Michael Grossman

English

Research Papers in Economics

This PDF is a selection from an out-of-print volume from the National
Bureau of Economic Research
Volume Title: The Demand for Health: A Theoretical and Empirical Investigation
Volume Author/Editor: Michael Grossman
Volume Publisher: UMI
Volume ISBN: 0-87014-248-8
Volume URL: http://www.nber.org/books/gros72-1
Publication Date: 1972
Chapter Title: THE PURE CONSUMPTION MODEL
Chapter Author: Michael Grossman
Chapter URL: http://www.nber.org/chapters/c3487
Chapter pages in book: (p. 31 - 38)III
THE PURE CONSUMPTION MODEL
In the previous chapter, I explored the effects of variations in the shadow
price of health in the context of the pure investment model. These varia-
tions were traced in turn to shifts in the rate of depreciation, market
efficiency, and nonmarket efficiency. In this chapter, I utilize the pure
consumption model to examine the effects of age, the wage rate, and
education.' The purpose of the analysis is to indicate the major differences
between the investment and consumption models rather than to develop
the latter in detail. Consequently, a formal presentation of the consump-
tion model, including derivations of all formulas, has been relegated to
Appendix C.
1. LIFE CYCLE PATTERNS
If the cost of capital were large relative to the monetary rate of return on
an investment in health and if =0,all i, then equation (1-13) could
be approximated by
Uh,G1—UH,—ir(r+
(3-1)
A—A(1+r)t
Equation (3-1) indicates that the monetary equivalent of the marginal
utility of health capital in period i must equal the discounted user cost
of H, •2Divisionof the equilibrium condition for by the equilibrium
1Themodel of life cycle behavior presented in this chapter is similar to Gilbert R. Ghez's
analysis of life cy.cle demand for durable consumer goods. See "A Theory of Life Cycle
Consumption," unpublished Ph.D. dissertation, Columbia University, 1970, Chapter 1.
2Solvingequation (3-1) for the monetary equivalent of the marginal utility of healthy
time, one has
Uh1—ir(r+ 53/G1
A (I+r)'
where ir(r + is the undiscounted price of a healthy day. Given diminishing marginal
productivity of health capital, this price would be positively correlated with H or h even if
iv were constant. Therefore, the consumption demand curve would be influenced by scale
effects. To emphasize the main issues at stake in the consumption model, I ignore these
scale effects essentially by assuming thatand henceare constant. The analysis would
not be greatly altered if they were introduced.32 The Demand for Health
condition for H1 generates the basic equation for the analysis of life cycle
demand:
(3-2)
Condition(3-2) simply statesthatthe marginal rate of substitution between
H1 and must equal the ratio of the discounted user cost of
to the discounted user cost of H..
To characterize the life cycle path of health capital in a precise
manner, certain restrictions must be placed on the utility function. In
particular, it must be assumed that this function is weakly separable in
H. and That is, the marginal rate of substitution between H. and
depends only on these two stocks and is independent of the other
H's and all other commodities. It is also assumed for the present that
there is no "health time preference." This means that 1/UH1 =m=1
when =H,.Of course, since indifference curves are convex to the
origin, a reduction .in H,÷1 relative to H along a given indifference curve
would increase relative to UH1.
Suppose the rate of interest were zero and the rate of depreciation
were independent of age. Then the discounted user cost ratio and, there-
fore, the marginal rate of substitution between H, and would equal
unity. Given no time preference, this implies H,H,÷1. If the rate of
interest were positive, the discounted user cost ratio and, hence,
1/UH1 would be less than unity. Convexity of indifference curves
implies H.÷1 >H.in this situation. Therefore, under the stated conditions
of no time preference and constant depreciation rates, H would rise
over time if r> 0 and would be stationary if r =0.Either of these
two life cycle patterns suggests that individuals would choose to live
forever.
As in the investment model, a positive correlation between the rate
of depreciation and age generates a stock of health pattern that is
consistent with a finite life. To see this, first let the rate of interest equal
zero. Then the discounted user cost ratio would equal which is
clearly greater than one. It follows that 1/UH1 must exceed one,
and this implies H,<H1.Thus, the stock of health would fall through-
out the life cycle because the price of the next period's health in terms
of its present period price is always greater than one.
Now let the rate of interest be positive. In this case, might
exceed H, even if >ö,.But if cS grew at a constant rate, the dis-The Pure Consumption Model 33
counted user cost ratio would rise over time.3 Since this price ratio
increases over time, so must the marginal rate of substitution between
and H,. Convexity of indifference curves dictates that must
fall with age. Health capital might increase for a while but would peak
when the depreciation effect began to outweigh the interest rate effect.
After the peak is reached, H, would decline until death is "chosen."
A formula for the percentage rate of change in health capital over
the life cycle is given by4
=a[lnm + ln (1 + r) — (3-3)
In this formula, o is the elasticity of substitution in consumption between
H, and
—
—ô(ln
Equation (3-3) includes a time preference effect as well as interest and
depreciation effects. If there were time preference for the present, the
marginal utility of H, would exceed the marginal utility of H,÷1 when
H. = Hence ln m <0, and H would fall faster and death would
occur sooner given preference for the present. On the other hand,
preference for the future makes in m > 0 and prolongs the time interval
during which H increases. Equation (3-3) also indicates that H. reaches
its maximum quantity when in m + In (1 + r) =
Although the demand for health capital declines after some point
in the life cycle, gross investment would tend to be positively correlated
with age if the elasticity of substitution between present and future
health were less than unity.6 Put differently, if present and future health
were relatively poor substitutes, individuals would have an incentive to
If twereconstant, then the derivative of the natural logarithm of the discounted
user cost ratio would be
r(ö,.,.1—ö,)
L(r + ö1)(r+
since5,+1
For a derivation of equation (3-3), see Appendix C, Section 1.
When In m + in (1 + r) = =0.This stationary point gives a maximum since
=—s1(1 —
Theformula for 11,, assumes o andare constant.
For a proof, see Appendix C, Section 1.34 The Demand for Health
offset part of the reduction in health caused by an increase in the rate of
depreciation by increasing their gross investments. In fact, there is reason
to believe that the elasticity of substitution is relatively small, at least in
the vicinity of the death stock. To see why this, is the case, redefine a as
ö(ln +
where H = H. —Hmjn.7I have shown that rises with age,
which increases Since —' as —,0
and since this condition must be satisfied at death, small increases in
must have large effects on the ratio of marginal utilities around
the death age.8
In both the consumption and investment models, biological factors
associated with aging cause individuals to substitute away from future
health until death is chosen. There are two major differences between.
the two models. First, even if the depreciation rate rises continuously
with age, the existence of time preference for the future or a positive rate
of interest might cause health capital to increase for a while in the
consumption model. Second, the elasticity of substitution between present
and future health, rather than the elasticity of the MEC schedule,
determines (1) the responsiveness of health to a change in the rate of
depreciation and (2) the life cycle behavior of gross investment.
2. MARKET AND NONMARKET EFFICIENCY
To study the effects of variations in the shadow price of health among
individuals of the same age, a cross-sectional consumption demand curve
must be specified. The simplest specification is
H = H(R*, Q*), (34)
where R* = R/Q, real full wealth; Q* = ir(r + ä)/Q, the relative user cost
or shadow price of health; ln Q = w ln ir(r+ö) + (1 —w)ln q, the
natural logarithm of a weighted geometric price level of health and the
7 definitionimplies the death time utility function is
U = Z1).
If both consumption and investment aspects of the demand for health were considered,
and Z1 would equal zero at death. Hence, total utility would be driven to zero provided
U(O, 0) =0.
8Ifthe elasticity of substitution were constant, it would have to be small at all stages
of the life cycle and not just in the vicinity of the death stock.The Pure Consumption Model 35
aggregate commodity Z, where the weights w and (1 —w)aresharesof
these commodities in full wealth, and where q is the price of Z. A reduction
in the relative shadow price of health would lead consumers to substitute
H for the aggregate commodity Z. Moreover, if health were not an inferior
commodity, an increase in real full wealth would increase demand. The
magnitudes of responses in H to changes in relative price and real wealth
are summarized by e,,, the own price elasticity of demand, and the
wealth elasticity.
Wage Effects
Differentiation of equation (3-4)withrespect to the wage rate
yields9
eH,W =—e11(K—K). (3-5)
In this equation, K is the fraction of the total cost of gross investment
accounted for by time and K is the average time intensity of nonmarket
production. The derivation of (3-5) holds real wealth constant so that
the equation shows the pure substitution effect of a change in the wage
rate. Since eH is positive by definition, eHw0 as KK.
The sign of the wage elasticity is ambiguous because an increase in
the wage raises the marginal cost of gross investment in health and the
marginal cost of Z. Hence, both it and the price level are positively
correlated with W. If time costs were relatively more important in the
production of health than in the production of a typical nonmarket
commodity, the relative price of health would rise with the wage rate,
which would reduce the quantity demanded. The ambiguity of the wage
effect here is in sharp contrast to the situation in the investment model.
In that model, the wage rate would be positively correlated with health
as long as K were less than one.
The Role of HumanCapital
To study the effects of variations in nonmarket productivity
associated with education, note that since E influences productivity in all
nonmarket activities, it alters the marginal costs of all home-produced
commodities. Given factor-neutrality, the percentage reduction in the
For a proof, see Appendix C, Section 2. The corresponding equation for the wage
elasticity of medical care is
eM,w= Ka,,—(K—K)eH.36 The Demand for Health
marginal cost of the aggregate commodity Z would be —re,whereis
the percentage increase in either the marginal product of Z'sgoodsinput
or time input as Eincreases.Therefore, human capital's effect on the
weighted geometric price level is given by
Q =—= — [wrH+(1—w)r
With money full wealth fixed, the term rE can. be viewed as the percentage
change in real full wealth due to the change in nonmarket productivity
associated with education. It indicates the nonmonetary return to an
investment in education.1°
If education improved productivity, then the last equation suggests
that an increase in this variable would reduce the absolute shadow
prices of all home-produced commodities, increase real wealth, and also
alter relative prices provided the improvements in productivity were not
the same for all commodities. Therefore, a shift in Ewouldset in motion
wealth and substitution effects that would alter the demand for health.
Differentiating the demand function (3-4) with respect to E,holding
money full wealth and the wage rate constant, one obtains"
I?= +eH(rH—rE). (3-6)
The first term on the right-hand side of equation (3-6) reflects the wealth
effect, and the second term reflects the substitution effect. If E's produc-
tivity effect on the gross investment function were the same as its average
productivity effect, then r11 =rE,and A would reflect the wealth effect
alone. In this situation, a shift in education would be "commodity-
neutral." If rH > rE, E would be "biased" toward health, its relative
price would fall, and the wealth and substitution effects would both
operate in the same direction. Consequently, an increase in E would
definitely increase the demand for health. If rH <rE, E would be biased
away from health, its relative price would rise, and the wealth and
substitution effects would operate in opposite directions.
The human capital parameter in the demand curve for medical care
is given by
M =rE(71H—1)+ (rH —rE)(eH—1). (3-7)
'°Foran exhaustive discussion of the above method of specifying the nonmonetary
benefits of education, see Robert T. Michael, TheEffect of Education onEfficiency in Con-
sumption, New York, NBER, Occasional Paper 116, 1972, Chap. 1.
"For derivations of the human capital parameters given by equations (3-6) and (3-7),
see Appendix C, Section 2.The Pure Consumption Model 37
If shifts in E were "commodity-neutral," then medical care and education
'would be negatively correlated unless￿1. If, on the other hand, there
were a bias in favor of health, these two variables would still tend to be
negatively correlated unless the wealth and price elasticities both exceeded
one.
The preceding discussion reveals that the analysis of variations in
nonmarket productivity in the consumption model differs in two impor-
tant respects from the corresponding analysis in the investment model.
In the first place, wealth effects are not relevant in the pure investment
model. This follows because an increase in wealth with no change in the
interest rate and the rate of depreciation would not alter the equality
between the cost of capital and the rate of return on an investment in
health, Note that health would have a positive wealth elasticity in the
investment model if wealthier people faced lower rates of interest.12 But
the analysis of shifts in education assumes money wealth is fixed. Thus,
one could not rationalize the positive relationship between education and
health in terms of an association between wealth and the interest rate.
In the second place, if the investment framework were utilized, then
whether or not a shift in human capital is commodity-neutral would be
irrelevant in its effect on the demand for health. As long as the
rate of interest were independent of education, H and E would be
positively correlated. Put differently, if individuals could always receive,
say, a 5 percent rate of return on savings deposited in a savings bank,
then a shift in education would create a gap between the cost of capital
and the marginal efficiency of a given stock.
3. GLOSSARY
UH,Marginal utility of H.
m Index of time preference
C Elasticity of substitution between and
R*Real full wealth
Q*Relative user cost or shadow price of health
Q Weighted geometric price level
12Ifthe rate of interest depends on full wealth and if health does not enter the utility
function, then
—(1 —
sisthe share of interest in the cost of health capital andisthe wealth elasticity
of the interest rate.38 The Demand for Health
eHOwn price elasticity of demand for health
Wealth elasticity of demand for health
K Average time intensity of nonmarket production
rz Percentage change in either the marginal product of goods or time
in the Z production function for a one unit change in E
rEPercentage change in real full wealth for a one unit change in E

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