Theoretical Economics Letters, 2013, 3, 18-25
http://dx.doi.org/10.4236/tel.2013.35A3003 Published Online October 2013 (http://www.scirp.org/journal/tel)
Why Are Children Impatient? Evolutionary
Selection of Preferences
Department of Economics, Meikai University, Urayasu, Japan
Received August 1, 2013; revised September 1, 2013; accepted September 10, 2013
Copyright © 2013 Junji Kageyama. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This study aims to explain why children are impatient. Using a biological framework called the life history theory, the
study investigates the evolutionary root of time preference, paying particular attention to childhood. The results show
that the biologically endowed rate of time preference is equal to the mortality rate not only in adulthood but also in
childhood, reflecting the ch ang e in the b iolog ical valu e of su rviv al. Mortality is the b aselin e fo r time pr efer en ce through
the entire course of life. These results are consistent with the findings in previous empirical and experimental studies
that the discount rate is U-shaped in age, and account for why young children, in particular, are impatient. In addition,
the difference in time preference between adults and children provides a biological explanation for the parent-offspring
conflict, in which the higher discount rate among children causes parents and their children to disagree over intertem-
poral allocation of resources in collective decision-making particularly within the household.
Keywords: Time Preference; Value of Survival; Life History Theory; Mortality; Parent-Offspring Conflict
The decision-making process within a household typi-
cally involves multiple individuals such as the wife, hus-
band and children. As a result, collective decisions made
in the household depend on preferences of household
members. For example, Dauphin et al.  showed that
children as well as parents influenc e household econo mic
Nevertheless, children, who sometimes make up the
larger portion of the family, seldom receive much atten-
tion in the study of economics. Considering that even an
infant is capable of tilting household decisions toward
his/her preferred choices using his/her limited but pow-
erful strategies, this gap needs to be addressed.
As a step toward this objective, this study investigates
the intertemporal choice of children, focusing on the
cause of their impatience. As found in Bettingera and
Slonim  and Steinberg et al. , patience increases
with age during childhood, suggesting that young chil-
dren are particularly impatien t2.
To be more specific, this study searches for the evolu-
tionary root of impatience of children, employing a bio-
logical framework. An increasing body of literature uses
this method to search for the biological basis of prefer-
The basic idea of this literature comes from the bio-
logical finding that preferences or, more broadly, geno-
types and associated phenotypes (strategies), are the end-
products of natural selection. Since preferences that are
successful in reproduction spread over the population in
the evolutionary time scale, preferences in the current
human population can be deduced from the preferences
that maximized fitness in the environment where we
evolved. Such environment is considered to be the Afri-
can savannah where our genus Homo appeared two mil-
lion years ago and stayed for most of our history as
hunter-gatherers. Human-specific characteristics are con-
sidered to have evolved in this ancestral environment.
Time preference is a leading topic in this literature.
Hansson and Stuart  showed that the marginal rate of
substitution in utility is equ al to the marginal rate of sub-
stitution in fitness, suggesting that resource allocation
maximizing utility corresponds to allocation maximizing
fitness. Later, Rogers  applied this idea to the in-
tertemporal allocation of resources, and explained the
1To justify the unitary model in which the unitary decision maker
maximizes household utility, we often require restrictive assumptions.
The rotten kid theorem (Becker ) is such an example. See Bergstrom
 for details.
2For other psychological studies, see Teuscher and Mitchell  for a
opyright © 2013 SciRes. TEL
J. KAGEYAMA 19
evolutionary origin of time preference. More recently,
technical similarities for studying aging in bio-de-
mography and the intertemporal allocation in economics
have allowed biologists, demographers and economists to
enter this hybrid field, leading to the ex amination of how
the time-discounting behavior relates to senescence (So-
zou and Seymour ) and to intergenerational transfers
(Chu et al. ), why it is hyperbolic (Robson and
Samuelson ), the rationale for social discounting
(Sozou ), how it depends on age (Kageyama ),
and how it relates to extrinsic mortality (Chowdhry
To investigate the impatience of children, the present
study follows this literature and extends Kageyama 
by incorporating childhood, i.e., the growth period to
maturity. In previous studies, Chu et al.  built a bio-
logical model incorporating the growth period and
showed that the impatience relates to the productivity
growth in childhood. Similarly, Robson et al.  ex-
amined the impatience of children, but paid more atten-
tion to adulthood and did not explicitly consider the role
of the growth period.
Despite these differences, both of these previous
studies concluded that the impatience of children does
not relate to the absolute level of mortality. This pre-
sents a sharp contrast to the result that mortality is a
major factor associated with time preference in adult-
The present study, on the other hand, finds that the
impatience of children relates to the mortality rate, as
does time preference in adulthood. The same logic ap-
plies in both childhood and adulthood.
Explaining time preference with evolutionary biology,
however, does not negate the relationship between time-
discounting behavior and non-biological factors. As
Becker and Mulligan  argue, social factors such as
culture and education affect time discounting. This paper
aims to assess the biological basis of time preference,
referred to as the “endo wed discount factor” in the abo ve
The rest of the paper is organized as follows. Section 2
examines the evolutionary optimal strategies using a
biological framework. Based on the results in Section 2,
Section 3 provides a biological explanation for time
preference, and shows that the optimal rate of time pref-
erence is equal to the mortality rate in the entire life
course. This implies that the biolog ically endowed rate of
time preference is U-shaped in age, as is the morta l it y rate,
and that children and old adults are, by nature, less pa-
tient than young adults. Section 4 concludes.
2. The Model
2.1. The Basic Structure
The model in this study is based on life history theory,
i.e., an analytical framework in biology to study spe-
cies-specific life-history strategies such as the age-tra-
jectories of fertility and mortality, presuming that life-
history traits are the end-products of natural selection.
Technically, it solves for the fitness-maximizing strate-
gies under given constraints to deduce species-specific
In particular, as in a standard life-history model, I con-
sider a model with the following properties. First, the
population is stationary at the upper limit of the carrying
capacity. This is to follow the carrying-capacity argu-
ment that human population in our evolutionary past was
confined by the environmental capacity5. Second, repro-
duction is asexual. This is simply to avoid complexities
related to matching between females and males.
With these specifications, the measure of fitness is
given by the expected number of offspring at the begin-
ning of life, which is expressed as
where l(x) is the survival probability to age x and m(x) is
the reproductive output at age x6. With the pressure of
natural selection, genotypes and associated phenotypes
that generate a higher value of R(0) spread over and fix
in the population. Note that, in a stationary population
where the p opulation growth rate converges to zero, R(0)
is equivalent to the reproductive value at birth and con-
verges to one. The lifetime exp ected number of offsp r ing is,
ex post, just sufficient to replace the current individual.
Survival and reproduction depend on age and the
amount of energetic resources respectively invested in.
Once consumed, resources are physiologically allocated
to either survival investment or reproductive investment.
Therefore, given that
is the mortality rate at age x, the depend-
ence of survival and reproduction on resources can be
where w(x) and v(x) are respec-
tively survival and reproductive investments. To avoid
unnecessary technical complexity, I assume that both
3Acharya and Balvers  examined time preference in an economic
framework, assuming that utility captures the effect of consumption on
mortality. Their model can also be interpreted as a biological model
that assumes that reproductive success solely depends on the length o
4See Stearns  for the general introduction of life history theory, and
Perrin and Sibly  for the technic a l i n t roduction.
5It is technically possible to examine the case that the population
growth rate takes non-zero constant values. See Taylor et al. .
6Equation (1) does not imply that lifespan is infinite.
Copyright © 2013 SciRes. TEL
investments exhibit diminishing marginal returns, satis-
fying , ,
ww wx x
where the sub-
script indicates a partial derivative, and also that the in-
dividual will certainly die without survival investment
and will not have any reproductive output without re-
In addition, reproduction depends on growth invest-
ment, z(x), in the earlier stages of life. Investing in
growth enhances the reproductive capacity and increases
the reproductive efficiency in the later stages.
To incorporate this relationship, I focus on determinate
growers, i.e. , organisms that stop growing at maturity, as
in the case of humans7. Thus, denoting the age of matur-
ity by α, we have for all x < α and z(x) = 0 for
Besides, I assume that
can be separated
into the age-dependent reproductive capacity, A(x), and
the contribution of resources,
, such that
is concave in v(x) as is
. In this setting, the development of the re-
productive capacity during the growth period can be ex-
where A(0) is the innate reproductive capacity and B(x) is
the age-dependent growth efficiency. Here, the contribu-
tion of resources to growth follows the same law as the
one to reproduction, and, thus, both reproductive and
growth investments share the same
Turning to the budget constraint, the model incorpo-
rates resource transfers. The extensive use of resource
transfers, such as intergenerational transfers between
parents and children and food sharing between house-
holds, is one of the most distinctive human characteris-
tics8. In this case, the budget constraint does not neces-
sarily hold at each point in time, and is given by the life-
time budget constraint,
where y(x) is the amount of resources that the individual
obtains at age x. Note that y(x) is exogenous, and that th e
interest rate is equal to zero since keeping resources does
not generate any return.
With these conditions, we can solve for the optimal
allocation of resources, expr essed by v(x), z(x), w(x), and
α, using the Lagrangian method. Given the objective
function (1), the budget constraint (5), and the Lagran-
gian multiplier, λ, the Lagrangian is defined as
where l(x), m(x), and A(x) are respectively specified by
Equations (2), (3), and (4). Using the Volterra derivative
(see Ryder and Heal ) to examine the effect of
changes in the investments around a particular age x, we
obtain the first-order conditions:
represents the benefit of an increment in the
reproductive capacity, and R(x) and k(x) express the
benefits of an increase in survival at age x respectively
for reproduction and production.
At the same time, the age of maturity is determined by
the returns of growth and reproductive investments.
These returns are given by
, which respectively represent the
increase in the future reproduction due to a greater re-
productive capacity and the increase in immediate re-
production. The individual switches from the growth
phase to the reproductive phase when the return for re-
production overtakes the one for growth. Therefore, the
age of maturity is implicitly given by
7To account for determinate growth, we can, for example, assume that
the transition from the growth phase to the reproductive phase, or the
other way around, incurs significant costs. This is consistent with the
fact that determinate growers go through some sort of metamorphosis
8See, e.g., Wiessner [21,22] and Gurven et al.  for the importance
of food sharing in modern foraging p op ulat ions .
Copyright © 2013 SciRes. TEL
J. KAGEYAMA 21
2.2. Optimal Life-History Strategies
These conditions can be interpreted as follows. From
here on, I will suppress the age notation when no confu-
First, Equation (7) indicates that the marginal produc-
tivity of reproductive investmen t,
fv, is equal to the
shadow price of resources and is constant across ages.
For example, if A depreciates after maturity,
increases and, thus, reproductive investment decreases
with age. The decline in the reproductive efficiency is
offset by the decline in reproductive investment.
Second, by rewriting Equation (8) as
Equation (15) shows that the marginal return of growth
investment conditional on survival to age x is equal to
the shadow price and is constant across ages. It is con-
ditional on survival because, while the timing of start-
ing to reap the return on growth investment is fixed by
the age of maturity that is endogenously determined, the
chance of reaching maturity changes with age. This
asymmetry makes the return of growth investment age-
dependent. As maturity approaches, the expected return,
, increases, and, to take advantage of the
higher return, the marginal productivity,
Bfz , de-
creases. For example, if B is constant across ages,
decreases and growth investment increases toward ma-
Third, by letting
Equation (9) can be rewritten as
The upper bar indicates that the value is adjusted by the
survival probability at the corresponding age. Thus,
Rx represents the expected reproductive output in the
remaining lifetime for the individual survived to age x,
kx expresses the expected productive surplus in
the remaining lifetime for the same individual. Therefore,
given that λ is the value converter of productive surplus
to reproductive contribution, the terms in the brackets in
, represents the value of
survival that includes both reproductive and productive
contributions. With resource transfers, the value of sur-
vival consists of not only reproductive contribution but
also productive cont ri b ut io n.
With these results, Equation (18) shows that the mar-
ginal benefit of survival investment is equal to the
shadow price and is constant across ages. This implies
that the marginal productivity of survival investment,
, depends on the change in the value of survival,
and reaches its lowest level at the prime of life at which
the value of survival is the highest. Assuming, for exam-
is independent of age, we can
expect that the mortality rate reaches its lowest level at
the prime of life.
An example of such a mortality curve is presented in
Figure 1. It shows that the mortality rate reaches its
lowest level in age-class two and increases thereafter,
corresponding to the result that the value of survival hits
its highest level in age-class three.
3. Implications on Intertemporal Allocation
3.1. Intertemporal Marginal Rate of Substitution
The intertemporal marginal rate of substitution (MRS)
measures the importance of resources at one point in time
over another. Specifically, it is defined as the rate to
compensate for a loss of resources at one point in ti me in
exchange for resources at another point in time while
keeping R(0) constant. In a continuous-time setting, MRS
at age x can be calculated as
Figure 1. Age-trajectory of Mortality. The maximum age-
class to which the individual can possibly survive is set to
nine, and the functional forms and parameter values are
given as follows:
, γ = 0.25, q = 0.5, B = 0.15, A(0) =
0.18, y(0) = 0, y(1) = 2, y(2) = 4, y(3) = 6, y(4) = 6, y(5) = 6, y(6)
= 6, y(7) = 6, y(8) = 4, and y(9) = 4.
Copyright © 2013 SciRes. TEL
where , , and are respectively the
marginal effects of reproductive, growth, and survival
investments on R(0). Note that, with respect to survival
investment, the change in resource surplus must be taken
These equations show that MRS depends on various
factors. Equation (19) illustrates that, in adulthood , MRS
consists of three components; the mortality rate, the
change in the reproductive efficiency, and the change in
the marginal contribution of resources. This result is
consistent with Sozou and Seymour , Chu et al. ,
and Kageyama .
In childhood, on the other hand, Equation (20) shows
that MRS consists of two components; the change in the
growth efficiency and the change in the marginal contri-
bution of resources. As in Chu et al. , the mortality
rate does not appear in MRS in childhood.
Furthermore, Equation (21 ) shows that, throughout the
life course, MRS can be calculated as the sum of three
components; the change in the marginal contribution of
resources, the change in the value of survival, and the
Despite these variations, however, the values of MRS
derived in Equations (19)-(21) are all equal to the mor-
tality rate when resources are optimally allocated. In
Equation (19), the second and the third terms cancel out
on the optimal path as indicated in Equation (7), and
MRS is reduced to the mortality rate. Similarly, as in
Equation (8), the two terms in Equation (20) sum up to
the mortality rate. Turning to Equation (21), it is also
reduced to the mortality rate since, as indicated in Equa-
tion (18), the first two terms cancel out.
This result is intuitive in the reproductive period. The
change in the reproductive efficiency is neutralized by
the change in reproductive investment on the optimal
path. However, giving up a unit of current reproductive
investment for future investment still accompanies the
mortality risk. Thus, to keep R(0) constant, the mortality
risk needs to be compensated.
By contrast, the result that MRS is equal to the morta-
lity rate during the growth period might be counter-in-
tuitive. As children do not yet reproduce, the timing of
growth investment seems irrelevan t as long as they build
up the same level of reproductive capacity at maturity.
The reason for this result is that the marginal produc-
tivity, , changes with age. As shown in Subsec-
tion 2.2, the marginal productivity decreases as the
chance of reaching maturity increases, and its changing
rate is exactly equal to the mortality rate. This is in tuitive
considering that the change in the marginal productivity
is originated by the change in the survival probability to
maturity. For example, if B is constant across ages, the
marginal productivity decreases with age to satisfy
. To give up a unit of current
growth investment for future investment, the reduction in
the marginal productivity must be compensated, and this
compensation rate is given by the mortality rate. For this
reason, MRS in childhood is, as in adulthood, equal to
the mortality rate.
3.2. Growth Process, Mortality, and MRS
The result that MRS in childhood on the optimal path is
equal to the mortality rate does not depend on the type of
growth process. For example, consider as in Chu et al.
 that the amount of resources, y(x), is determined by
body size, which in turn depends on growth investment
up to age x. I assume, for simplicity, that y(x) is equal to
body size at age x.
In this case, the growth process can be described as
yxygsx x x
where s(x) is growth investment allocated for increasing
body size and
is the contribution of the
investment at each point in time. Here, I assume that
is sufficiently concave in s(x) to have an
Incorporating this size effect into the lifetime budget con-
straint in Equation (5), we obtain the first-order conditions:
for all x where
, i.e., the re-
maining life expectancy at age x. This shows that the
marginal return of investing in body size is equal to its
cost. The return is measured by the amount of resources
that the individual is expected to accrue in the future, and
the cost is equal to one, representing a unit of resource
necessary for the investment.
Equation (23) further illustrates that the change in the
s, depends on the change in
the remaining life expectancy. While the remaining life
expectancy is high, the marginal produ ctivity is low so as
to take advantage of the higher expected return. There-
fore, if the growth efficiency is independent of age,
s increases and s decreases with age10. This is con-
sistent with Chu et al. .
The next step is to calculate MRS. By applying the
same method as before, it is given by
9We can also assume that y(x) depends on knowledge learned in the
ast and interpret
as the increment of knowledge. See
g  for how knowledge affects fitness.
10The results that the two types of growth investment, s and z, change
in the opposite direction suggests that the age-trajectory of whole
growth investment is indeterminate.
Copyright © 2013 SciRes. TEL
J. KAGEYAMA 23
The first term represents the change in the marginal pro-
ductivity, the second term expresses the change in the
remaining life expectancy, and the third term is the mor-
tality rate. As indicated in Equation (23), the first two
terms cancel out on the optimal path. As a result, MRS in
childhood is again given by the mortality rate.
3.3. Time Preference
While the intratemporal allocation of resources is invol-
untary depending mostly on physiology, the intertempo-
ral allocation is voluntary. It primarily depends on be-
havior since we can deliberately control the allocation by
deciding how much to consume at present.
This implies that, to further analyze the intertemporal
allocation, we need to move on to the economic frame-
work that allows us to study voluntary behavior. We
need to remember that humans do not intend to maximize
fitness but rather behav e in a utility-maximizing manner,
and that those who had preferences that coincided with
the fitness-maximizing behavior spread over and fixed in
To do this, I employ a standard economic model in
which the objectiv e function is given by lifetime utility,
is the discount factor for future utility,
is a strictly concave instantaneous utility
function, and is consumption, which can be re-
garded as the sum of growth, reproductive, and survival
investments. Note that the lifetime budget constraint
presented in Equation (5) is still relevant.
Here, the instantaneous utility directly d epends on age.
Given that preferences were shaped by natural selection,
the instantaneous utility should reflect the effect of con-
sumption on reproductive success at the corresponding
age. To incorporate this aspect into the model, I assume
that the marginal utility on the fitness-maximizing con-
sumption path is constant across ag es. This type of utility
induces the individual to consume more while its mar-
ginal effect on fitness is high.
In this setting, MRS in utility is given by
The individual applies this rate to evaluate future con-
sumption and determines the intertemporal allocation.
Thus, for the voluntary allocation to match the fitness-
maximizing allocation, this rate must be consistent with
MRS in fitness obtained in Subsection 3.1, and must be
equal to the mortality rate when it is evaluated on the
fitness maximizing consumption path. In particular, when
the instantaneous utility depends on age as described
is equal to zero on the fitness-
maximizing path, and, thus,
, i.e., the pure
rate of time preference, is equal to the mortality rate11.
These results further imply that, considering that time
preference is psychologically embedded, the endowed
rate of time preference is equal to the mortality rate in
our evolutionary past when we existed as hunter-gather-
ers. As a result, presuming that the age-trajectory of
mortality in our evolutionary past is similar to the one in
modern hunter-gatherer populations, we can predict that
the endowed rate of time preference is U-shaped in age,
reaching its lowest level in early adulthood, as is the age-
trajectory of the mortality rate in modern hunter-gatherer
populations (e.g., Hill and Hurtado ; Hill et al. ).
This is consistent with empirical findings. As dis-
cussed in Section 1, the discount rate decreases in child-
hood. In adulthood, on th e other hand, while still empiri-
cally inconclusive, Trostel and Taylor  and Read and
Read  found that the discount rate increases with
senescence12. The present study focuses on the biological
discount rate ignoring the effect of social factors such as
education, and may overestimate the discount rate in
adulthood in modern human populations. Nevertheless,
the age-trajectory of time preference predicted in the
present analysis is still in line with the empirical find-
4. Concluding Remarks
This paper aims to understand the mechanism that coor-
dinates intertemporal choice, paying particular attention
to the impatience of children. To do this, it examines
human life-history strategies, incorporating the growth
period to maturity. The results show that the endowed
rate of time preference is equal to the mortality rate in th e
entire life course and, thus, is U-sh ap ed in ag e, ind ica ting
that it is higher for children than for young adults.
At the behavioral level, this provides a biological ex-
planation as to why parents and children often have con-
flicts in the allocation of resources when they make col-
lective decisions. Due to the difference in the discount
rate, parents, who follow their own time preference,
would allocate less to the present than their children
would. Namely, the difference in time preference gener-
11If we correctly perceive the survival probability and discount future
utilities accordingly, the pure rate of ti me preference w ould be equal to
zero. However, there is no biological basis to suppose that we are able
to correctly perceive the survival probability. Given that time dis-
counting is not limited to humans, it makes more sense to consider that
time preference is psyc h olo gica lly embedded.
12These studies, however, differ in the age at which the discount rate
reaches the lowest level. While Trostel and Taylor  found that the
discount rate is lowest among individuals in their twenties, Read and
Read  argued that it is lowest in their forties. Thus, the age at
which time preference reaches its lowest level is still an open question.
Copyright © 2013 SciRes. TEL
ates parent-offspring conflict over the intertemporal al-
location of resources, affecting the decision-making
process particularly within the household13.
One limitation of this study, however, is that it treats
the growth period as one state and compresses the re-
productive capacity into one parameter. This is equiva-
lent to assuming that a unit of growth investment is ex-
changeable between any points in time withi n the growth
period although such a transaction may incur additional
In reality, it is often not the case. A deficiency in a
particular type of growth investment at a certain age, for
example, may not be compensated by the same kind of
investment at a later age, and may have a permanent ef-
fect. In such cases, intertemporal choice would not be as
simple as the present analysis suggests, and the impa-
tience of children, in particular, would be underesti-
Nevertheless, as the first-order approximation, this
study provides an explanation for the age-trajectory of
time preference, connecting it to mortality in both child-
hood and adulthood. The future is discounted because
survival is uncertain (Yaari ), and survival is uncer-
tain because the future is discounted (Kirkwood ;
Kirkwood and Rose ). Both propositions hold in the
entire life course because time discounting and survival
uncertainty are two sides of the same coin, reflecting the
change in the value of survival. There are other factors,
including the variability of the environment, sexual re-
production and genetic relatedness, that can possibly
affect time preference, but mortality is the baseline for
time preference through the entire course of life.
This study also suggests that the traditional view that
time discounting is somethin g unfavorable is not entirely
accurate. Time discounting generally carries negative
connotations and has been described as cognitive defi-
ciency, impatience, shortsightedness, myopia, irrational-
ity, and so forth. For example, Ramsey  regarded
time-discounting behavior as “a practice which is ethi-
cally indefensible and arises merely from the weakness
of the imagination (p. 543)”. However, as discussed in
this paper, being impatient is not a deficiency, but an
optimal trait in the biological sense that was acquired in
the course of evolution. Our surrounding environment
has changed since our evolutionary past, and the en-
dowed rate of time preference is no longer optimal in the
economic sense. Nevertheless, being too patient is not
necessarily favorable either as it would cause other prob-
lems such as dynamic inefficiency in macroeconomics
and the postponement of reproduction that, together with
the decline in reproductive efficiency, results in below-
replacement fertility. Given our human nature, an appro-
priate level of impatience may in fact be beneficial to our
I thank Akihiko Matsui and Takashi Shimizu for their
helpful suggestions in developing ideas for this paper,
and Kazuhiko Kato for his helpful comments on the ear-
lier draft of this paper. Any remaining errors are my own.
This study is supported by Grant-in-Aid for Challenging
Exploratory Research from the Ministry of Education,
Culture, Sports, Science and Technology in Japan (No.
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