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American Economic Association
Life Cycle Consumption and Labor Supply: An Explanation of the Relationship between Income
and Consumption Over the Life Cycle
Author(s): James Heckman
Source: The American Economic Review, Vol. 64, No. 1 (Mar., 1974), pp. 188-194
Published by: American Economic Association
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Life Cycle Consumptionand Labor Supply:
An Explanationof the Relationship
Between Income and Consumption
Over the Life Cycle
By JAMES HECKMAN*
In a recent paper in this Review, Lester
Thurow presents empirical evidence in apparent
contradiction with the conventional
life cycle consumption theory enunciated by
Franco Modigliani and Richard Brumberg,
Menahem Yaari, and James Tobin. That
theory predicts no necessary relationship between
consumption and income receipts at
any age, but Thurow demonstrates a strong
relationship and shows that income and consumption
expenditure both peak in the age
interval 45-54. Thurow's principal explanation
for his finding is that credit market restrictions
prevent consumers from borrowing
as much against their future income as they
desire at the going interest rate. As long as
income tends to increase with age, and discounted
future income cannot be fully transferred
at the borrowing rate, a consumer's
effective net worth increases with age which
causes increasing consumption with age.
Based on this argument, Thurow recommends
government intervention into the
consumption loan market to allow for optimal
adjustment of consumption.
Keizo Nagatani explains the same facts by
building a model based on the uncertainty of
future income. By adjusting expected future
income for risk, a "typical" consumer will
buy less than he would in a riskless environment
with the same expected income stream.
However, being the typical consumer, he
realizes his expected income, and he successively
revises his consumption plan upward
since his realized income exceeds his risk
adjusted income forecast. For this reason,
his consumption expenditure and income
streams are closely related.
Both authors relax a standard neoclassical
assumption to obtain their theoretical results:
Thurow assumes imperfect credit markets
while Nagatani invokes uncertainty.'
However, their different explanations lead to
different policy implications, since Nagatani's
results provide no basis for government
intervention to break down institutional
barriers in the credit market.2
In this paper, we present an alternative
neoclassical model which can explain
Thurow's results without resort to either
credit market imperfections or uncertainty.
Rather than treating income as exogenously
given, we view earnings as resulting from a
life cvcle labor supply decision. If individuals
are free to set their hours of work, and if
wage rates change systematicallv over the
life cycle, the path of consumption of market
goods will depend on the wage rate at each
age unless goods and leisure are independent
of each other in utility.
There is strong empirical evidence that* Columbia University and the National Bureau of
Economic Research. This research was sponsored by a
IU.S. Department of Labor Manpower Administration
dissertation grant. I am deeply indebted to Edmund
Phelps for comments, and to members of my dissertation
committee at Princeton: Orley Ashenfelter, Stanley
Black, Richard Quandt, Albert Rees, and Harry Kelejian.
I retain responsibility for all errors. This paper is
not an official National Bureau publication since the
findings reported herein have not yet undergone the full
critical review accorded the National Bureau's studies,
including approval of the Board of Directors.
1 Both authors also discuss alternative explanations
such as family composition effects, shifts in preferences,
and measurement errors.
2 One might argue that some portion of the risk adjustment
of income in the Nagatani model is due to
"market imperfection." However, in the presence of
uncertainty, market imperfection is not a well-defined
operational concept and specific policy recommendations
are more difficult to obtain. I am indebted to
Phelps for this point.
188
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VOL. 64 NO. 1 HECKMAN: LIFE CYCLE CONSUMPTION 189
wage rates vary over the life cy-cle.3 We
demonstrate below that if the interest rate
equals the rate of pure time preference, the
level of consumption by age moves exactly
with the path of wage rates if time and goods
are substitutes in utility. Allowing for a difference
between the rate of time preference
and the interest rate, an association remains
between wage rates and consumption but it
is not as precise.
In the first section of this paper, we present
an informal statement of the mlodel.
The second section is devoted to a rigorous
derivation of these propositions.
I
Standard life cycle models assume that the
consumer's preferences for goods are the
sanmeat each age, and independent of goods
consumption at other ages, with future utilitv
discounted at an exponential rate.4 These
models either ignore the consumption of
leisure or assume that hours of work are institutionally
fixed so that a given lifetime
wage path implies an exogenously determined
income stream.5 The only factor creating
differences in goods consumption by
age is the interest rate net of pure time preference.
If the rate of interest exceeds the rate
of time preference, a consumer has an incentive
to postpone his consumption of goods to
later ages.
In our model, we introduce an explicit
labor-leisure choice at each age, maintaining
the assumption that the utility at one age is
independent of the consumption of goods and
leisure at other ages. The rates of interest
and tinmepreference continue to operate on
consumption and leisure in the usual way,
but a new factor is added. If the price of
leisure is high at certain ages, individuals
tend to consume less leisure at those ages.6
To focus on this effect, suppose that the
interest rate and rate of time preference are
zero, and that the price of goods is the same
at all ages. If market goods are complements
with leisure in the sense that the marginal
utility of leisure increases with increments in
the consumption of goods,7 the consumer has
an incentive to economize on both his coInsumption
of goods and leisure at ages where
the price of leisure is high, since there are
gains in utility from consuming time and
goods jointly. In this case, he works more
and saves more at ages with higher wage
rates than at other ages. If market goods are
substitutes for leisure in the sense that a reduction
in the consumption of leisure raises
the marginal utilitv from consuming goods,
at ages where the price of leisure is high relative
to other ages, the consumer has an incentive
to economize on his leisure but spend
more on goods. In this case, at ages where
wage rates are high, consumers work more,
earn more, and consunme more than at ages
where wages are lower.
Upon introducing the effect of time preference
and interest rates, and assuming that
the rate of interest exceeds the rate of time
preference, one finds that the consumption of
goods and leisure tends to be pushed towards
later ages, but the wage-induced pattern
of consumption remains, albeit somewhat
blurred. XVefornmalize these intuitive statements
below.
I See for example the work of Michael liird.
4 The objections to this utility, specification are well
known, but it is widely used (see Yaari, ModiglianiBrumberg,
Nagatani). For a statement of those objections,
see J. Hicks, p. 261.
Frank Ramsey explicitly considers an intertemporal
model of consumption and work effort. However, he
assumes both iintertemporal and contemporaneous
additive separability of the preference function in goods
and leisure. We demonstrate below that the latter assumnptionleads
to the same predictions as standard
models of life cycle consumption which exclude the work
decision from consideration.
6 The same intuitive model is suggested by Milton
Iriedman, p. 206, and is applied by Robert Lucas and
Leonard Rapping in their analysis of the Phillips curve,
p. 266.
7 This definition of complementarity, used by F. Y.
Edgeworth, Irving Fisher, and Vilfredo Pareto, differs
from the more conventional definition which refers to
the sign of substitution effects resulting from the effect
of a price change of one good on the consumption of
another good. As Paul Samuelson, p. 183, notes, this
"direct" definition depends on a cardinal specification
of the utility function. Since we follow ModiglianiBrumberg,
Yaari, and Tobin in assuming additively
separable intertemporal preferences, we have in fact
assumed a cardinal specification for the utility function
at each age.
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190 THE AMERICAN ECONOMIC REVIEW MARCH 1974
II
We assume that the consumer has a
strictly concave twice differentiable utility
function U(L(t), X(t)) which is the same at
each age (t). X(t) is his consumption of market
goods, and L(t) is his consumption of
leisure. Two commodities are employed as a
simplifying device. Invoking the composite
commodity, theorem, the argument remains
valid if there are many goods whose relative
prices do not change with age, and if we introduce
the many,uses of leisure discussed by
Gary Becker. Continuous time is used to
facilitate the derivations. All of our results
remain valid if time is segmented into discrete
periods.
Allowing for time preference at rate p, a
consumer with horizon T has a lifetime utility
function
rT
(l) J ~e-PtU(L(t),X(t))dt
At each age, he has a fixed amount of time
M available so that if he consumes L(t)
units of leisure, his work time is M-L(t).
The price of goods at age t is defined to be
P(t) while the price of time at age t is WT(t),
and his money earnings at each age are
W(t)(M-L(t)) 8
Letting r be the rate of interest, A(O)
initial assets, and assuming that no constraints
are imposed on borrowing or lending
except that all loans must be repaid, the consumer's
lifetime budget constraint in the absence
of bequests is
rT
(2) A(O)+ f e-rt[W(t)(M - L(T))
- P(t)X(t)]dt = 0
The consumer is assumed to maximize (1)
subject to (2). Letting Ui be the partial derivative
of U with respect to its ith argument,
the necessary conditions for an interior
maximum are
(3) U1(t)- Xe(P-r)IW(t) = 0
(4) U2(t) - Xe(P-r)tP(t) = 0
and (2), where X is the Lagrange multiplier
associated with constraint (2) and is positive
if this constraint is effective. From the strict
concavitv of U( ), these conditions are also
sufficient for a maximum. (See G. Hadley
and Murray Kemp, p. 228.)
One inmmediate implication of the model is
that the discounted lifetime marginal propensity
to consume goods out of initial net
worth need not be unity. Thus it is not necessary
to introduce bequests to achieve this
result as Yaari has done.9 To see this, differentiate
equation (2) with respect to A(0) to
obtain
T aX(t)
1=1 e-rtP(t)_- dt
J0 ~~AA(0)
rT &L(t)
+ I e-rt(t)---- dt
The first term on the right is the discounted
lifetime marginal propensity to consume
goods out of initial net worth, and it need
not be unity as long as aL(t) aA(0) X 0.
It is useful at this point to collect wellknown
results about strictly concave functions
which are needed below. Letting U1j be
the second partial derivative of U( ) with respect
to its ith andjth arguments, strict concavity
implies
(5) iUl U12 > U11<?, U22<0
while U12 is indeterminate in sign. U12= U21
from the twice differentiability of U( ).
From the strict concavity of U( ), we may
solve for L(t) and X(t) as functions of
Xe(p-r)tP(t), and Xe(p-r)ItJV(t):
(6) L(t) = L[Xe(P-r)tWV(t),Xe(P-r)tlP(t)]
I It is possible to introduce human capital accumulation
so that more work at one age raises future wages.
In this case, the price of time includes the conventional
money wage rate and the effect of an extra unit of
work effort on discounted future earnings. These considerations
complicate, but do not alter, the essential
arguments of this paper. A retirement period may also
be introduced without altering any of the essential conclusioins.
For a more complete treatment of these
issues, see my dissertation, Essay I.
9 Ralph Pfouts noted the same point in a one period
model.
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VOL. 64 NO. 1 HECKMAN: LIFE CYCLE CONSUMPTION 191
(7) X(t) = X[Xe(P-r)tW(t), Xe(P-r) tp(t)]
Letting Li and Xi indicate partial derivatives
with respect to the ith argument of L
and X, respectively,
(8) L1 < 0, X2 < 0
while L2 and Xi are equal and ambiguous as
to sign.
To prove these propositions about the partial
derivatives, substitute equations (6) and
(7) into equations (3) and (4), and differentiate
with respect to the arguments of equations
(6) and (7). This manipulation leads to
the matrix equation
r711/U12 LI L21 1
LuT21UT22JLXi X2 1
Multiplying by the inverse of the first matrix
on the left which is guaranteed to exist by the
strict concavity of U, we reach
[L 2 1 U[ 22 - LU12
Xi X2J D L-U12 U,,
where
D Ull U12 > 0 by condition (5).
U12 U22
It is immediately apparent that L2= X1, and
that if, and only if, U12=0 (X1=L2=0) will
the path of consumption be independent of
the wage pattern.
To facilitate the analysis, it is convenient
to characterize the wage and price paths by
(9) Xe(P-r)tIW(t) = [IV(0) + bm(t)Ieb(p-r)t
(10) Xe(P-r)tp(t) = XP(O)eb (p-r) t
defining m(O)= 0. We assume that the price
of market goods (P(t)) does not change with
age, but we allow for wage growth by introducing
the term m(t).
A consumer stationary state is defined as
the case where p= r, and b=0. In this state,
it is obvious from equations (6) and (7) that
the individual consumes the same leisure
(L(O)) and market goods (X(O)) at all stages
of his life cycle.
Suppose we disturb this state by imposing
wage growth keeping p equal to r. Since we
seek to characterize consumer life cycle profiles,
it is convenient to normalize all values
of leisure and consumption with respect to
their initial values L(0) and X(0), respec-
tively.
The effect of this displacement on normalized
demand may be written as
(L(t)\
_\L(o)/ ab
b=O
ax
iLi(t) R(0)- + Xm(t)
L(0) ab
22W+ [ L2(t)P(O) -]
L(t) Fax axi
--- L1(0)W(0)- + L2(0)P(0) - I
L2(0) Lab abj
a(X(t))
ab b=O
ax
(0)<X1(t) L - + XM(t)
r ~~ax-~
+LX2(t)P(O)
ab-J
X(t) r ax+ X2(0)P(0)ax1
x2(0)L ab abThe
partial derivatives are evaluated in the
neighborhood of b=O, the stationary state
position. Since at the stationarv state L(t)
= L(0), Li(t) = Li(O), X(t) = X(0), and Xj(t)
= Xj(0), these expressions may be simplified
to
aL(+) L1(t)
L- =x m(t)
ab b=O L(0)
IX(t)\
\X(X(O) Xl(t)
____=x
- m(t)
ab b=O X(0)
Since L1<0, the age with the highest wage
rate (i.e., the age with the largest m(t)) is the
age of minimal consumption of leisure and
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192 THE AMERICAN ECONOMIC REVIEW MARCH 1974
hence the age of maximal work and earn-
ings.10
The sign on X1 is indeterminate. If goods
and leisure are direct complements (U12>0),
the age for peak wage rates is the age for
minimum goods consumption. If time and
goods are substitutes (U12 <0), the age of
peak wages and earnings is also the age of
peak goods consumption. Only if leisure and
goods are independent in utility (U12=0)
would goods consumption remain the same
at all ages when wage rates differ by age, and
in this case the predictions of standard life
cycle consumption models remain intact.
Simple as this model is, it can account for
the observed relationship between earnings
and consumption if goods and time are direct
substitutes in utility (U12<0). Recent empirical
work suggests that male hourly wage
rates rise to a peak in the age range 45-54
and fall off afterward.1"
In this simplified model, the only force
causing differences in the consumption of
leisure and goods by age is the pattern of
wage growth. The introduction of a difference
between the interest rate and the rate of
time preference sets other forces in motion.
To fix ideas, suppose wage rates rise monotonicallywith
age. If the interest rate exceeds
the rate of time preference, the consumer has
an incentive to consume more goods and
leisure at older ages.12But since wage rates
increase monotonically with age, the consumer
has an incentive to consume less
leisure as he ages, and, if goods are complements
with leisure, fewer goods with advancing
age. In this tug of war, any result can
emerge, and in particular it is possible that
hours of work and the consumption of goods
will reach a peak at an interior age in the life
cycle.
To see this more clearly, we again differentiate
the normalized demand functions
with respect to b in a neighborhood of the
stationarv state. WVenow let p and r be unequal.
After some manipulation we reach
(L(t)
L(__ XL1(O)
(11) -n(t)
( Lb() b=O L(O)
X(P -r)t
+ Xp-L(O)t[L1W(o) + L,P(O)]
L(O)
(
rt))
_____
XX1(O)
(12) - m(t)
-ab b=O X (O)
X(P- r)t
+ X(p [X1W(o)
+ X2P(O)]
X(O)
The term in brackets in each expression
must be negative if X(t) and L(t) are normal
goods.'3 From equation (11), it is seen that
10 In order to cast the theory into observable phenomena,
we must replace differentials with differences
from the initial stationary state values. Equivalently,
we apply the mean value theorem to L(t)/L(O) in a
neighborhood of the stationary state path. Thus
(d/db)(L(t)/L(O))db becomes A(L(t)/L(O)).
1' Hurd finds a peak for the wage rates of white males
in the age range 45-54 using the one in a thousand
Census data for 1959, and the Survey of Economic Opportunity
(SEO) data for 1966. See his Table 3, p. 194.
Using the SEO data, we regressed hourly wage rates on
schooling, age, and age squared, allowing for interactions
between schooling and age. We found a peak for
hourly wage rates at ages 48-50 for males with 10-12
years of schooling with an approximate standard deviation
of four years. Thurow reports a similar age peak for
income.
12 See H. G. Lewis.
1"To see this, differentiate equations (6) and (7) with
respect to A (0) to obtain
A(t)= e(P-r)t[L,W(t) + L2P(t)] aA(O)
aX(t)=
e(P-r)t[XIW(t) + X2P(t) ] ax
aA(0) aA(O)
Since 9X/OAA(O)<O,and the assumption of normality
implies that the partial derivatives on the left must be
positive, the expressions in brackets must be negative.
To see why 8X/8A (0) <0, we note that
AA(O)[JTe-PtU(L(t), X(t)dt] = X
Then
ax X T aL(t) aX(t)1 Ul1 U12 1
aA(0) = 3 A(0) AA(0) UL12 U2 i
L AL(t) 1
aA(0) dt < 0
ax(t)
AA(0) i
since the strict concavity of U implies that the quadratic
form inside the integral is negative definite.
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VOL. 64 NO. 1 HECKMAN: LIFE CYCLE CONSUMPTION 193
the age of peak wage rates is not necessarily
the age of peak work effect, unless p= r.
To pursue the example of monotonic wage
growth, suppose that r>p, and that m(t) increases
with age (t). It is possible that the
age of peak work effort occurs at a boundary
of the life cvcle, i.e., 0 or T. But if a peak
comes in the interior of this interval, the age
of peak work effort comes before the age of
peak earnings since earnings are the product
of hours worked and wage rates, and the latter
increase monotonically with age by as-
sumption.
In this example, only if X1 is negative
(U]2>0) and suitably strong will we observe
a peak in the consumption of goods at an interior
age of the life cycle. If r<p, we observe
an interior peak only if Xi is positive
(U12<0).
If wage rates rise to a peak and decline
after a certain age, as Hurd's research suggests,
the analysis is only slightly more complicated.
Again, it is possible to observe
peaks for hours worked and the consumption
of goods only at the boundaries of the life
cycle. If these peaks occur at interior ages,
and r>p, the peak age for hours worked
comes before the age of peak wage rates, and
the age of peak earnings occurs between these
peak ages.'4
It is instructive to compare the ordering of
the peak ages for hours worked, consumption,
earnings, and wage rates, in a simplified
model where the interest rate equals the rate
of time preference, to a model where they
differ. In the first model, the peak age is the
same for all variables if Xl>O(U,2<0). Allowing
for a difference between the interest
rate and the rate of time preference, and assuming
r>p, the peak age for hours of work
comes before the peak age for wage rates if
X1>o.
In fact, it is observed in the Survey of
Economic Opportunity (SEO) data that the
peak age for hours of work occurs in the age
interval 39-44 (Heckman, Essay II, p. 24).
This comes before the peak age for wage
rates and earnings which in these data occurs
in the age interval 45-54. Given
Thurow's finding on the peak age for the
consumption of goods, the data appear to be
broadly consistent with a model of r>p,
X1>O(U12<0), with a peak in wage rates occurring
in the age interval 45-54.
However, this is only one possible explanation
of Thurow's facts. We have already
shown that even if wage growth is monotonic,
it is possible to have a peak in consumption
and hours of work in the middle
years of the life cycle. More exotic patterns
for wage rates, interest rates, and time preferences
can easily produce the same results,
as can changes in the pattern of goods prices
bv age.
Without a more extensive empirical analysis
of wage patterns it is impossible to be
more precise about the exact set of assumptions
about preferences necessary to explain
Thurow's results. Nevertheless, we can unequivocally
assert that Thurow's findings are
consistent with a model of perfect certainty
and perfect credit markets if consumers face
anticipated changing wage rates over their
life cycle, and are free to choose their hours
of work.
III. Conclusions
In this paper, we have shown that
Thurow's empirical results are consistent
with a neoclassical model of life cycle consumption
and labor supply. It is not necessary
to resort to credit market restrictions or
uncertaintv to explain his facts.
With this in mind, we join Nagatani in
questioning Thurow's conclusions in favor of
government intervention in the loan market
to ensure optimal consumption patterns.
Since labor supply behavior and uncertainty
can also explain Thurow's empirical results,
we must sort out the relative importance of
each of these factors in determining observed
14 To prove this, assume that wage rates, W(t), and
hours worked, h(t), are continuous functions. From
equation (11) it is clear that if r>p the peak age for
hours of work cannot occur after the peak age for wage
rates. The age of peak wage rates (t3) is implicitly defined
by W'(t3)= 0, where X denotes the derivative with
respect to t. The age of peak hours worked (t,) is
defined by h'(t1)=0. The age of peak earnings (t2) is
defined by
[w(t)Iz(t)]'= W'(t) Iz'l(t)
W(t) h(t)
Then clearly tl < t2< t3.
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194 THE AMERICANECONOMIC REVIEW MARCH 1974
consumption patterns before firmpolicy prescriptions
are possible.
Throughout this paper, we assume exogenous
wage growth, and a single earner.
The same predictions developed in this paper
emerge from a more general model with wage
growth due to human capital investment. In
such a model, the "shadow price" of time
plays the role of the market wage in this
paper, and differs from the observed market
wage because work at one age may affect
future earnings. (See Heckman, Essay I.) It
is also relatively straightforward to generalize
our results to multiple workerhouseholds,
although few new analytical insights emerge.
The resulting life cycle consumption path
depends on the wage paths of all earners in
the household.
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