Finance a úv r-Czech Journal of Economics and Finance, 59, 2009, no. 1 71
JEL: D9, I2, O1
Keywords: growth, human capital, education, time discounting, poverty
Time Discounting, Education, and Growth:
Evidence and a Simple Model
Michal BAUER – Institute of Economic Studies, Charles University (bauerm@fsv.cuni.cz)
Julie CHYTILOVÁ – Institute of Economic Studies, Charles University (chytilova@fsv.cuni.cz)*
Abstract
In the paper we propose a new channel through which education may promote economic
growth: education makes people more patient, and more patient people are more likely to
save, invest, or send children to school. The paper first summarizes the results of field studies
which we conducted in two very different poor countries: Uganda and India. In both
countries the findings are consistent with the causal effect of education on the subjective
discount rate. In the second part of the paper, we build a simple two-period human-capital-driven
growth model where the subjective discount rate depends on the level of human
capital. This new assumption gives rise to the possibility of multiple development regimes
and the model illustrates a wider role of education in tackling possible development traps.
1. Introduction
In recent decades, much of the attention of economists has been directed at
understanding the determinants of economic growth. The results of many cross-country
empirical studies underline the role of education, and it has been shown that
the growth rate of GDP per capita and the education level are highly correlated
across countries (Benhabib and Spiegel, 1994; Barro and Sala-i-Martin, 2004). However,
the correlation between education and growth appears to be too high to be fully
reconciled with micro-level estimates of the effects of education on individual productivity
(Bils and Klenow, 2000; Banerjee and Duflo, 2005). Bils and Klenow (2000)
calibrated a simple neoclassical growth model where the impact of education on individual
productivity was consistent with the average coefficients of Mincerian private
returns to education that are commonly estimated in the labor literature. According to
their results, the impact of education on productivity explains less than one-third of
the high correlation between education and growth in the aggregate cross-country re-
lationship.1
Similarly, World Bank (2006) acknowledges that “the education impact
on economic growth is well-established; precisely how this happens is less well-under-
stood”.
*
We would like to thank O. Bandiera, R. Filer, I. Gang, W. Greene, J. Hlavá ek, O. Knot, M. Mejst ík,
J. Morduch, D. Munich, A. Ortmann, D. Ray, A. Schotter, M. Sko epa, P. Streblov, anonymous referees,
and participants at various conferences and seminars for valuable comments and discussions about
the empirical part of the paper. We thank BPKS, UCDT, and Caritas Prague for collaboration on the field
work and J. Kabatová, D. Mascarenhas, S. Crasta, and L. Perreira for excellent research assistance. We
appreciate the comments of T. Cahlík on the model. We gratefully acknowledge financial support from
the IES Research Institutional Framework 2005–2010 (MSMT 0021620841), the CERGE-EI Foundation
under a program of the Global Development Network, the Fulbright Foundation, and the Charles University
Mobility Fund. All errors are the authors’ own.
1
The explanation of Bils and Klenow (2000) relies on the reverse causality, i.e., that a high expected
growth rate is reflected in higher returns to education and thus leads to increased demand for education.
72 Finance a úv r-Czech Journal of Economics and Finance, 59, 2009, no. 1
In addition to the role of human capital as a factor in the aggregate production
function, there might be other effects that contribute to economic growth. For example,
education may stimulate growth by inducing a lower fertility rate (Barro, 2001).
Psacharopoulos (1994) summarizes other, less measurable channels such as the effect
of education on transaction costs, quality of democracy or health.
This paper focuses on an additional pathway between education and growth:
education may lower the individual subjective discount rate. Time discounting may
be a fundamental factor for explaining a number of puzzling phenomena behind dismal
economic performance, particularly in countries with less developed economic
institutions. If people are impatient, they may be reluctant to form savings or invest
in profitable opportunities. They may also be less inclined to educate themselves and
their children, since education may be understood as a long-term investment in future
income.
The paper first discusses our recent empirical studies from two completely
different developing country environments – Uganda and India. We show that the results
are consistent with the proposition that education lowers the discount rate. We
then build a human capital growth model with endogenous time discounting which
illustrates the wider role of education in the process of development. The economic
dynamics, the possibility of multiple development regimes, and the policy implications
are discussed.
2. Education and Time Discounting: Empirical Evidence
2.1 Literature Review
Many psychological studies from developed countries (e.g. Metcalfe and Mischel,
1999) argue that humans are born impatient, as patience can be regarded as
a component of a broader set of non-cognitive skills. The abilities to imagine future,
to structure problems, and to plan scenarios help people to become more future-oriented
(Becker and Mulligan, 1997). Learning to be future-oriented and to choose
actions whose reward is postponed in time is essential part of our upbringing and
educational process. Without such learning, people would live solely in the present
and would forego the pleasures that future events can bring.
In a major study that elicits the discount rate for a representative sample in
a developed country, Harrison, Lau, and Williams (2002) showed on a sample of
Danish households that highly educated adults have subjective discount rates as low
as two-thirds of those less educated. In the least developed countries, the difference is
likely to be even more profound, as there may be additional pathways through which
education makes people more patient. The evidence confirms that education enhances
life expectancy through putting a higher emphasis on prevention and healthier
lifestyle (Mirowski and Ross, 1998). Social norms embedded in traditional cultures
may also direct individuals to focus on present gratification and rely on a “higher
will” with respect to the future (Harrison and Huntington, 2000). Education typically
reduces the role of traditional beliefs in an individual’s decision-making.
The emerging evidence from field experiments in developing countries also
shows a negative correlation between the subjective discount rate as a measure of patience
and education. Kirby et al. (2002) studied the discount rate in remote villages
in the Bolivian rain forest. They found a high average discount rate and that a big
Finance a úv r-Czech Journal of Economics and Finance, 59, 2009, no. 1 73
proportion of its variance can be explained by years of schooling, parental education,
and measures of qualitative education achievement in terms of mathematics and language
proficiency. In Vietnam, Anderson et al. (2005) measured the stability of subjective
discount rates when the delay varies. They also refer to the importance of
education, although they do not specifically measure it. On the other hand, Pender
(1996) in Indian villages finds a correlation between the discount rate and recent income
and credit constraints, but not with education.
In addition to the empirical evidence summarized above, our proposition about
the relationship between the discount rate and education builds on the results of two
surveys we conducted in Uganda and India in 2005 and 2007, respectively. The next
two subsections describe the methodology, the sample, and the results of these sur-
veys.
2.2 Evidence from Uganda
The study from Uganda (Bauer and Chytilová, forthcoming) is based on data
from a questionnaire survey that was conducted in November 2005 in ten village
areas in the southern part of the country. The data were collected in cooperation with
UCDT, a Czech-Ugandan NGO. We used a combination of the random walk method
and quota sampling in our sampling procedure. The target population was restricted
to literate individuals above 15 years. The questionnaires were bilingual – in English
and Luganda – to allow us to approach less educated people who speak only Lu-
ganda.2
Our sample size is 856 respondents.
The area is primarily rural with an economy based on small-scale subsistence
farming and local market trade. People usually grow maize, vegetables, plantain bananas,
cassava, vanilla, and coffee. The vast majority of households in this region are
very poor by any standards, with an average per capita income of less than 300 USD
a year and a life expectancy at birth of only 46 years, largely due to the effects of
malaria and HIV/AIDS.
We used the so-called choice task method with hypothetical questions to elicit
the discount rate. Individuals were asked to express their preference between receiving
a smaller reward immediately and receiving a larger reward with some delay.
Starting with the question, “Would you prefer to receive 200,000 USh today or
250,000 USh in one year?”3
we posed five questions, each time increasing the amount
offered in the future. As we increased the future amount, we expected more individuals
to select the future option. The point at which a respondent switched from
the current amount to the future amount provides the range for her discount rate, and
the midpoint of the range is our best estimate of the person’s discount rate. We also
collected information on individual characteristics, specifically regarding education
level, age, marital status, clan linkage, and profession. The level of education was
measured by the number of school classes completed by the respondent.
We identified a close association between education and the discount rate,
which suggests that more educated people make more future-oriented choices.
The higher is the level of education completed, the lower is the discount rate. This re-
2
The official language in Uganda is English. The different local languages reflect the tribal divisions
within Uganda. Luganda is the language of the Baganda people – the largest tribe in Uganda.
3
In November 2005, the exchange rate was 1,830 USh to the U.S. dollar.
74 Finance a úv r-Czech Journal of Economics and Finance, 59, 2009, no. 1
sult also holds if we control for cohort effects and all other observable characteristics
(Table 1). This correlation, however, does not help us determine the direction of
the causal relationship between these two factors. It is also possible that people with
a lower – perhaps innate – discount rate are more willing to invest in schooling,
which has delayed payoffs, and study longer. Alternatively, there could be an omitted
variable, such as intelligence, which would account for both the discount rate and
the differences in education level.
To our knowledge, as yet there has been no empirical study of the discount
rate that has tested the causal direction in the observed correlation between education
and the discount rate. In order to do so, we exploit two independent exogenous sources
of variation in schooling: across villages and over time. The first instrumental variable
is based on differential access to secondary schools in different villages. The second
type is based on the number of individual school-age years that overlap with
the rule of Idi Amin (1971–1979) – a period of overall instability characterized by
a drastic decline in the quality of education. For reasons described elsewhere (Bauer
and Chytilová, forthcoming) both identification strategies are more suitable for men
than for women. For men, both instrumental variables strongly predict individual
education in the expected direction and the 2SLS estimates demonstrate a significant
effect of education on men’s discount rate (Table 2). The two types of instrumental
variables deliver estimates of similar magnitude. The results are robust to the inclusion
of a set of dummies for cohort effects and individual characteristics such as
profession and family status.4
2.3 Evidence from India
The study from India (Bauer and Chytilova, 2008, 2009) is based on data from
a series of lab experiments in the field and a questionnaire survey organized in June
2007 in cooperation with the Indian NGO, BPKS, in 18 villages in the south-western
Indian state of Karnataka. In each village, 35 people older than 15 years
were selected using the random walk method. This study was organized in such a way
that illiterate respondents could also participate; their proportion in the sample is almost
40 %. Our sample size is 540 respondents. Although the selection strategy was
not intended to generate a sample representative of the entire rural Karnataka populaTable
1 Education and the Discount Rate (Uganda) – OLS
Dependent variable Discount rate
(1) (2) (3)
all male female
Education (years) -0.170 -0.168 -0.171
(0.048)*** (0.066)** (0.035)***
Observable characteristics yes yes yes
Observations 856 465 391
R-squared 0.06 0.06 0.11
Notes: Standard errors corrected for clustering at village level.
** significant at 5%, *** significant at 1%.
4
The discount rate elicited is, of course, a noisy measure of patience. It is therefore not surprising that
the proportion of unexplained variance in the OLS regressions is quite high, similarly as in other studies
related to this topic. In any case, a low R-squared is not an indicator of model misspecification.
Finance a úv r-Czech Journal of Economics and Finance, 59, 2009, no. 1 75
tion, it captures most of its variety, and the averages for the socio-economic characteristics
measured are not statistically different from the averages for the whole of
Karnataka.
To elicit the discount rate we again used the choice task method, this time
with real incentives build into the experiment. The respondents were asked to choose
between receiving a smaller monetary amount earlier in time or a larger amount with
a three-month delay – for example: “Would you prefer 250 Rs. tomorrow or 300 Rs.
three months later?” 5
We posed five such questions to each individual, each question
increasing the future amount while keeping the earlier amount constant. The same
series of binary choices were made in a further time frame: “Would you prefer 250 Rs.
in one year’s time or 300 Rs. in one year and three months?” The time frame was shifted
by exactly one year to avoid the possibility of confounding factors due to the seasonality
of agricultural incomes or the regularity of local celebrations. At the end of
each experimental session, 20 percent of the respondents, selected randomly, were
paid according to one of their choices. The future payments were guaranteed by cash
certificates signed by the chief of the NGO, a local leader, and a social worker familiar
to the community.
In line with the previous findings of others and our findings from Uganda,
the results suggest a negative correlation between these two variables. Table 3 provides
the results of OLS regressions, which show that more educated men are significantly
more patient.6
For women, we also find a negative correlation with respect
to education, although it is not statistically significant. This finding may be due to lowTable
2 The Effect of Education on the Discount Rate for Men (Uganda) – 2SLS
IV: Number of secondary
schools
IV: Number of school-age
years during Amin era
(1971–1979)
(1) (2) (3) (4)
Education (years) -0.600 -0.948 -0.636 -1.020
(0.234)** (0.320)*** (0.288)** (0.443)**
Number of SS 4.949 3.125
(0.728)*** (0.759)***
At PS age during war (years) -0.301 -0.171
(0.111)** (0.056)**
Cohort effects yes yes yes yes
Observable characteristics no yes no yes
Underidentification test (p-value) 0.070 0.069 0.053 0.026
Weak identification test (F stat.) 46.273 16.933 7.365 9.311
Notes: * significant at 10 %, ** significant at 5 %, *** significant at 1 %. Standard errors corrected for clustering
at village level. The lower panel reports results from the first-stage regression, where the dependent variable
is years of schooling. The upper panel shows the results for the second stage, where the dependent
variable is the discount rate. In columns 1 and 2, the instrumental variable is the number of
secondary schools per thousand inhabitants in the particular village area when an individual was at
the age of 15 years. In columns 3 and 4, the instrumental variable is the number of school-age years
that overlap with the era of Idi Amin (1971–1979). The underidentification test statistic is the Kleibergen-Paap
rk LM statistic. The weak identification test statistic is the Kleibergen-Paap rk Wald F sta-
tistic.
5
In July 2007, the exchange rate was 1 USD = 40.2 Indian Rupees. In the area of our study, 250 Rs. is approximately
a week’s wage.
76 Finance a úv r-Czech Journal of Economics and Finance, 59, 2009, no. 1
er variance in women’s education, as 45.1 % of the women in our sample are illiterate,
compared to 34.2 % of men. The results hold for both measures of discount
rates that were elicited. For the Indian dataset, we did not have suitable data to construct
a valid instrumental variable and therefore we cannot run 2SLS regression to
test the causal effect.
3. Model
3.1 New Feature of the Model: An Endogenous Discount Rate
This paper aims to integrate these empirical findings into a human-capital-driven,
overlapping-generations growth model. We understand human capital to mean
skills and abilities that increase the productivity of an individual. These skills and
abilities can be inherited from parents and transmitted from parents to children during
the process of upbringing and education. We will abandon the usual assumption
of growth models about a constant, exogenously given, subjective discount rate.
A key component of our model is the assumption that the subjective discount rate is
high for low levels of human capital and decreases with own and parental education.
Our model theoretically illustrates how this relationship may generate a stable low-human-capital
equilibrium (poverty trap), where people have a strong preference for
current consumption and do not save and invest, and hence the economy remains
poor. As is common in the related literature (e.g. in the survey article of Azariadis
and Stachurski, 2004, on poverty traps), we define a poverty trap as being a self-reinforcing
mechanism which causes poverty to persist. It is a result of individually
optimal choices, but at the same time it depends on the initial conditions of the endogenous
variables, which can shape the long-run outcomes and prevent sustained
growth.
There are a number of poverty trap models based on a varying propensity to
save (for a survey, see Azariadis and Stachurski, 2004). It has typically been assumed
that poor people cannot save due to their low income. Recently, this argument has
6
Table 1 and Table 3 both show a significant negative association between education and the discount
rate. However, note that the magnitude of the particular coefficients is not directly comparable. In
the Ugandan study, the dependent variable is the one-year discount rate, whereas in the Indian study it is
the three-month discount rate. The set of observable characteristics that we control for differs slightly as
well.
Table 3 Education and the Discount Rate (India) – OLS
Current discount rate Future discount rateDependent
variable all male female all male female
Education -0.013 -0.018 -0.007 -0.015 -0.022 -0.007
(0.003)*** (0.004)*** (0.005) (0.003)*** (0.003)*** (0.005)
Observable
characteristics
yes yes yes yes yes yes
Observations 538 272 266 538 272 266
R-squared 0.09 0.15 0.03 0.11 0.17 0.04
Note: *** significant at 1 %. Standard errors corrected for clustering at village level. For columns 1, 2, and 3,
the dependent variable is the “current discount rate” calculated from the binary choices between
the amount the next day and the amount after three months. For columns 4, 5, and 6, the dependent
variable is the “future discount rate” calculated from the binary choices between the amount after one
year and the amount after one year and three months.
Finance a úv r-Czech Journal of Economics and Finance, 59, 2009, no. 1 77
been further elaborated in models which introduce endogeneization of life expectancy
by allowing the probability of survival to depend on the level of development of
the economy itself (Haaparta and Puhakka, 2004; Chakraborty and Das, 2005). It is
argued that individuals (or governments) in poor countries cannot afford to invest in
their patience by investing in health, and a shorter lifespan consequently distorts savings.
The complementarity between the level of development and health may produce
multiple equilibria.
We believe the following model differs from the existing ones in the sense
that a high discount rate does not necessarily have to be related to an income constraint.
Although we are still far from a full understanding of patience formation in
poor countries, from the available evidence it seems that being future-oriented can be
learned at school and transmitted from parents. We construct a model where the discount
rate is treated as a characteristic rooted in individual human capital, which encompasses
the level of education and parental influence.7
3.2 Human Capital and Growth: Standard Model
We begin with a simple human-capital-driven model with overlapping generations.
The agents are homogenous and the model abstracts from the potential role of
human capital spillover effects at the country level. The individuals live for two periods.
We distinguish two types of individuals: “young” and “old”. In our model,
parents exhibit the standard type of perfect altruism as in mainstream dynastic models.
Parents evaluate their children’s life prospects from the standpoint of their own
preferences and induce them to make the schooling choices they regard as desirable.
An individual is young in period 1 and old in period 2. She maximizes her expected
lifetime utility over consumption in the two periods. We assume that individuals
have the following utility function:
1 2 1 2
1
, , ln ln
1
U C C C C (1)
where 1C denotes consumption when the individual is young, 2C is consumption during
old age, and is the subjective discount rate. The utility function is increasing
and concave in consumption and Inada conditions hold.
The young generation invests in education and produces and consumes the output.
The old generation acts only as a producer and consumer. An agent combines
her initial human capital with labor to produce output ty :
1t
t t ty Ah l
where 0A is a productivity parameter. In the first period of life, agents divide
their time between schooling and labor supply. The total time available to an agent is
normalized to one and is spent only on working and schooling. The time spent on
7
In line with the usual literature on economic growth we do assume that less educated individuals are able
to maximize utility over time; only the weight they assign to future utility varies. Admittedly, some of
the psychological arguments for why education makes an individual more future-oriented are also compatible
with the notion of bounded rationality of less educated people. An analysis of bounded rationality
could be an interesting extension of this issue. We thank T. Cahlík for this point.
78 Finance a úv r-Czech Journal of Economics and Finance, 59, 2009, no. 1
education 1s is therefore 11 l , where 1l is the time spent working. We can rewrite
the first-period production function as follows:
0
1 1 1(1 )y Ah s
where s1 is the endogenous level of schooling, chosen so that individual utility is
maximized, and 0
1h is the initial level of human capital. The subscript 1 indicates
the level of human capital in the period when an individual is young, while
the superscript 0 indicates that the level of human capital was determined one period
earlier. In general, we assume that human capital can be either transmitted from
parents through innate abilities and upbringing, or accumulated through education. In
the first period, the level of human capital 0
1h is derived only from parental human
capital, because schooling is not completed. In the second period, the inherited human
capital is combined with the accumulated human capital gained by attending
school.
Given this, an agent produces her human capital for the second period, 1
2h , in
accordance with
1 0
2 1 1h Bh s (2)
where 0B is the productivity parameter in human capital accumulation. For 1
2h ,
the subscript 2 indicates the level of human capital in the period when an individual
is old, while the superscript 1 indicates that the level of human capital was determined
by the level of human capital and choices made in the period when an individual
was young.
Income in the second period is of the same form as in the first period.
The agent no longer invests in education through schooling. She spends all time
working and utilizes the new human capital level.
1
2 2y Ah (3)
By combining (3) with human capital formation (2) we get
0
2 1 1y ABh s
In each period, each agent consumes what she produces. There are no savings
and bequests in the form of physical capital.
t ty C
Individual utility (1) is maximized subject to the constraints 0
1 1(1 )tC Ah s
and 0
2 1 1C ABh s .
The maximization problem 1 2,MaxU C C can be rewritten as follows
0 0
1 1 1 1
0
1
ln 1 ln
1s
MAX Ah s AB h s (4)
Individuals compare the opportunity costs of education with the discounted
future benefits of education in the form of higher income. By differentiating (4) with
Finance a úv r-Czech Journal of Economics and Finance, 59, 2009, no. 1 79
respect to the level of schooling we get the optimal decision rule for schooling, which
depends on the subjective discount rate
*
1
1
2
s (5)
The higher is the discount rate, the less are individuals inclined to study. A lower
has a positive effect on human capital accumulation by increasing the amount of
time devoted to education. This can be easily verified by combining (2) and (5) to
obtain the general formula
1
1
1
2
t t
t th B h
This expression is crucial for generating growth and it is the major building
block of the remainder of our analysis, as it describes the equilibrium path of development.
In the simplest version of this model, the productivity parameter of human
capital accumulation B and the subjective discount rate determine if an economy
develops perpetually along a constant positive growth path ( 2B ) or if it stagnates
( 2B ).
Figure 1 illustrates the former situation. An increase in the efficiency of schooling
or a decrease in the subjective discount rate makes the condition for long-run
growth more likely to be satisfied. The reason is straightforward. The discounted future
benefits of education outweigh the earnings forgone during schooling, which
motivates parents of young individuals to invest a higher proportion of time in education
and accumulating human capital.
3.3 Human Capital and Time Discounting
The implications of the model become more interesting when we link the subjective
discount rate and the level of human capital. It is this feature that distinguishes
this model from the existing ones. The effects of education on promoting
planning abilities and disease prevention and the cultural influences mentioned earlier
suggest a negative relationship between the level of human capital and the subjective
discount rate. We will therefore assume that 0h .8
We further assume
that the strength of this effect is diminishing, so that '' 0h .
Figure 1 Human Capital and Growth
th
1th
/ (2 )tB h
450
80 Finance a úv r-Czech Journal of Economics and Finance, 59, 2009, no. 1
Again, the schooling choices are made in the first period under the influence
of perfectly altruistic parents. The subjective discount rate, which depends on parental
human capital, enters the utility function of the young individual:
1 2 1 20
1
1
, , ln ln
1
tU C C C C
h
Using the same procedure as above, we find the optimal decision rule for
schooling to be *
1 0
1
1
2
s
h
. The lower is the human capital in the family, the less
patient is the young individual and the lower is the optimal level of schooling perceived
by the individual and her parents. The transition function of human capital
then looks as follows:
1
1 1
1
2 ( )
t t
t tt
t
h B h
h
(6)
The role of human capital is now enlarged for its effect on the discount rate. Figure
2 shows the dynamics of the economy. This non-convexity gives rise to an S-shaped
transition curve and hence the possibility of multiple equilibrium development
paths, and a poverty trap emerges.
A low-income equilibrium appears if the efficiency of the education system is
too low compared to the individual discount rate, specifically if 1
2 ( )t
th B . In
such a situation, the opportunity costs of education are higher than the discounted future
benefits of education in the form of higher income, and there is negative human
capital accumulation. In an environment of low human capital, people discount future
income heavily and accumulate an insufficient level of human capital. However,
when the threshold level of human capital is overcome, autonomous growth gets gen-
erated.
Figure 2 Human Capital and Time Discounting
th
1th
t
threshold
h
2 th
ththreshold
th
B
8
Admittedly, we could have included two parameters, one for the subjective discount rate and one for
the probability of survival – in our model it would be the probability that an agent survives to the second
period. These two sources determining the weight assigned to future pleasures are practically difficult to
distinguish. As both are dependent on human capital, we find it convenient to simplify the analysis by including
only one variable encompassing both. The major results remain unchanged.
Finance a úv r-Czech Journal of Economics and Finance, 59, 2009, no. 1 81
An endogenous subjective discount rate poses difficulties for welfare analysis.
As a consequence of varying time discounting, the form of the utility function changes
for different levels of human capital. In such a case, it is not possible to compare
an individual’s well-being in alternative situations.
3.4 Human Capital, Time Discounting and Population Growth
The likelihood of a low-income equilibrium in the model may be further reinforced
by considering factors that influence the efficiency of schooling B, which
we have so far treated as a constant. A variety of evidence (e.g. Greenwald, Hedges,
and Laine, 1996) suggests that schools with more resources per student deliver more
efficient outcomes to the studying individuals. We approximate the functional form
as ,t tB B T n , where T represents public resources invested in the education
system, which are not paid by individuals, and nt is population growth. Per capita
public expenditure on education determines class size, training of teachers, and availability
of books and other school equipment, or may reduce the financial exposure of
individuals by eliminating school fees. In poor countries, public education expenditures
are usually under constant pressure from rapid population growth. For example,
the pupil-teacher ratio in primary schools in sub-Saharan Africa is three times
higher than in high-income countries (Glewwe and Kremer, 2005). Hence, it is plausible
to assume a positive relationship between the level of expenditure on education
and the efficiency of education, such that
,
0
tB T n
T
, and a negative relationship
with respect to population growth:
,
0
t
t
B T n
n
.
In addition, empirical studies based on the World Fertility Surveys and the Demographic
and Health Surveys establish a strong negative correlation between education
and fertility in the developing countries (e.g. Martin, 1995). This correlation is of
even greater magnitude than the correlation with per capita income. Recent micro-level
studies have found a significant causal impact of education on lowering fertility
(Breierova and Duflo, 2002; Bauer and Chytilová, 2007). The relationship has also
been illustrated theoretically in a model of human capital and endogenous fertility by
Becker, Murphy, and Tamura (1990).
In line with this literature, we assume that population growth in period t depends
negatively on parental human capital and is given by a non-increasing concave
function: nt = n ( 1t
th ), n´(h) < 0, and n´´(h) <0.
The transition path (6) emerges now as:
1 1
1 1
1
,
2
t t t
t t tt
t
h B T n h h
h
(7)
Again, the resulting transition function is compatible with the possibility of
multiple equilibrium development paths. There are now two non-convexities with
respect to human capital built into the model. First, for low levels of initial human
capital, people discount the future heavily. Secondly, the efficiency of schooling is
82 Finance a úv r-Czech Journal of Economics and Finance, 59, 2009, no. 1
low due to rapid population growth, which is also rooted in low levels of human
capital. These two effects jointly undermine the motivation of the young generation
to attend school. For countries with an uneducated population, the dynamic behavior
of society does not generate a sufficient level of human capital for growth, because
the optimal level of schooling is too low (see the Appendix for the algebraic derivation).
The two charts in Figure 3 illustrate the dynamics graphically. On the lefthand
side, the transition curve (7) is depicted. The chart on the right-hand side goes
further behind the slope of the transition curve by showing how t and tB are
associated with human capital.
If 1 1
, 2t t
t tB T n h h , the economy is placed on a low development
path, as 1
1
t t
t th h . People discount future pleasures heavily and have high numbers
of children, the efficiency of schooling is low, and as a consequence people decide
on a low level of education. If 1 1
, 2t t
t tB T n h h , the economy is on a positive
development path. People are future-oriented, fertility is low, the efficiency of
schooling is high, and people invest in their human capital.
The outcome of the economy may then depend on the initial conditions in
terms of human capital. An economy with human capital below threshold
th converges
to the low equilibrium, whereas an economy with an initial human capital stock above
threshold
th grows perpetually.9
The threshold level of human capital is an unstable
equilibrium. In this situation, , 2threshold threshold
t tB T n h h .
Our model shows that in an environment of low human capital, a high level of
time discounting, and low efficiency of schooling, economic agents make choices
such that poverty may become persistent. The model implies a particularly important
role of public action in education systems in the least developed countries. The threshold
level of human capital can be reached by introducing mandatory school attendance.
In poor countries, where parents may not have enough resources to send their
children to school, alternative actions may stimulate children’s school attendance.
Higher public expenditure focused on increasing quality or reducing the direct costs
of education (T) increases the efficiency of schooling as perceived by individuals (B).
This can be achieved, for example, by spending more on training teachers, buying
more books and other school equipment, or eliminating school fees. The higher “returns”
from attending school are then more likely to outweigh the high subjective discount
rate in individual schooling choices. A higher T can be illustrated as an upward
shift of the , tB T n h curve (the right-hand side of Figure 3) or as an anti-clockwise
rotation of the transition curve (the left-hand side of Figure 3). As a result,
the threshold level of human capital decreases and autonomous growth is more likely
to appear.
9
If we assumed decreasing returns to scale of human capital above a certain level of human capital,
the economy would converge to a high steady state and would not grow perpetually. However, for the sake
of simplicity we keep constant returns to scale over the whole interval.
Finance a úv r-Czech Journal of Economics and Finance, 59, 2009, no. 1 83
4. Conclusions
There is no doubt that the subjective discount rate is an important factor in determining
the life-cycle behavior of individuals. A high level of time discounting may
help to explain widely observed phenomena in the poorest countries, such as low
savings, school attendance or investment rates. At the same time, it is usual to presume
that the discount rate is wholly exogenous and independent of personal characteristics
and economic conditions.
This paper complements recent micro-level evidence from field studies in
the least developed countries demonstrating a close negative relationship between
education and the individual discount rate. We describe in greater detail the evidence
from our two recent studies in Uganda and India. Both environments are characterized
by a high prevalence of poverty, though they are different across many other
dimensions. Interestingly, we find the results from both studies to be consistent with
the causal effect of education on the subjective discount rate.
In accordance with this evidence, the paper integrates an additional assumption
about the negative impact of human capital on the individual subjective discount
rate into a simple human-capital-driven growth model. In this model, an individual
optimizes consumption over two periods by dividing her time between schooling and
work in the first period. The endogenous level of schooling represents individual savings
between the two periods. It is demonstrated that in a low-human-capital environment
the individual discount rate may be too high compared to the efficiency of
schooling. Consequently, the individual optimal schooling choices may result in negative
or low dynamics of human capital accumulation, and a low human capital equilibrium
may emerge. The theoretical predictions are further reinforced when we
model school efficiency in the context of population pressure, which largely stems
from low education levels, as demonstrated by a large body of empirical evidence.
The analysis points to education investment as a prerequisite for sustained economic
growth, as its wider impacts are particularly important in countries with less
developed economic institutions. The relationship between education, time discounting,
and fertility is consistent with slower growth in poor countries than predicted by
standard theory and may even result in a development trap, where the initial conditions
in terms of human capital matter and threshold effects appear.
Figure 3 Human Capital, Time Discounting, and Population Growth
th
1th
t
threshold
h ththreshold
th
2 th
, tB T n h
84 Finance a úv r-Czech Journal of Economics and Finance, 59, 2009, no. 1
As noted earlier, it has been widely observed in cross-country analyses that
standard estimates of the contribution of additional schooling to economic growth,
based on productivity differences associated with differences in the level of schooling,
cover only a small portion of the total correlation between education and subsequent
growth. The paper may help to reconcile this important puzzle by analyzing
an additional channel through which education promotes economic growth – by
shaping individual time discounting.
APPENDIX
The first derivation of the transition function10
(7) underlines the broader role
of human capital in this model and shows three channels through which human
capital influences growth dynamics, especially in less developed countries:
1 '1
2
1 ( 1) 1
2 2(2 )
t t
t t t t t
t t t tt
h B
B B h h
h h
The first item on the right-hand side is the standard result. It shows how optimal
schooling time translates into new human capital. The second item represents
the effect through a decreased discount rate. The third item represents the impact
through decreased population growth and hence increased education efficiency. All
three items are positive, ensuring a positive slope of the transition function.
The second derivation of (7) after rearranging looks as follows:
2
' '1
2 2 2
2
2 ''
3 2 2
1 ( 1) ( 1)
2 2 2
2 (2 ) (2 )
2 ( 1) 1
( ')
2(2 ) (2 )
t t t
t t t t
t t tt t t
t
t t t t t t t
tt t t
h B B
B h
h hh
B
B h B h h
h
(8)
The necessary condition for the existence of a low-human-capital equilibrium
is a convex transition function for low levels of human capital, or equivalently
2
1
2
0t
t
h
h
. Equation (8) confirms that the transition function is very likely to be
convex in our model. The first four items on the right-hand side are positive, whereas
the last two items are negative, and the relative magnitudes differ with the level of
human capital. However, to ensure a convex transition function, we would need to
make additional assumptions about the specific functional forms of B(.), (.) , and
n(.).
10
For simplicity the superscripts are omitted in the derivations.
Finance a úv r-Czech Journal of Economics and Finance, 59, 2009, no. 1 85
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