heal th to decl ine. This difference arises be-

cause the price of tim e influences the value of the m arginal p rodu ct of

health c apita l while the price of m edical care does no t.

B

he Role of H um an Capital

Up

to now, no systematic al lowance has been made for variat ions 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

l ike technology and entrepreneurial capaci ty, forces that shift product ion

funct ions or that al ter the environment in which fi rms operate. Reason-

ing by analogy, one can say that certain environmental variables influ-

ence product ivi ty in the nonmarket sector by al tering the marginal

products of the direct inputs in household production functions. This

study is part icularly concerned with environmental variables that can be

associated with a partic ular person-his or her race, sex, stock of hum an

capital, etc. Tt 'hile the analysis that follows could pertain to any environ-

mental variable, i t is wel l documented that the more educated are more

efficient producers of money earnings. Consequently, it is assumed that

shifts in human capi tal , measured by educat ion, change product ivi ty in

: "or a pro of, see P ar t

C

of t he Mathemat i ca l Appendix . The corresponding equa-

t ion ior the wage elast ici ty of the own t ime input i s

c r ~ a - ( 1

K ) E

0 , .

This elasticity is positive only i o .

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44 JOURNAL O F P O L I T I C A L E ONOMY

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.33

T o determine th e effects of e duca tion on productio n, 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. I t follows th at the marginal product

of

El

the index of human capital, would be

where g tg is the marginal product of medical care and g is the mar-

ginal product of time.31 If a circumflex over a variable denotes a per-

centage change per u nit change in E the last equation can be rewritten as

1

( gi-tg P)

THg

I( g - t g ) ]

i .

? H Z

aE I =[

I

tg

( 2 8 )

Equat ion

( 2 8 )

indicates that the percentage change in gross investment

supplied to a consumer by a one-unit change in

E

is a weighted average

of the percentage chan ges in the m arginal produ cts of

M

and

T H . 3 5

If

E increases productivity, then

r~

> 0 Provided E raises both mar-

ginal products by the same percentage, equation ( 2 8 ) would simplify to

? I3The model developed here is somewhat s imilar to the one used by Michael ( 1 9 6 9 ) .

: i I is homog eneous of degree in M and T H , then i rom Euler ' s theorem

Differentiation of this equation with respect to E holding M a n d T H constant, yields

the marginal product of hum an capi ta l .

Unstead of put t ing educat ion in the gross investment product ion iunct ion, 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 t reatment of environm ental var i-

ables like education must permit these variables to influence all household commodities.

Since deprecia t ion ra tes and s tock-f low rela t ionships are re levant only i i a par t icular

commodity is durable , a symmetr ical development oi the role of environmental var i-

ables requires that they affect household product ion funct ions and not deprecia t ion

rates or stock-flow re lationships. In a more c omplicated version of the m odel, the gross

investment iunct ion, the ra t e of deprecia t ion, and the marginal produ ct ivi ty of heal th

capi ta l might a l l depend on educat ion. But the basic implicat ions of the model would

not change.

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C O N C E P T

OF

H E L T H C P I T L 45

In this case, education would have a neutral imp act on the marginal

produ cts of all factors. T h e rest of the discussion assumes factor neu-

trality.

Because education raises the marginal product of the direct inpu ts, it

reduces the quantity of these inputs required to produce a given amount

of gross investm ent. Hence, with no change in inp ut prices, an increase

in E lowers average or marginal cost. In fact, one easily shows that

where is the percentage change in average or marginal 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 co nst ant, an increase in education would

raise the marginal efficiency of health capital and shift the MEC schedule

to the right.3i In figure

5,

an increase in E f rom El to

E

shif ts the MEC

curve from MEC l to MEC 2

If

the cost of c apital were indep enden t of

E there would be no change in the supply curve, and the more educated

would demand a larger optimal stock (compare

H1

and

H 2

in fig. 5 ) .

The percentage increase in the amount of health demanded for a one-

unit increase in

E

is given by38

Since

Y H

indicates the percentage increase in gross investment supplied

by a one-unit increase in

E

shi f ts in th i s variable w p l d not a l te r 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 .4ppendix, where the human capital

formulas are developed in more detail.

fl

It should be stressed th at the model of nonma rket p roductivi ty varia tions pre-

sented here examines the partial effect of a n increase in educatio n with th e wage rate

held constant. Although these tw o 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 ) .

: X

If and are fixed and if

G

depends only on

H

then

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J O U R N A L O F P O L I T I C A L E C O N O h l Y

schooling might supply him with

3

percent more health. Given this con-

di t ion, both persons would demand the same amounts of

M

and

TH

AS

this example illustr ates, any effect of a change in

E

on the demand for

~ e d i c a l are or t ime reflects a posit ive or negative difference between

H

and

Y ~ ~ ~

E q u a t i o n

( 3 2 )

suggests tha t, if the elasticity of t he

MEC

schedule

were less than un i ty, the more educated would dem and m ore heal th bu t

less medical care. Put differently, they would have an incentive to offset

part

of the increase in health ca used by an increase in education by re-

ducing their purchases of medical services. Kote that if

~

were negative

and were less tha n 1

H

would be negat ive and

M

would be positive.

Since educat ion improves market product ivi ty,

I

have examined the im-

plications of the hypothesis th at I . is positive. But the model is appli-

cable whether YH is positive or negative and gives empirical predictions

in either case.

V Summary

and onclusions

T h e main purpose of this pa per ha s been to construct a model of the

dema nd for the commodity "good heal th." T he central proposi tion of th e

model is that heal th can be viewed as a durable capi tal s tock that pro-

duces an output of heal thy t ime. A person determines his optimal stock

of hea lth cap ital at a ny age by equa ting the m arginal efficiency of this

capita l to its user cost in term s of the price of gross inve stme nt. Gra phi-

cal ly, each person has a negat ively incl ined demand curve for heal th

" 'The t e rms 9 a n d

TH

are equal because,

by

the defini t ion of factor neutral i ty,

has no effect on the rat io of the marginal product of 1 t o t he marg ina l p roduc t of TH

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ON EPT O F

HE LTH

CA P IT A L 47

capital, which relates the m arginal efficiency of capital to th e 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 1 969 ) , these studies have not developed behav -

ioral models that can predict the effects that are in fact observed. Con-

sequently, the f ramework I have developed is important because of its

ability to b ridge th e existing gap betm~een theory and empiricism in t he

anzlysis of health differentials. J l y model explains variation s 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 ra te of d epreciation on th e

stock of health with ape, and i t has traced upward shif ts in the demand

curve to increases in the wage rate and education.

One predictio n of the model is th at if the rat e of dep reciation incre ases

with age, at least af ter some point in the l i fe cycle, then the quanti ty of

health capital demanded would decline over the life cycle. At the same

tim e, provided the elasticity of the marg inal efficiency of capit al sched ule

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 th at one of th e advanta ges of the

model is tha t it enables one to stud y the effects of d emographic variables

like age and education without assuming that these variables are posi-

tively or negatively correlated with consum ers' tastes for hea lth. In -

stead, these var iables enter the analysis through their impact on ei ther

the cost of health capital or its marginal efficiency, and one can make

strong predictions concerning their effects on health levels or medical

care.

I t must be admitted th at this paper has made a number of simplifying

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. Mo st imp orta nt of all, it would m odify the assum p-

tion that consumers ful ly anticipate inter temporal var iat ions in depre-

ciation rates an d, therefore, know their age of d eath w ith certai nty . Since

in the real world length of life is surely not known with perfect foresight,

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48

JOURNAL OF POLITICAL ECONOMY

i t might be postulated that

a

given consumer faces a probabi l i ty dis t ribu-

tion of depreciation rates in each period. This uncertainty would give

persons an incentive to prote ct themselves against the losses associated

with higher than average depreciat ion rates by purchasing various types

of insurance and perh aps by holding an excess stock of

heal th.40 But

whatever modifications are made, it would be a mistake to neglect the

essential features of the model hav e presented in this pap er. Any model

must recognize that heal th is a durable capi tal s tock, that heal th capi tal

differs in important respects from other forms of human capi tal , and th at

the demand for medical care must be derived from the more fundamental

deman d for good health.

ppendix A

Mathematical ppendix

A .

Utilit y Mnxinzization-Discrete Tim e

To maximize utility subject to the full wealth and production function con-

straints, form the Lagrangian expression

U($oHo, . , ,H,, Zo, . Z,)

where

Ci P,MI W I T H L

and

C I i V i X i W i T i .

Differentiating

L

with re-

spect to gross investment in period 1 and setting the partial derivative

equal to zero, one obtains

ahi dHi dhi+l

Uhi uhi+i--

d ~~ d r i l d H i+ l

Wn dTL,/dH,)/ dHn/a~i-1)

. . . - I -

1 T)

B u t

dhi aH , aHi+l dHn

G,, 1, (1 &),---

a~~ a ~ ~ ~ dI -

I

I L - ~

*')For

an a t t empt t o i n t roduce uncer t a in ty in to a model t ha t v i ews heal th as a

durable capi t a l s tock , see Phelps ( i n pr epara t ion) .

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C O N C E P T O F H E A L T H C A P IT A L 249

dC,-

1

d

TL,

I h l ) . . .

1

? i m - 1 ) ) - - -

n,-,,

and

-Ga.

dl&-

1

dH1

Therefore ,

B.

Utility

2axi~~zisatio~z Colzti~~oz~sime

Let the u t i l i ty funct ion be

U J nzi f(+ iHi , i)di,

(A41

where

mi

is the weight attached to utili ty in period i. Equat ion

iA4 )

defines an

addi t ive u t il i ty funct ion, but an y monotonic t rans format ion of th is funct ion

could he employed.- Let all household production functions he homogeneous of

degree

1 .

The n C ,

rr,Ii.

C,i q iZi?%nd ful l weal th can be wri t ten as

R J e- (x ,I , q,Z, W,TL,)di .

(A51

By definition,

1,

w,,

(A6)

where

H

is the instantaneous rate of change of capital s tock. Substitution of

A6

in to

As)

yields

R J e- (x16,H,

, H ,

q,Z,

W,TL,)di. (A71

To maximize the u t i l i ty funct ion. form the Lagrangian

L

hR

J Int,f(+,H,,Z,) he- (x,G,H,

n ~ q ,z ,WLTL,) di, (A81

where

Euler 's equat ion for the opt im al pa th of H i is

Strotz (1955-56) has shown , however , that cer ta in res tr ic t ions must be placed on

the

mi I n

particular, the initial consumption plan will be fulfilled if and only if

ma

( m o ) .

* he variable q , equals the marginal cost of Zi

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2 5

JOURNAL OF POLITICAL ECONOMY

I n the present context

Consequent ly

which is the continuous time analogue of equation

13).

C .

W a g e

Ejects

T o obtain the wage elasticities of medical care and the time spent producing

heal th three equat ions mus t

be

partially differentiated with respect to the wage.

These equat ions are the gross inves tment product ion funct ion and the two f i rs t -

order condi t ions for cos t minimiza t ion:

l ( M , T H ; E ) M g ( t ; E ) 6 ) H ,

ng ,

P n ( g - t g ) .

Since is linear homogeno us in

M

a nd TH

( g- tg )g

0

I ( [ d ( g t g ) ] d T H )

Ther efore the following relationships hol d:

g ( g - t g l g

~ T H

t

l o ,

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CONCEPT O F HE LTH C PIT L

Carrying out the differentiation, one gets

d T H d M

H ( Zf )F

gf---

g t g )

d W dl.V t

djt dg d T H dg d M

1

= g

dJV

djt d ( g t g ) d T H d ( g t g ) d M

= g t g )

n

-+

dl.V

[

d T H d W d M W

Using the cost-minimization conditions and A141 and rearranging terms, one

has

d n d T H d M I E X

I F

W

P

=

d W d W d W W

Since (A151 is a system of three equations in three unknowns-dTH/d6Z7,

dLI I drl., an d dn, dl17-Cramer s rule ca n be applie d to solve fo r, say, diM/dT.t .

Ihe determinant in the den ominator reduces to

l a D ~ r 2 Z 2 )

T H J I . T h e d e -

terminant in the numerator is

( z j t o , T B M X E

T H M

Therefore

d M T H M

dll

I x

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C O N C E P T O F H E A L T H C A P I T AL

H M

r ~ .

Solving for M and noting that H = r H c , one gets

M = Y H E

1 .

ppendix B

Glossary of Mathematical Terms

n

. . . . . . . . . . . . .

Total length of life

. . . . . . . . . . . . . . .Age

H

. . . . . . . . . . . .

Inherited stock of health

Hi

. . . . . . . . . . . . Stock of health in period i

H,,,

. . . . . . . . . .

Death stock

C

. . . . . . . . . . . .

Service flow per unit stock or number of healthy days per unit

stock

. . . . . . . . . . . . . Total number of healthy days in period

Z . . . . . . . . . . . . Consumption of an aggregate commodity in period

i

i . . . . . . . . . . . . . Gross investment in health

i . . . . . . . . . . . . . Rate of depreciation

. . . . . . . . . . . . Medical care

TH . . . . . . . . . . . Time input in gross investmect function

X i

. . . . . . . . . . . .

Goods input in the production of i

Ti . . . . . . . . . . . . Time input in the production of Zi

Ei . . . . . . . . . . . . Stock of human capital

g t,g

. . . . . . . . Marginal product of medical care in the gross investment

production function

g

Marginal product of time

Pi

. . . . . . . . . . . .

Price of medical care

V i

. . . . . . . . . . . . Price of

Xi

Wi . . . . . . . . . . . . Wage rate

A . . . . . . . . . . . . Init ial assets

r . . . . . . . . . . . . . Rate of interest

TW . . . . . . . . . .

.Hours of work

TL

. . . . . . . . . . .

Sick time

. . . . . . . . . . . . . Constant length of the period

R . . . . . . . . . . . . Full wealth

i

. . . . . . . . . . . . Marginal product of health capital

U h i

. . . . . . . . . . .

Marginal utility of healthy days

?

.Marginal utility of wealth

X . . . . . . . . . . . . . Marginal cost of gross investment in health

xi

. . . . . . . . . . . . .

Percentage rate of change of marginal cost

y i . . . . . . . . . . . . Monetary rate of return on an investment in health or

marginal efficiency of health capital

a . . . . . . . . . . . . . Psychic rate of return o n an investmen t in health

. . . . . . . . . . . .

A

tilde over a variable denotes a percentage time derivative

s,

. . . . . . . . . . . . . Share of depreciation in the cost of health capital

. . . . . . . . . . . . . Elasticity of the MEC schedule

K

. . . . . . . . . . . . . Fraction of the total cost of gross investment accounted for

by time

a

. . . . . . . . . . . .

Elasticity of substitution betwen medical care and own time in

the production of gross investment

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54

J O U R N A L O F P O L I TI C A L E C O N O M Y

e

Elastici ty of

EI

with regard to

W

e

Elastici ty of

3 1

wi t h r ega rd t o

W

ci rcumflex over a var iable denotes a percentage change per

uni t change in

E

Percentage change in gross investment for a one uni t change

in

E

'Total cost of gross investm ent i n health in period 2

Total cost of

Z

W ei ght a t t ached t o t o t a l u t i l i t y i n pe r i od i

Marginal cost of

Z

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