WPS4025
BACKGROUND PAPERTOTHE2007 WORLDDEVELOPMENT REPORT
Economic Growth, Education, and AIDS in Kenya:
A Long-run Analysis¤
Clive Bell, Ramona Bruhns, and Hans Gersbach§
Abstract
The AIDS epidemic threatens Kenya with a long wave of premature adult mortality, and thus
with an enduring setback to the formation of human capital and economic growth. To investigate
this possibility, we develop a model with three overlapping generations, calibrate it to the
demographic and economic series from 1950 until 1990, and then perform simulations for the
period ending in 2050 under alternative assumptions about demographic developments, including
the counterfactual in which there is no epidemic. Although AIDS does not bring about a
catastrophic economic collapse, it does cause large economic costs and very many deaths.
Programs that subsidize post-primary education and combat the epidemic are both socially
profitable the latter strikingly so, due to its indirect effects on the expected returns to education
and a combination of the two interventions profits from a modest long-run synergy effect.
Keywords: HIV/AIDS, Growth, Education, Kenya, Policy Programs
JEL Classification: I10, I20, O11, O41
World Bank Policy Research Working Paper 4025, October 2006
The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the
exchange of ideas about development issues. An objective of the series is to get the findings out quickly,
even if the presentations are less than fully polished. The papers carry the names of the authors and should
be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely
those of the authors. They do not necessarily represent the view of the World Bank, its Executive Directors,
or the countries they represent. Policy Research Working Papers are available online at
http://econ.worldbank.org.
¤We thank Emanuel Jimenez, Mamta Murthi and participants in seminars at University of California,
Berkeley, Mannheim University and the World Bank for their valuable comments and suggestions.
South Asia Institute, University of Heidelberg, INF 330, 69120 Heidelberg, Germany.
South Asia Institute, University of Heidelberg, INF 330, 69120 Heidelberg, Germany.
§Alfred-Weber-Institut, University of Heidelberg, Grabengasse 14, 69117 Heidelberg, Germany.
1 Introduction
This paper analyses the prospects for the formation of human capital and economic
growth in Kenya, even as the AIDS epidemic threatens that country with a long wave
of premature adult mortality. It focuses, naturally enough, on the young. For young
people in Kenya, as in most parts of the world, are confronted with a real choice
between continuing their education and entering the labour market, and they usually
have some say in that decision. At roughly the same time, they also become sexually
active, with all the gratifications and risks that entails, including unplanned parenthood
and contracting sexually transmitted diseases. Somewhat later, as young adults, they
choose partners, have children and take on the responsibilities of family life.
The long wave of premature adult mortality is not the only factor imposing the
time frame. What is called human capital is built up slowly, through a combination of
child-rearing within the family and formal education during childhood and adolescence,
and is then extended in some ways through experience on the job, especially in young
adulthood. This gestation process far exceeds that connected with investments in
virtually all forms of physical capital. Yet viewed statistically, the capital embodied in
young adults is also strikingly durable unless the hazards posed by accidents, violence
and lethal communicable diseases are substantial.
Our approach assigns a central role, first, to the formation of human capital through a
process in which child-rearing and formal education combine to produce the wellspring
of long-term economic growth and, second, to premature adult mortality as the primary
threat to that process. The victims of AIDS are overwhelmingly young adults or those
in their early prime years, the great majority of them with children to raise and care
for, so that the nature of the ensuing long-term threat to social well-being is clear.
For if parents sicken and die while their children are still young, then all the means
needed to raise the children so that they can become productive and capable citizens
2
will be greatly reduced. The affected families' lifetime income will shrink, and hence
also the means to finance the children's education, whether in the form of school fees
or taxes. On a parent's death, moreover, the children will lose the love, knowledge
and guidance which complement formal education. AIDS does much more, therefore,
than destroy the existing abilities and capacities or human capital embodied in its
victims; it also weakens or even wrecks the mechanism through which human capital
is formed in the next generation and beyond. These ramifications will take decades
to make themselves fully felt: like the course of the disease in individuals, they are
long-drawn out and insidious. For these reasons, expectations about the future course
of mortality have a potentially decisive influence on investment in education.
History plays an important role in any long-run analysis, and there is a fairly com-
plete account for Kenya from the middle of the 20th century onwards. The salient
features are the onset of rapid population growth and falling mortality (even before
World War II), the acceleration, slowing and then decline of individual productivity,
and the substantial but fitful expansion of education. The corresponding series provide
much of the material that is indispensable, not only to the calibration of the model in
Section 3, but also to the demographic projections in Section 4, the setting up of the
benchmark cases in Section 5 and the analysis of policy interventions in Section 6.
The central emphasis placed here on the role of human capital is fully in keeping with
Azam and Daubr´ee's (1997) finding that its accumulation was the main force that drove
economic growth in Kenya between 1964 - 1990. Their econometric analysis also reveals
a sharp fall in the growth of total factor productivity in the second half of that period,
which is consistent with the results of our calibration. It emerges from the latter that
the slowdown and then subsequent decline in the economy after 1980 are attributable
to two factors. First, there was a fall in the efficiency of the `technology' for producing
human capital young from inputs of schooling and the quality of child-rearing in the
family (as measured by the parents' own level of human capital). Second, and with a
3
lag, there was a fall in the productivity of existing human capital in producing aggregate
output. Both bode ill for Kenya's prospects, quite independently of the outbreak of
the AIDS epidemic.
Section 5 analyses two benchmark cases for the period 1990 to 2050, namely, the
counterfactual in which the epidemic never arose and the world in which breaks out.
Neither involves any explicit form of intervention, though the latter rests heavily on the
assumptions adopted by the researchers at the U.S. Bureau of the Census, whose pro-
jections form the basis of the benchmarks. AIDS does not bring about a catastrophic
economic collapse, but it does inflict large economic costs and very many deaths.
Independent, though indirect, support for our conclusion that the epidemic will exact
a heavy economic toll is provided by micro-economic evidence that premature adult
mortality in rural areas of Kenya has strongly adverse effects on school attendance
and advancement from one grade to another. Yamano and Jaynes (2005) find the
effect to be especially adverse among poorer households and young girls. Evans and
Miguel (2005), who use a still larger and in some ways more detailed panel, albeit one
restricted to a single district in which the HIV-prevalence rate is high, confirm the
former finding but not the latter. They also establish that maternal death produces a
bigger setback than a paternal one, and that children whose test scores were low before
bereavement are more likely to be withdrawn from school. The cross-country evidence
for sub-Saharan Africa yields the same general finding where schooling is concerned;
see, for example, Kalemli-Ozcan (2006).
The question of how public policy can alter the course of the epidemic or mitigate its
impact upon the accumulation of human capital is taken up in Section 6. Since health
and education are clearly intertwined in this setting, the design of public policy should
be concerned with both domains. The relationship between spending on combating
the epidemic and the profile of age-specific mortality is speculative territory, for it
involves the efficacy of various policy interventions that seek to reduce premature adult
4
mortality. There is little research that addresses this relationship as we have formulated
it, so we have drawn on work for South Africa (Bell et al., 2006), which turns out to be
transferable to Kenya. On this basis, we find that separate programmes to subsidise
post-primary education and combat the epidemic are both socially profitable the
latter strikingly so, due to its indirect effects on the expected returns to education
and that a combination of the two profits from a modest synergy effect.
No single paper can hope to deal with all aspects of such a topic, so it will be helpful
to say at the outset what will be left out. The intimate connection between risk-taking
and demography in the context of AIDS is recognised, but not analysed: infection and
demography are treated here as exogenous processes, in contrast to Young's (2005)
study of the South African epidemic. Nor do we go into the bargaining that takes
place between the young and their parents. Since some rule for determining outcomes
is unavoidable, we adopt a simple one. We also steer clear of learning on the job as a
contributing factor to the accumulation of human capital, and of youth unemployment
as a barrier thereto. For contributions to the macroeconomic literature on AIDS that
use an OLG framework in which agents can also invest in physical capital, the reader
is referred to Corrigan, Glomm and M´endez (2004, 2005).
2 The model
The model is a substantial extension of Bell, Devarajan and Gersbach (2006): there
are now essentially three overlapping generations within an extended family, in which
all surviving adults care for all related children. This arrangement reflects Kenya's
social structure reasonably well. There are two levels of the educational system, with
a corresponding elaboration of the process through which human capital is formed. In
applying the model, moreover, we choose a still finer unit time period than a genera-
tion, namely, the inter-censal interval of a decade. This permits a relatively detailed
5
treatment of demography, which plays a central role in the analysis.
Children are defined to be aged 5 to 14 (that is, centered on 10). This definition
corresponds closely to the official span of primary education. The minimum school
age in Kenya has been six for generations, and the period of primary schooling has
likewise comprised eight years, with the exception of the interlude between 1966 to
1985, when it was a year shorter. Leaving out the five-year olds, assuming that many
children, especially boys, will not be enrolled until their seventh birthday, and allowing
for the fact that there has always been a fair amount of grade-repetition, full primary
schooling according to schedule will nicely exhaust the decade in question.
An important object of this paper is to incorporate the effects, costs and benefits
of human capital formation, especially among what can be called the `youth', who
are defined to be individuals aged 15 to 24. On reaching 15, the child has become a
youth, put his primary education behind him and augmented his original endowment
of human capital. A maximum of ten years is then available to build on this foundation
before he settles into the duties and routines of mature adulthood.
2.1 Human capital and output
We begin by introducing some notation.
Nt : the number of individuals in the age-group a in year t (a = 0 for 0-4 year-olds,
a
a = 1 for 5-14 year-olds, ...., a = 6 for 55-64 year-olds, a = 7 for those over 64),
at: the human capital attained by an individual in age-group (a = 2,3) in year t on
completing the decade corresponding to primary and higher education, respectively,
: the human capital of a school-age child,
t: output per unit of human capital input in year t,
e1t: the proportion of their school-age years (6-14) actually spent in primary school
6
by the cohort of children who are aged 5-14 years in year t,
e2t: the proportion of their youth (15-24) actually spent in secondary and tertiary
education by the cohort aged 15-24 years in year t,
zt: the transmission (or efficiency) parameter associated with the educational tech-
nology,
Yt: GDP in year t,
yt: GDP per adult of working age (15-64) in year t.
Some of these series and the associated sources have been discussed and set out in Bell
et al. (2004). Those for t, , zt and t must be estimated as part of the calibration.
Two difference equations govern the system' dynamics. They involve the level of
parents' human capital and the educational technologies, one for primary, the other
for secondary and higher education. The evidence suggests that parents in Kenya have
most of their children when they are in their twenties. Since the cohort of children in
primary school is centred on 10-year olds, we form the level of their parents' human
capital by taking the weighted average of the human capital of the age-groups a = 2,3
in period t. The human capital attained by a child on becoming a youth (a = 2) in
period t is given by
2t = 2zt1
-10f1(e1t
-10) · [(Nt
2
-10 t-10
2 + Nt
3
-10 t-10
3 )/(Nt 2
-10 + Nt3
-10 )] + 1, (1)
where f1(·) represents the educational technology at the primary school level and the
transmission factor zt for the cohort of school-age children in period t is defined with
1
respect to the parents' combined human capital, normalised to a representative couple.
In the secondary and higher stages of the educational process, a plausible, general
relationship is
3t = (e2t-10,2t
-10,3t-10),
7
where it is only the fully mature adults who influence the inter-generational trans-
mission process at this stage. For the purposes of calibration, however, one needs a
specification that is closely anchored to (1). It is arguable that, in the present frame-
work, the transmission parameter zt does not vary much across educational levels
a
and anyway, there is no chance whatsoever of calibrating the model with the data
available without the restriction zt = zt . Where the educational technology is con-
1 2
cerned, the assumption that f1(·) continues to hold at higher levels of education, with
argument e1t -20 + e2t
-10, seems rather implausible for the following reasons. The cohort
of school-going age in 1980 had completed, on reaching adulthood, just over six years
of education on average, and earlier cohorts fewer still. It is also known that net en-
rollment rates in secondary and tertiary education were very low until recently (they
are still modest at best). Up to 1990, therefore, the position was e1t < 1 and e2t = 0.
Given the strong diminishing returns that set in when f1(·) is strictly concave and e1t
approaches unity, this implies that the private returns to secondary education would
be much lower than those to primary education. In fact, the opposite was the case in
the 1990s (Appleton et al., 1999; Manda, 2002; Manda et al., 2004; Wambugu, 2003).
Faced with this difficulty, we proceed as follows. First, we note that completion of
primary schooling, in the form of passing the State Examination at the end of Standard
8, is a necessary condition for entry into Form 1 in a secondary school: formally, e2t > 0
only if e1t
-10 = 1. Secondly, we impose two conditions on the function f2(·), namely,
that, given e190 = 1 and the expectation of full primary education thereafter, it yield the
observed net secondary enrollment rate in 2000 and that the asymptotic rate of growth
of 3t when all cohorts enjoy full primary and secondary education (e1t = e2t = 1 t) be
the long-run rate estimated by Ndulu and O'Connell (2002), which is based on cross-
country regressions (see Section 3 for the numerical derivation of the parameters).
8
Then, corresponding to (1), we have
< 1
3t = 2t
-10 if e1t
-10 (2)
2zt
1
-10· f2(e2t
-10 ) · 3t
-10+ 2t-10 if e1t
-10 = 1
where f2(0) = 0.
At this point, a brief remark is needed on the asymptotic behaviour of 2t when there
is full primary education but never any investment in secondary education. Observe
from eqs. (1) and (2) that, starting from a sufficiently large value of the parents'
combined human capital, unbounded growth of 2t is possible only if 2z1f1(1) 1
and that the growth rate then approaches 2z1f1(1) - 1 from above. If, however,
2z1f1(1) < 1, then 2t will approach the stationary value 1/[1 - 2z1f1(1)].
Turning to aggregate output, this takes the form of an aggregate consumption good.
We assume its level is proportional to inputs of labor measured in efficiency units. A
natural normalization is that an adult who possesses human capital in the amount
at is endowed with at efficiency units of labor, which he or she supplies completely
inelastically. Let a child supply (1 - e1t) efficiency units of labor when it works 1 - e1t
units of time, where (0,1), i.e., a full-time working child is less productive than an
uneducated adult. Ignoring unemployment, we then have the identity
6
Yt t 0.9Nt (1 - e1t) + Nt (1 - e2t)2t +
1 2 Nt t
a 3
+10(3-a), (3)
a=3
where the scalar 0.9 reflects the fact that among the group a = 1, five year-olds can
do very little in the way of useful work. The advantages of (3) are that it is free of
any other behavioural assumptions and, if t is specified exogenously, it is linear in
at (a = 2,3) and . It also fully reflects premature mortality among adults, whatever
be its causes.
9
2.2 The family's preferences and decisions
The next step is to specify the (extended) family's preferences and its decision-making
rules. Some additional notation is needed.
ct: the consumption of the aggregate good by each young adult (a = 2,3,4),
: the proportion of a young adult's consumption received by each child under the
age of 15,
: the proportion of a young adult's consumption received by each adult over the
age of 44,
t : the direct costs per child of each unit of full-time schooling (a = 1,2),
a
nt: the number of children born to a representative couple who survive to school age
in period t,
x t
q : the probability that an individual aged t will die before reaching the age of t+x,
qt: the probability that a young adult in period t will die before reaching the third
phase of life,1
t: the poll tax levied on each young adult.
The parameters and are viewed as contractually binding by all concerned and they
are rigorously enforced by appropriate social sanctions.
A brief remark on the introduction of a poll tax is in order. Taxation and public
expenditures in the period up to 1990 are already implicitly included in the calibration
procedure in Section 3. For the period from 2000 onwards, we shall be concerned
with measures to combat the spread of the epidemic and treat its victims, so that the
means of financing them must enter into the reckoning. To keep matters simple, and to
preserve consistency with the calibration in section 3, we use the time-honoured device
1In the present structure, this statistic corresponds to q , the probability that an individual will
20 20
die before 40, conditional on surviving until 20.
10
of a poll tax, whose proceeds are used solely for the purpose of financing the measures
in question. It should be noted that, by definition, the poll tax is zero not only in the
historical period 1950-90, but also in the period 1990-2050 in the counterfactual case
in which there is no epidemic. It is zero also in the base case in which the epidemic
simply runs its course without any specific intervention by the government within the
framework of the model.
The extended family's expenditure-income identity involves expenditures on the ag-
gregate consumption good and education, where the latter include both the direct
expenditures and the opportunity costs of the students' time. In what follows, it will
be convenient to normalize the identity by the number of individuals in the first phase
of adulthood in period t, namely, by
4
Nt
y Nt .
a (4)
2
The said identity may then be written as:
Pt(,) · ct + Q1t(t,,t ) · e1t + Q2t(t,2t,t ) · e2t + t t · (t + 0.9Nt )/Nt , (5)
1 2 1 y
where
Pt(,) 1 + (Nt + Nt + Nt )/Nt + (Nt + Nt )/Nt
5 6 7 y 0 1 y (6)
is the `price' of ct,
Q1t (0.9t + t )Nt /Nt
1 1 y (7)
is the `price' of a unit of full primary education, and
Q2t (t2t + t )Nt /Nt
2 2 y (8)
can be interpreted as the `price' of a unit of full secondary and tertiary education
11
combined, all prices being normalised by Nt . The quantity
y
6
t Nt t + Nt t +
2 2 3 3 Nt t
a 3
+10(3-a) (9)
a=4
is the aggregate human capital of all individuals over the age of 14. The RHS of (5) is
the level of so-called (normalised) full income in year t. It is seen that the demographic
structure plays a potentially important role in determining both relative prices and the
level of full income.
We turn to the question of who decides how to allocate current full income. As
noted above, Kenya's parents have most of their children in their twenties and early
thirties, so it is plausible that the age groups a = 2,3 have the greatest say in making
the decisions. Since the unit time period is a decade, the younger ones will be parties
to a partial revision a decade later. To keep matters manageable, we assume that
they take no heed of this fact when contributing to the current outcome. Observe also
from (5) that those aged 35-44 also enjoy ct although, by assumption, they have
no say in the matter. That those aged 15-24 should have a strong say in their own
education at this stage of their lives seems perfectly natural. Given these assumptions,
the decision variables comprise the triple (ct,e1t,e2t), whereby the social rules governing
the distribution of consumption among the three generations, as expressed by and
, are regarded by all participants as permanent.
The young adults' preferences are, in principle at least, defined over four goods: the
levels of consumption in young adulthood and old age, ct and ct +20, respectively, and
the human capital attained by their school-age children on attaining full adulthood
(3t+20), which they may appreciate in both phases of their own lives. Investment in
the children's education therefore produces two kinds of pay-offs, namely, in the form
of altruism, as expressed by the value directly placed on 3t +20 , and selfishly, inasmuch
as an increase in 3t +20will also lead to an increase in ct+20 under the said social rules.
12
Although the pooling arrangement implicit in the extended family structure and the
law of large numbers combine to eliminate some risks, others remain. A young adult
still faces uncertainty about whether he or she will actually survive into the last phase
of life, and whether his or her children will do likewise, conditional on their reaching
adulthood in their turn. There is also arguably uncertainty about future demographic
developments, which will influence the realised levels of ct +20 and 3t
+20.
To make the analysis tractable, we resort to a number of further assumptions. First,
and plausibly, the parents' altruistic motive makes itself felt in their preferences only
when they themselves are young and actually make the sacrifices. Second, their prefer-
ences are additively separable in ct, ct +20 and 3t+20, and also conform to the expected
utility hypothesis. Third, the partners making up the representative young couple in
the extended family regard consumption in both phases as a private good, but the
human capital attained by each of their children on attaining adulthood as a public
good within the partnership. Fourth, in keeping with common practice in the macroe-
conomic literature, the sub-utility functions for consumption are logarithmic in form.
Fifth, the sub-utility function representing altruism through 3t +20belongs to the iso-
elastic family with parameter :
(3t +20) = 1 - (3t+20)- /,
(10)
where the fact that is to be estimated lends the system substantial flexibility, espe-
cially by allowing the two types of sub-utility functions to exhibit different degrees of
curvature. The preferences of a young adult at time t are then represented as follows:
EtU = b1 lnct + b2(1 - qt)Et(lnct +20) + (1 - qt+20)nt(3t+20), (11)
where the taste parameters b1 and b2 are also to be estimated by calibration. It should
be noted, first, that the `pay-offs' in the event that the parent should die prematurely
13
(with probability qt), or that the children, in their turn, should die prematurely in
adulthood (with probability qt +20), have been normalized to zero, and second, that
ct+20is also a random variable for those who survive into old age, by virtue of the
fact that its level depends on a whole variety of future economic and demographic
developments.
The individual takes all features of the environment in periods t and t + 20 as
parametrically given. It will be helpful to summarize the latter in order to distinguish
between what the individual knows and what he or she must forecast. The current
environment is described by the vector
Zt (Pt,Q1t,Qt,Nt (a = 0,...,7),nt,t,t,t).
2 a (12)
This assumed to be known. The future environment, as described by the vector
Zt (Nt
e a
+20 (a = 0,...,7),t +20,qt,qt
+20,t+20 ), (13)
must be forecast, where it should be noted that forecasting Nta
+20 on the basis of know-
ing Nt implicitly involves forecasting cohort-specific mortality rates over the period
a
from t to t + 20. For simplicity, the individual's forecasts of all elements of the fu-
ture environment are assumed to be point estimates: equivalently, there is uncertainty
about individual premature mortality, but none about the future mortality profile itself
or future fertility. The world being what it is, this assumption is a lot to swallow; but
the calibration will be virtually impossible without it.
Investments in education at the secondary and tertiary levels in period t will yield
pay-offs to both age groups in future periods, albeit tempered by the chances of an
untimely death, and these expectations will enter into their calculations. Since imple-
menting the whole under full rational expectations would entail a research project in
14
itself, we choose a (comparatively) simple procedure. As assumed above, consumption
in the second phase of adult life, namely, ct +20, is determined by a social rule under
which the decisions of neighbouring cohorts of young adults have a strong influence on
the outcome. Given the complexity of the associated forecasting problem with which
young adults are confronted, we assume that they approach it as follows.
First, ct +20 is taken to be a fixed fraction, , of full income per adult in the first
phase of life in period t + 20, net of taxes:
ct +20= t +20(t +20+ 0.9Nt1
+20 - t +20)]/Nt
y
+20. (14)
Observe that the terms Nt 2
+20 t+20
2 and Nt 3
+20 t+20
3 in t +20 make up the greater part
of the new human capital that will arise in period t + 20, and these depend on the
decisions of young adults not only in period t, but also those in t + 10. The term
Nt4
+20 t+10
3 , in contrast, depends only on current decisions given those made earlier
in periods t - 10 and t - 20. To bring home this point, it is useful to define
t +20(e1t,e2t,e1t
+10,e2t
+10 ) Nt 2
+20 t+20
2 (e2t,e1t +10 ) + Nt3
+20 t+20
3 (e1t,e2t +10)
+ Nt 4
+20 t+10
3 (e1t -10,e2t) + Nt 3 + Nt 3 .
5 6
+2 t +2 t-10 (15)
The results of decisions made by cohorts of young adults in preceding periods are, of
course, known to the young adults at t, but what pair (e1t +10,e2t
+10 ) the cohort that
follows will choose is something that the young adults at t must guess when deciding
on (e1t,e2t). This brings us to the second step: the young adults at time t are assumed to
have stationary expectations concerning (e1t +10,e2t
+10 ) and more distant future choices,
to the extent that these are relevant. That is, they employ the rule
Eteat
+10= eat, a = 1,2 (16)
15
In arriving at their decision at t, however, they take Eteat +10 as parametrically given,
that is, they view t +20 as only directly influenced by (e1t,e2t) when they calculate the
optimum. Only after the optimum has been found does (16) come into play. Having
cut this particular Gordian knot, we can then solve the model sequentially, one period
at a time.
The said individual's decision problem takes the following form:
max EtU (17)
((ct, e1t, e2t)|Zt,Zt )
e
subject to: ct 0, eat [0,1], a = 1,2, (1), (2), (5), (14).
If f1(·) is strictly concave, EtU(·) will be strictly concave in (ct,e1t,e2t). Hence, problem
(17) has a unique solution and the first-order necessary conditions are also sufficient.
Observe that the optimum always involves ct > 0; but the corner solution in which the
children are not educated at all can also be ruled out when f1(·) satisfies the lower
Inada condition, since 2t +10/e1t is then unbounded at e1t = 0. What cannot be ruled
out, however, is the possibility that e2t = 0 indeed, as e2t is defined on the basis of the
data, this was precisely the state of affairs until 2000.
The associated Lagrangian is
Lt = EtU + µt[t(t + 0.9Nt )/Nt - Ptct - Q1te1t - Q2te2t - t].
1 y (18)
The first-order conditions are
bt/ct - µtPt = 0, (19)
1 - qt ct +20 (1 - qt
b2 · · + +20 )nt 3t · +20 (20)
ct+20 e1t (3t+20 ) +1 e1t - µtQ1t 0, e1t 1 compl.
and
1 - qt ct +20 (1 - qt
b2 · · + +20 )nt 3t · +20 (21)
ct+20 e2t (3t+20 ) +1 e2t - µtQ2t 0, e2t 0 compl.
16
As can be seen from (1), (2) and (15), the derivatives of ct +20and 3t +20 involve very
long and messy expressions, which are suppressed here.2
To close this section, a remark on the (non)-uniqueness of the expectations path
is called for. Although the solution to problem (18) is unique for any given set of
expectations, only when the solution satisfies (16) is it admissible as an equilibrium
under stationary expectations. The imposition of (16) when solving the first-order
conditions therefore opens up the possibility that there is more than one pair (e1t,e2t)
that will satisfy them in effect, there may be more than one equilibrium path.
3 Calibration
To determine the parameters and past variables of interest, we need to go back deep
into the past century, and since censuses were carried out in 1948, 1962, 1969, 1979,
1989 and 1999, it is natural to choose the time-points, or periods, as t = 10, 20,
..., 100, ..., where 100 denotes the year 2000. Morbidity and mortality from AIDS
became significant after 1990, so that some aspects of the calibration of the model may
be restricted to the period 1910 to 1990. We proceed in two steps.
3.1 Step 1: Human Capital and Output
Eqs. (1) and (3), together with the specialisation of f1(·) to the iso-elastic form (e1t) ,
provide the basis for the first stage of the calibration of the system's parameters and
the historical values of at. The first, preliminary step is to smooth out the considerable
fluctuations in the Kenyan economy. In keeping with the unit time period of a decade,
the values of Yt are obtained by forming 5-year moving averages of GDP per caput
from the Penn World Tables (hereafter, PWT) and multiplying them by the population
2They are available upon request.
17
series. Since the PWT series run from 1950 to 2000, the resulting estimates for those
boundary years contain an element of further guesswork, which is based on inspection
of the entire series. The smoothed estimates are reported in Table 1.
Starting with t = 50 in (3), we need a10, ..., a50: this is the deepest foray back into
the 20th Century. At t = 10, apart from a few colonial administrators, missionaries and
white settlers, not even the youngest adults had any education at all (Sheffield, 1973;
Thias and Carnoy, 1972). This may be termed a `state of economic backwardness',
in which, by definition, a = 1 for all age-groups. Hence, we anchor the system to
a10 = 1. The history of education in Kenya also leaves no doubt that 220 was scarcely
above unity, so we shall set it exogenously, at a value to be determined in the light of
the series for eat (a = 1,2).
To obtain at for t = 30 onwards, only (1) need come into play. For the fact that
e2t = 0 up to and including 2000 implies that 3t +10 = 2t t 100. Since we have set
a10 and a20 exogenously and have estimates of e1t for t = 20 onwards from the Censuses,
we can employ (1) starting with t = 30 if we have estimates of the relative sizes of the
age-groups 15-24 and 25-34, respectively. In 1950 and 1960, the said ratio was 57:43,
which we assume to have held also in the earlier decades of the 20th century. Thus,
with the above assumptions, we can indeed obtain 2t from t = 30 onwards by choosing
the plausible value 220 = 1.01.
Since (1) is a purely technical relationship and (3) is effectively an identity, we can
employ both for the entire historical period 1950-2000, even though AIDS had begun
to make itself felt by 1990. Eqs. (1) and (3) then yield 14 equations in the following
26 unknowns: (30, ..., 100); (50, ..., 100); (z20, ..., z90); , . Some restrictions
1 1
on the latter are therefore unavoidable if we are to solve the system.
We employ the following criteria to define the restrictions and select the solution.
First, we use additional information to identify the possible points in time when struc-
18
tural breaks in t and zt occurred. Second, we vary exogenously and the timing of a
possible structural break in zt between 1920 and 1970, and then reject any values of
1
that imply infeasible economic values such as negative human capital. This produces
an admissible range of . Third, we vary exogenously by a grid search within this
range and the timing of the structural break in zt between 1920 and 1970 until we find
1
solutions in which b1 and b2, which will be estimated in step 2, are in keeping with the
results of other studies in which the pure rate of time preference plays a key role. In
other words, the value of and a possible structural break are anchored by well-known
time preference relationships in the literature through systematic sensitivity analysis.
Inspection of the series for the level of GDP per adult aged 15-64,
6
yt Yt/ Nt ,
a (22)
a=2
which is set out in Table 1, reveals that Kenya suffered, first, a marked slowdown
in the growth of yt in the 1980s, and then a further turn for the worse in the form
of an actual fall in yt the 1990s. Since e1t continued to rise somewhat, even up to
1990, this slowdown and subsequent decline must have arisen from one or more sharp
reductions in t or zt after 1980. With a peak to be explained, it is hard to pin
1
the blame on the productivity factor t alone. Rather, intuition suggests that the
transmission factor zt may have declined first, thereby braking the growth of t, with
1
a subsequent fall in t to produce the retreat in yt. There are good grounds for
supposing that the quality of schooling declined substantially in the late 1970s, with a
surge in enrolments, overcrowded classrooms and an influx of poorly trained teachers.
As a working hypothesis, therefore, we impose the following restrictions:
50 = 60 = 70 = 80 = 90; and z80 = z90,
1 1 (23)
with the possibility of an earlier break in zt between 1920 and 1970. These restrictions
1
19
leave us with the following 15 unknowns:
30, ..., 100; 50, 100; z20, z80; , ; and one structural break in z30,...,z70,
where the superscript on zt can now be suppressed without introducing ambiguity. In
the light of the results of the calibration procedure described above, we confined to
the range 0.4 to 0.6. Outside this range, the solutions have unacceptable features, for
example, some parameters or levels of human capital (including ) become negative.
3.2 Step 2: Preferences
In a purely formal sense, the first stage of the calibration can be carried out quite
independently of the second. It turns out, however, that the choice of and the
degree of freedom conferred by the structural break in z30,...,z70 combine to produce
a substantial effect upon the results from the second stage, which involves the preference
parameters b1,b2 and . We continue, therefore, by deriving a condition from which
b1,b2 and can be estimated, given the results from the first stage; only then do we
choose and report some combined results.
Recalling that e2t = 0 during this period, we may drop (21) and seek an expression
in e1t alone. Some manipulation using (14), (15), (19) and (20) yields
1
·2t +10 (1 - qt+20 )nt (1 - qt)b2
· +
2t +10(e1t) e1t (2t+10 (e1t)) (t +20+ 0.9Nt 1
+20)/Nt 3
+20 t+10
2 (e1t)
b1Q1t (24)
t[(t + 0.9Nt )/Nt ] - Q1te1t - t
1 y
where the first term on the LHS is obtained from (1) and, apart from a limiting case,
(24) holds as a strict equality when e1t < 1. Observe that (24) is independent of the
social sharing rules, as expressed by the parameters and , a feature that stems from
the choice of a logarithmic sub-utility function for consumption.
The first stage of the calibration yields complete descriptions of the current envi-
20
ronment for the years 1950, 1960, 1970, 1980 and 1990. Where the associated future
environments are concerned, we assume that agents possessed perfect foresight, except
for the timing of the outbreak of the epidemic and its immediate consequences. In this
regard, the data in Table 1 and the demographic data up to 1990 suffice for 1950 and
1960. Thereafter, we need progressively more. In view of (11), we require q70+20 for
1970, which involves the population pyramid for 2010; for 1980, we require the entire
population pyramid for 2000 and q80+20, whereby the latter involves N120; and we re-
3
quire the corresponding items for 1990. All these elements are estimated in Bell et al.
(2004), and qt is reported in Table 1. As for the question of whether the outbreak of
the epidemic featured in expectations about the future at that time, it is surely safe
to rule out this possibility except, possibly, for 1990. The cumulative number of
deaths due to AIDS was still rather small in relation to the population in 1990, and
the prevalence rate at that time was also low. It seems very plausible, therefore, that
the overwhelming majority of Kenyans had not, at that time, come to recognise the
dimensions of the wave of mortality that was about to befall them. In any case, that
is the position we adopt for the purposes of calibration.
It still remains to estimate the series {t }t
1 t=90
=50. According to a survey of some 6000
farm households, representatively drawn from districts in six of Kenya's eight provinces
in 1981-82, no less than 20 percent of total expenditure on average was devoted to
education (Evenson and Mwabu, 1995: 12-15). For the purposes of calibration, we
assume that this proportion stayed constant from 1950 until the mid 1980s, though it
seems a bit questionable in the early years, when school-places were rather rationed.
By the mid 1990s, there is reliable evidence that this budget proportion had fallen
substantially (Kenya, 1996). In calibrating the model, therefore, no use will be made
of the year 1990.
The number of equivalent, full-time schoolchildren in period t is e1tNt . Using the
1
budget share 0.20 and recalling that, by assumption, the whole of GDP accrues to
21
households in the first stage of the calibration, we have t etNt = 0.2Yt, or
1 1 1
t = 0.2Yt/e1tNt .
1 1 (25)
Substituting for t in (20) and rearranging, we obtain
1
1
·2t +10 (1 - qt
+20 )nt (1 - qt)b2
· +
2t+10 (e1t) e1t (2t+10(e1t)) (t +20 + 0.9Nt 1
+20)/Nt 3
+20 t+10
2 (e1t)
1
· t + 0.9Nt - et b1, (26)
[0.9 + 0.2Yt/(etNt t)]Nt
1 1
where the poll tax t is zero throughout the period 1950-90. Since only interior solutions
are observed in this period, (26) will hold as a strict equality for the purposes of
calibration.
3.3 Selection
As noted above, the values of the three parameters b1, b2 and are determined by the
values of other parameters and t that emerge from the first stage of the calibration.
For each of a wide variety of results from the first stage, we obtained the corresponding
triple (b1,b2,) by selecting three from the four years 1950, 1960, 1970 and 1980,
and then applying (26) to the years in question. The only apparent reservation about
deriving the triples (b1,b2,) from (26) is that there is some evidence that school-places
were rationed somewhat in 1950 and 1960. In judging the constellation of values from
both stages combined, we note that the quantity 1-b2/b1 is the rate of pure impatience,
for the probability of prematurely dying as an adult, qt, is already allowed for in the
specification of EtU. Although we are unaware of any attempts to estimate this rate
for Kenya, two well-known studies for the U.S. suggest that it lies between 0.25 and 0.5
per cent per annum (Altig et al., 1996; Fullerton and Rogers, 1993). On this basis, we
reject any constellation for Kenya that yields an implied rate of pure impatience much
22
exceeding 1 per cent per annum. Tables 1 and 2 report two plausible constellations
that meet this criterion. In the first, the implied rate is 1.23 per cent, in the second,
1.26 per cent per annum. It is also noteworthy that (·) is roughly midway between
the logarithmic ( = 0) and the Ramsey ( = 1) cases, which confirms the importance
of not forcing the entire preference structure into a logarithmic straitjacket.
The adverse developments in the production of human capital and output that began
around 1980 express themselves in the behaviour of zt and t. In solution 1, the first
break, in 1960, is a small one; it is followed by a very sharp fall in 1980, which leaves
z at barely half its previous value. The story in solution 2 is very similar, albeit with
a much earlier and sharper first break, in 1940. The fall in z between 1970 and 1980
leads to a corresponding fall in 2 between 1980 and 1990, in both solutions of almost
20 per cent. These troubles are compounded by a decline in between 1990 and 2000,
also by just under 20 per cent.
A particularly depressing feature of both solutions is that the value of zt lies some
way below the critical value of 0.5 from 1980 onwards. Recall from section 2 that
unbounded growth through primary education alone is possible only if 2z1f1(1) 1.
Since f1(1) = 1 under the specialisation adopted for the purposes of calibrating the
model, it follows at once that the said condition is violated in both solutions, so that
a continuation of these values of zt without investments in secondary education imply
1
that 2t will converge, at best, on the stationary values 1/(1 - 2 × 0.41) = 5.56 and
1/(1-2×0.43) = 7.14, respectively. This finding indicates the pressing importance of
improving the performance of the primary school system and of promoting secondary
schooling in order to ensure long-run growth. To complete the picture, note that the
productivity of a school-aged child, , is estimated to be about 70 per cent of that of
an uneducated adult in both solutions. This seems a little high, perhaps, but it is not
outlandish.
Summing up, there is little to choose between solutions 1 and 2, except that the
23
Table 1: Calibration: solution 1
1910 1920 1930 1940 1950 1960 1970 1980 1990 2000
First stage
Exogenous:
Yt(107) 436 642 1089 2014 3076 3633
yt 1286 1506 1930 2567 2716 2295
e1t 0.063 0.107 0.179 0.268 0.489 0.611 0.693 0.760
Endogenous:
2t 1.00 1.01 1.35 1.57 1.93 2.42 3.31 4.52 3.69 3.81
t 732 732 732 732 732 610
zt 0.84 0.84 0.84 0.84 0.79 0.79 0.41 0.41
0.69 0.69 0.69 0.69 0.69
0.57 0.57 0.57 0.57 0.57 0.57 0.57 0.57 0.57 0.57
Second stage
Exogenous:
t1 2022 1133 1057 1175
nt 4.00 4.00 4.00 3.85
qt 0.175 0.163 0.154 0.141 0.127 0.113
Endogenous:
b1 3.05 3.05 3.05 3.05 3.05 3.05 3.05 3.05 3.05 3.05
b2 2.39 2.39 2.39 2.39 2.39 2.39 2.39 2.39 2.39 2.39
0.57 0.57 0.57 0.57 0.57 0.57 0.57 0.57 0.57 0.57
24
Table 2: Calibration: solution 2
1910 1920 1930 1940 1950 1960 1970 1980 1990 2000
First stage
Exogenous:
Yt(107) 436 642 1089 2014 3076 3633
yt 1286 1506 1930 2567 2716 2295
e1t 0.063 0.107 0.179 0.268 0.489 0.611 0.693 0.760
Endogenous:
2t 1.00 1.01 1.67 2.19 2.34 2.90 4.00 5.49 4.51 4.66
t 605 605 605 605 605 489
zt 1.32 1.32 0.81 0.81 0.81 0.81 0.43 0.43
0.71 0.71 0.71 0.71 0.71
0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50
Second stage
Exogenous:
t1 2022 1133 1057 1175
nt 4.00 4.00 4.00 3.85
qt 0.175 0.163 0.154 0.141 0.127 0.113
Endogenous:
b1 3.16 3.16 3.16 3.16 3.16 3.16 3.16 3.16 3.16 3.16
b2 2.46 2.46 2.46 2.46 2.46 2.46 2.46 2.46 2.46 2.46
0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.47
25
course of zt is noticeably smoother in the former, so we plump for 1.
3.4 Secondary Education
Recall that the function f2(·) is assumed to satisfy the conditions that it yield the
observed net secondary enrollment rate in 2000, namely, e2100 = 0.065, and that the
asymptotic rate of growth of 3t when all cohorts enjoy full primary and secondary
education (e1t = e2t = 1 t) be the long-run rate of growth of GDP per head, as
estimated by Ndulu and O'Connell (2002), namely, 2 per cent per annum. In view of
this modest growth rate, it is clear that f2(e2t) must be concave but flexible. Let it
take the following form, with two free parameters:
f2(e2t) = a1 ln e2t + 1 .
1 + a2e2t
Then (2) specialises to
2t-10 if e1t
-10< 1
3t = (27)
2zt 1
-10· a1ln (1+a2)e2t
-10 +1 = 1
1+a2e2t · 3t-10+ 2t
-10if e1t
-10
-10
The parameters a1 and a2 are obtained from the two conditions as follows. The house-
hold's first-order condition with respect to e2100, namely, (21) with e1100 = 1, holds
with equality when e2100 = 0.065. The second condition involves finding an expression
for limt 3t+10/3t. Noting that Nt Nt in steady-state growth with a stationary
2 3
population, some tedious manipulation yields
(2f2(1) + 1)zt
lim 3t+10 = . (28)
t 3t 1 - zt + 2f2(1)(zt)2
Given the fact that the system converges rather slowly to the asymptotic growth rate
from above, its growth rate will exceed Ndulu and O'Connell's estimate of 2 per cent
26
a year even over the later stretch of our time horizon. We have
lim 3t /3t = 1.0210 = 1.219.
t +10
This value, together with the lower branch of (27) for e2100 = 0.065, completes the second
condition. The resulting values of the parameters are a1 = 1.522 and a2 = 0.560.
4 Demographic Projections for 1990 - 2050
In order to estimate the effects of the epidemic and to evaluate public policies designed
to combat it, we need the corresponding demographic projections. These include the
natural counterfactual case in which the epidemic never breaks out and variations in
which it does so with or without policy interventions. In undertaking this rather
speculative task, we shall draw heavily on the projections made by the U.S. Bureau
of the Census (USCB) for the period 1985-2050. It should also be noted that the
formal economic analysis of the period 1990-2050 requires demographic forecasts for
the period 2000-2070.
In what follows, the counterfactual `without the epidemic' and the actual case `with
the epidemic' will be denoted by D = 0 and D = 1, respectively. Both projections
are based on a whole variety of assumptions, which are known to its authors at the
USCB, but are not reported in the data file made available to us.3 This omission is
not especially problematic when analysing D = 0 and D = 1 as benchmarks, since
the formal description is clear enough. It does, however, confront us with serious
difficulties when attempting to construct and analyse alternative policies, for they may
be partly or fully incorporated in the USCB's variant of D = 1 itself. In any event, the
relationship between the level of spending on combating the disease and mortality that
3This in no way reflects badly on the authors, of course, for such background material hardly
belongs in tabular files of this kind.
27
we construct in Section 6 is to be interpreted as representing the effects of measures
not contained in that variant of D = 1.
By 1990, it was apparent that the epidemic had started to take hold, though the
number of annual deaths was still very modest and the virulence of its course was
difficult to forecast. This date is therefore not only a natural `break point' in the
calibration of the model, but also a natural point of departure for the construction of
alternative demographic projections.
4.1 The projections
In view of the differences between our `revised' estimates of the demographic structure
for 1990 and the USCB's (see Bell et al. [2004]), and given the need to ensure a smooth
transition into the period 2000-2050 when the unit of time is a decade, we choose the
former population pyramid for both D = 0 and D = 1 for 1990 a small inconsistency.
Beginning with D = 0, a comparison of the age-specific mortality rates implied by
the `revised series' for 1950-90 with those implied by the USCB's estimates for 1990
and 2000 reveals that the former are so much higher than the latter that reconstruc-
tion of the USCB's counterfactual estimates for 2000-2050 rather than mere smoothing
is called for. This preliminary conclusion is reinforced by the fact that the USCB's
relatively low initial levels are also followed by steep projected declines in age-specific
mortality rates out to 2050. In reconstructing the USCB's estimates for the counter-
factual in the direction of higher mortality, we err, therefore, on the side of optimism.4
To anchor the system at the other end of the horizon, at 2070, we assume that in the
absence of the epidemic, Kenya would have enjoyed the profile of age-specific mortality
rates prevailing in the U.S. in 2000, as reported by the WHO (2002): that is surely a
4The UN's implicit estimates of mortality rates for the period 1990-2000 are even lower than the
USCB's, especially among young adults. For those aged 25 to 34 in 1990, for example, the USBC's
survival rate is 80.9 percent, whereas the UN's is 90.8 percent (UN, 2004).
28
sufficiently optimistic view of the long-term prospects for good health in Kenya. The
mortality profiles in the intervening period are obtained by simple linear interpolation.
The pyramid for the year 2000 is generated by applying the corresponding mortality
profile to the cohorts aged 5 to 54 in 1990, and then inserting the cohort sizes for
children under 15 and adults over 64 taken directly from the USCB (D = 0) series
itself. This process is then applied to the resulting pyramid for 2000 in order to yield
the pyramid for 2010, and the procedure is repeated up to 2050. The pyramids for
2060 and 2070 are obtained by making the assumption that the cohorts of 0-4 and
5-14 year-olds, as well as those over 64, remain at their sizes for 2050, which is in
keeping with the projection that the population will have almost reached a stationary
configuration by that date. It is readily admitted that this mixing of cohort sizes at the
top and bottom of the pyramids generated by one set of assumptions with mortality
profiles generated by quite another is inconsistent, but inspection of Table 3 in relation
to the USCB's counterfactual does not reveal any departures so egregious as to call
into question the implicit assumption of identical fertility.
The corresponding series for D = 1 from 2000 onwards is obtained in similar fashion,
except that the USCB's mortality profiles are taken over almost in their entirety. The
sole amendment concerns q in 1990, which appears to be implausibly high in view
10 10
of the fact that among all age groups, children of school-going age are the least likely
to die of AIDS. The said statistic is set instead at its level in 1980. The pyramids for
2060 and 2070 are generated using the mortality profiles for 2040. In this connection,
it should be remarked that the mortality rates for adults between 15 and 44 in the year
2040 are between 50 and 100 percent higher than those that would rule in the absence
of AIDS: the USCB's projections are based on the assumption of a continuation of the
epidemic beyond the middle of this century, albeit on a greatly weakened scale.
The above procedure above somewhat understates the mortality due to AIDS, as
estimated by the USCB at least. For the level of mortality in Bell et al.'s (2004) `revised'
29
Table 3: Projected population pyramids, No AIDS
1990 2000 2010 2020 2030 2040 2050 2060 2070
0-4 4458 4696 4602 4503 4537 4398 4336 4336 4336
5-14 7182 9006 9550 8995 8965 8968 8715 8715 8715
15-24 4715 6875 8666 9236 8744 8759 8807 8601 8644
25-34 2979 4447 6526 8277 8878 8458 8525 8624 8475
35-44 1833 2731 4116 6098 7809 8454 8129 8269 8442
45-54 1099 1634 2463 3755 5628 7290 7982 7761 7982
55-64 698 911 1379 2115 3280 4998 6581 7323 7234
65+ 511 864 1294 1969 3131 5266 8395 8395 8395
Source: Bell et al. (2004)
Table 4: Projected population pyramids, AIDS
1990 2000 2010 2020 2030 2040 2050 2060 2070
0-4 4458 4556 3874 3436 3238 3038 2972 2972 2972
5-14 7182 8612 8416 7197 6578 6258 5971 5971 5971
15-24 4715 6839 8209 8040 6945 6429 6186 5902 5902
25-34 2979 4187 5780 6912 6997 6360 6182 5948 5676
35-44 1833 2410 3052 4136 5266 5873 5878 5713 5498
45-54 1099 1487 1760 2173 3125 4372 5352 5357 5207
55-64 698 906 1149 1361 1746 2666 3956 4843 4847
65+ 511 821 1087 1362 1693 2353 3657 3657 3657
Source: Bell et al. (2004)
30
series, to which our modification of the USCB (D = 0) series is closely anchored until
2020, is somewhat higher than that in the USCB's (D = 0) series itself. To this extent,
the series for D = 1 in Table 4 represents a more conservative estimate of the level
of mortality in the presence of AIDS than the USCB's (D = 1), since by no means
all those who die of AIDS would have died at the same age of other causes. The
use of the series in Tables 3 and 4 as reference cases therefore involves a somewhat
more conservative stance than that implicit in the USCB series where both the size
of the initial `shock' to mortality and the general level of mortality are concerned.
With this reservation, the size of the shock to the level of premature adult mortality
can be represented by the difference between the summary statistics q (D = 1) and
20 20
20 20
q (D = 0). These statistics are reported in Table 5.
Table 5: Premature adult mortality compared: 20 20
q (D = 0) and q (D = 1)
20 20
1990 2000 2010 2020 2030 2040 2050
NO AIDS 20 20
q 0.127 0.113 0.099 0.085 0.070 0.056 0.041
AIDS 20 20
q 0.353 0.395 0.359 0.270 0.154 0.111 0.111
Inspection of this wave of mortality reveals that it is large indeed, but of compara-
tively short duration in relation to an inter-generational period of 20 years, in which,
by hypothesis, young adults make decisions with a horizon of 30 years. After the ini-
tial surge in 1990 (looking forwards), qt peaks in 2000, falls a little by 2010, then more
sharply by 2020, and again by 2030, by which point, the shock has essentially passed.
Since the chances that a mortality shock will bring about an economic collapse depend
on its duration as well as its size, it is desirable to do some sensitivity analysis. As a
variation on D = 1, therefore, we prolong the shock by a full decade in the following
way. In 2010, we repeat the age-specific mortality profile for 2000 and push those for
2010, 2020, ... out one decade into the future. The age pyramids in Table 4 are then
recalculated accordingly. This variant of the base case is labelled D = 1. Note that in
31
view of the fact that the calibration of f2(·) is anchored to e2100 = 0.065, the differences
in qt associated with the difference between the two `epidemics' D = 1 and D = 1 will
yield different values of a1 and a2.
5 The benchmark cases
The evolution of the entire system is determined by the extended household's decisions
(ct,e1t,e2t) at each point in time, which then determine, in turn, the formation of new
human capital through the educational technology, as specified in (1) and (2). Put
formally, given the current environment Zt and future environment Zt , as forecast by
e
the household's decision-makers, and given the assumption of stationary expectations
concerning future investments in education, the household solves problem (17) for
period t. In doing so, it determines some of the elements of Zt +10 and Zt e
+10 . The
associated demographic elements are provided by the forecasts in Section 4. The values
of t and t (a = 1,2) are assumed to remain constant from 2000 and 1990 onwards,
a
respectively. These complete the respective vectors, whereby the values 90 = 185 and
1
90 = 800 are drawn from Kenya (1996) and Appleton et al. (1999, Table 6), after
2
conversion into the units of Yt. One period later, in t + 10, the household solves its
problem anew, and so forth. The results of this process for D = 0 and D = 1 are set
out in Tables 6 and 7, starting in 1990.
A noteworthy feature of the counterfactual is that all children complete full primary
schooling (e1t = 1) from 1990 onwards, a result that does not square with the actual
value of e190 = 0.76 (see Table 1). The main reason for this discrepancy is that the
chosen value of 90 relates to the mid 1990s, at which time its level appears to have
1
been markedly lower than in the mid 1980s (Appleton et al., 1999, Table 6). Unable
to find any estimates for the intervening years, we choose the later of the two, thereby
erring on the optimistic side. With the promise of declining mortality and no further
32
Table 6: Benchmark 1: No AIDS (D = 0)
Year 1990 2000 2010 2020 2030 2040 2050
e1t 1.000 1.000 1.000 1.000 1.000 1.000
e2t 0.000 0.220 0.296 0.419 0.605 0.748
2t 3.69 4.29 4.33 4.83 5.32 6.03 6.96
3t 4.52 3.69 5.12 5.77 6.93 8.55 10.55
yt 2647 2183 2390 2654 2946 3403
Yt(109) 29.97 36.24 55.32 78.25 101.15 129.19
Population (103) 23475 31164 38596 44948 50972 56591
Table 7: Benchmark 2: AIDS (D = 1)
Year 1990 2000 2010 2020 2030 2040 2050
e1t 1.000 1.000 1.000 1.000 1.000 1.000
e2t 0.000 0.065 0.115 0.241 0.431 0.589
2t 3.69 4.29 4.33 4.63 4.91 5.40 6.11
3t 4.52 3.69 4.57 4.92 5.82 7.07 8.63
yt 2647 2347 2475 2571 2703 3016
Yt(109) 29.97 37.16 49.37 58.17 65.09 77.52
Population (103) 23475 29818 33327 34617 35588 37349
33
adverse shocks to zt and t, and with sufficient human capital accumulated in the past,
the expected pay-off to primary education rises with accumulation, and so ensures the
continuation of the process.
Where secondary education is concerned, the series broadly accords with intuition.
With relatively low current mortality and the prospect of falling levels in the future,
e2100 is considerably higher than the actual level of 0.065 under D = 1. Thereafter,
e2t accelerates until 2030, and slows only slightly thereafter, reaching 0.748 in 2040.
The factors at work here are the modest concavity of f2(·), which does much to offset
the rising opportunity costs of secondary education as 2t increases through primary
education. The end result is hardly cause for reproach: e1130 +e2140 = 1.748 corresponds
to about 16 completed years of education, on average, for the cohort in question.
The trajectories of 2t and 3t reflect these investments in education just as one would
expect. There is an early stumble in 2t, as the effects of the fall in z between 1970 and
1980 work their way through the system (see Table 1). By 2040, however, a primary
school graduate will be about 60 per cent more productive than her counterpart 50
years earlier, a improvement that owes not a little to her parents' secondary education.
The course of 3t is bumpier still, with a decline between 1990 and 2000 the combined
effect of the earlier fall in z and the fact that the upper branch of eq. (2) applies when
e2t = 0. Only after 2020 does the trajectory settle down. A secondary school graduate
in 2040 with e2130 = 0.605 has a productivity two-thirds greater than her counterpart
in 2010 with e2100 = 0.220.
These various lags make the road to recovery of GDP per adult after the fall in t
between 1990 and 2000 a long one. It takes about 30 years for yt to regain its level of
1990, but it then increases by almost 30 per cent in the following 20 years up to 2040.
As noted in Section 4, the AIDS epidemic is forecast to cause heavy mortality and
smaller cohorts of births for at least four decades from 1990 onwards. It is striking,
34
therefore, that investment in primary education fully withstands this shock. Investment
in secondary education, however, suffers a considerable setback in the period of peak
mortality, where it should be noted that the observed net enrollment rate in 2000,
e2100 = 0.065, is imposed exogenously in order to anchor the calibration of the function
f2(·) and the household is assumed to have perfect foresight. The level of e2t lies just
under 40 per cent of that in the counterfactual in 2010, rising to almost 60 and 80 per
cent, respectively, in 2020 and 2040.
This setback in secondary education is felt throughout the system. By 2040, a
primary school graduate will be 10 per cent less productive than her counterpart in the
counterfactual, despite e1t = 1 over the whole time span, a secondary school graduate
17 per cent less, and GDP per adult will be 11 per cent less. Another way of summing
up the effects of the epidemic is that it delays the attainment of the corresponding
counterfactual values of these central variables by about one decade.
The effects of the prolonged shock (D = 1) are not reported here. It turns out that
`recovery' in the sense defined in the previous paragraph is delayed by a mere additional
year or two. This, at first sight, rather surprising result stems from the fact that, given
the anchoring point e2100 = 0.065, the post-primary educational technology must be
somewhat more productive in order to offset the higher value of qt under D = 1. The
additional toll of mortality results in a still lower population in 2040: about 2 million,
or a little over 5 per cent, fewer.
6 Public policy
In this setting, there are three domains in which interventions can promote growth
and welfare, namely, health, education and the conduct of economic policy. Measures
designed to combat the epidemic are not only intrinsically valuable because they re-
duce the toll of suffering and death, but they also promote investments in education by
35
increasing the family's lifetime resources and the expected returns to such investments.
In contrast, measures that promote education directly, for example, school-attendance
subsidies, have no demographic effects in our model an assumption that demands
great care when making comparisons of policies across these two domains. Improve-
ments in the general conduct of economic policy manifest themselves in increases in
and thence in families' full income. They have no effect on the long-run rate of growth
in this framework, and are not considered here.
In what follows, the model is not `closed' where public finance is concerned; for
the budgetary outlays entailed by the interventions analysed below are financed not
by domestic taxes, but rather by outright grants from abroad. Kenya has been quite
heavily dependent on budgetary support of this kind in the past though it should
be added that its present prospects are clouded by the fact that corruption continues
unabated under the latest government. Be that as it may, we adopt this simplifying
assumption. Closing the model by introducing the government's budget constraint and
associated tax instruments would involve a substantial increase in complexity.5
6.1 Public spending and mortality
Interventions aimed at altering the course of mortality in the presence of the epidemic
produce variants of the benchmark D = 1. In order to formulate them precisely, it is
necessary to establish the relationship between the level of spending on measures to
contain the epidemic and treat those infected and the profile of age-specific mortality.
For this purpose, we draw heavily on both the procedure and estimates in Bell, Devara-
jan and Gersbach (2006). In essence, this involves choosing a functional form for the
relationship between the probability of premature death among adults and the level of
aggregate expenditures on combating the disease. Bell, Devarajan and Gersbach argue
5Some idea of what is involved can be obtained from the corresponding sections of Bell, Devarajan
and Gersbach (2006), whose two-generation OLG model for South Africa has a much simpler structure.
36
that when the main source of this mortality is AIDS, the efficacy of such spending
depends directly on the number of (sub-) families, Nt , within the fictional extended
F
family that makes up the whole society. Aggregate public spending is then written as
Nt Gt, where Gt is the level of spending per sub-family in period t.
F
For simplicity, and erring on the side of optimism, we assume that such aggregate
expenditures produce a pure public good, so that ridding the relationship of the size
of the population, we write
qt(D = 1) = qt(Gt;D = 1), (29)
where the function qt(Gt;D = 1) is to be interpreted as the efficiency frontier of the
set of all measures that can be undertaken to reduce qt in period t. We also assume
and this seems sensible that the effects of these expenditures last for only the period
in which they are made. Although very little is known about the exact shape of qt(·),
our definition of the benchmark D = 1 implies that qt(0;D = 1) should yield the
estimates in Table 5. A second, plausible, condition is that arbitrarily large spending
on combating the epidemic should lead to the restoration of the status quo ante, that
is, qt(;D = 1) = qt(D = 0) in all periods. It is also desirable to choose a functional
form that not only possesses an asymptote, but also allows sufficient curvature over
some relevant interval of Gt, so the natural choice falls on the four-parameter logistic:
1
qt(Gt;D = 1) = dt - . (30)
at + cte-btGt
With four parameters to be estimated, two additional, independent conditions are
required. One way of proceeding is to pose the question: what is the marginal effect
of efficient spending on qt in high- and low-prevalence environments, respectively?
Equivalently, we need estimates of the derivatives of qt(Gt;D = 1) at Gt = 0 and
37
some value of Gt that corresponds to heavy spending, when the scope for using cheap
interventions has been exhausted. The basis of the estimation and the steps involved
are set out in detail in Bell, Devarajan and Gersbach (2006). The said four conditions
yield the values of the parameters at, bt, ct and dt for the years 2000, 2010 and 2020 in
Table 8. The associated functions qt(Gt;D = 1) are plotted in Figure 1.
Table 8: The parameter values of the functions qt(Gt;D = 1)
2000 2010 2020
at 0.9073 0.9863 1.3852
bt 0.0169 0.0187 0.0266
ct 0.3124 0.3395 0.4768
dt 1.2152 1.1128 0.8066
Source: Bell et al. (2004)
It should be remarked that the approach employed in deriving these results rests on
a rather sharp distinction between preventive measures and treatment. Some would
argue, however, that such a distinction is not always easy to draw, especially in high-
prevalence environments. Granted as much, the general import for our approach is
clear: the absolute slopes of the functions in Figure 1 at Gt = 0 are too large; for in
these settings, there are many infected individuals other than prostitutes and truck-
drivers, and the cost-efficient bundle of interventions at low levels of spending will also
involve HAART, at least in some measure.
6.2 School-attendance subsidies
The two reference cases involve the following levels of the direct costs of education:
t = 185 and t = 800 t 90. The former proves to be no deterrent to full primary
1 2
education for all from 1990 onwards, even in the face of the epidemic. The size of the
latter, though clearly smaller than the opportunity costs of time spent in secondary
38
0,4
q (G ;D=1)
t t
0,3
0,2
t=10
0,1 t=11
t=12
0,0
0 250 500 750 1000
G
t
Figure 1: The effect of public spending on premature adult mortality
education, is surely a contributing factor to the very modest levels of e2t from 2000
until well towards the end of the horizon at 2040. This prompts consideration of a
substantial subsidy to secondary education, say, of 50 per cent. The resulting claims
upon the government's budget follow trivially from the corresponding course of e2t.
6.3 Health or education?
It would be unrealistic to expect indefinite budgetary support for such programmes.
Given the course of the epidemic, as expressed by qt(D = 1), the worst should be over
by 2030, so that one can imagine the possibility of a corresponding stream of grants
that ends in 2029. The exact stream, denoted by (A100,A110,A120), is generated by
the above proposal to cut t by a half from 2000 onwards. Since the course of demo-
2
graphic developments is exogenous under this particular proposal, the sums in question
are arrived at by repeating the simulation D = 1 with t = 400 and calculating the
2
39
Table 9: Education policy: school-attendance subsidy (D = 1)
Year 1990 2000 2010 2020 2030 2040 2050
e1t 1.000 1.000 1.000 1.000 1.000 1.000
e2t 0.000 0.127 0.194 0.329 0.455 0.605
2t 3.69 4.29 4.33 4.71 5.09 5.69 6.42
3t 4.52 3.69 4.81 5.30 6.34 7.53 9.19
yt 2647 2277 2430 2592 2863 3218
Yt(109) 29.97 36.04 48.49 58.64 68.94 82.70
Population (103) 23475 29818 33327 34617 35588 37349
Subsidya 0.94 1.29 1.82
a in % of the AIDS GDP level without any intervention.
product 0.5 · 90e2tNt (last row of Table 9). For the purposes of comparing interven-
2 2
tions in the two domains, the alternative is to allocate the same stream to measures
designed to combat the epidemic, as summarised in the function qt(Gt;D = 1), where
(G100,G110,G120) is suitably normalised to (A100,A110,A120) in order to allow for the
difference in the unit of observation (family and entire cohort of youth, respectively).
As can be seen from a comparison of Tables 7 and 9, the programme of subsidies to
secondary education produces a fairly substantial effect on e2t, and hence indirectly on
2t. In 2040, the productivity of both primary and secondary school graduates will lie
almost exactly half way between their respective levels in the reference cases D = 0,1.
The same holds for GDP per adult. Put somewhat differently, the programme reduces
the delay in attaining the counterfactual levels from ten years to five.
If these funds were allocated instead to combating the epidemic, not only would
there be a substantial fall in mortality which is desirable in itself but the associated
improvement in expectations would also promote investment in secondary education.
The latter effect is small at first, but then gathers pace, with a startling jump between
40
Table 10: Health policy (D = 1)
Year 1990 2000 2010 2020 2030 2040 2050
e1t 1.000 1.000 1.000 1.000 1.000 1.000
e2t 0.000 0.067 0.133 0.292 0.550 0.701
2t 3.69 4.29 4.33 4.64 4.95 5.52 6.38
3t 4.52 3.69 4.58 5.00 6.04 7.59 9.39
yt 2647 2346 2458 2554 2694 3058
Yt(109) 29.97 37.13 52.24 66.11 79.28 89.84
Population (103) 23475 29818 34633 37880 40937 41028
Subsidya 0.94 1.29 1.82
a in % of the AIDS GDP level without any intervention.
2020 and 2030. The reason for the latter is the reduction in Q2t and the increase in
full income per adult that stem from the cumulative reductions in premature adult
mortality up to 2030, since the ending of both spending programmes in 2029 leaves
them with identical mortality profiles thereafter. 3140 just exceeds the level attained
under the educational programme, while 2140 lies just below its corresponding level (see
Table 10). The lags involved are sufficiently drawn out, however, that y140 is still 5 per
cent lower. The reductions in mortality would lead to a more numerous population in
2040 (by about 10 per cent), and in 2050 new graduates are more productive. It is this
consequence of spending on measures to reduce premature adult mortality that make
them socially profitable against the alternative of spending directly on education at
least when the time horizon is long enough.
Yet another alternative is to combine the two programmes. Table 11 reports the
results from spending half the stream of educational subsidies paid out under the 'pure'
programme t = 400 (see Table 9) on measures to combat the epidemic and treat
2
its victims, while reducing the educational subsidy from 400 to 200. This combined
programme would cost 0.82 per cent of GDP in 2000, 1.20 per cent in 2010 and 1.79
41
Table 11: Combined Program (D = 1)
Year 1990 2000 2010 2020 2030 2040 2050
e1t 1.000 1.000 1.000 1.000 1.000 1.000
e2t 0.000 0.096 0.164 0.314 0.522 0.674
2t 3.69 4.29 4.33 4.68 5.02 5.62 6.44
3t 4.52 3.69 4.69 5.15 6.20 7.64 9.42
yt 2647 2313 2445 2574 2771 3138
Yt(109) 29.97 36.61 50.59 63.23 76.66 88.51
Population (103) 23475 29818 34070 36562 39172 39859
Subsidya 0.82 1.20 1.79
ain % of the AIDS GDP level without an intervention
per cent in 2020. As can be seen from comparing Tables 9, 10 and 11, the combined
program requires a smaller stream of subsidies, and yet results in a slightly higher level
of 3140 and then of both 2150 and 3150. This indicates a certain synergy between health
and education policy.
The full results for the variants with prolonged adult mortality D = 1 are not
reported here. Suffice it to say that `recovery' is only slightly delayed under all pro-
grammes. The levels of spending are somewhat lower than under D = 1, in keeping
with the lower levels of e2t (t = 100,110,120) that ensue from grimmer expectations.
The above-mentioned synergy is also evident in this variant, whereby the difference in
public spending streams is still larger than under D = 1.
6.4 Cost-benefit analysis
In view of the very different time paths of the streams of costs and benefits of these
three interventions, an evaluation of their social profitability demands that they be
discounted at an appropriate (real) rate. Four percent seems quite appropriate for
42
Table 12: The benefit-cost ratios of interventions
Intervention Aggregate GDP Per Capita GDP
r = 4% r = 5% r = 4% r = 5%
School-attendance subsidy 3.5 2.1 2.9 1.7
Health programme 16.3 13.5 14.6 11.7
Combined programme 13.3 10.6 11.7 9.0
public sector projects in Africa, for there is always the alternative of investing in
international paper of good quality. To be on the safe side, we also set the bar a
bit higher at 5 percent. Deciding on an appropriate welfare index is not at all
straightforward, because one of the policy instruments education subsidies will not
save lives, so that troubling ethical problems arise in comparing the above programmes.
One crude, but defensible, way of proceeding is to use per capita and aggregate GDP,
respectively, as (instantaneous) indices. Since only the latter gives any weight to the
number of individuals actually alive and then equally these two indices arguably
bracket the range of possibilities when output is the underlying measure of performance.
Table 12 reports the net present values of the streams of costs and benefits generated
by the three programmes, whereby the streams of benefits after 2040 are interpolated
from the differences in growth rates between 2030 and 2040 with and without the inter-
vention, respectively, and then discounted over an infinite horizon. Linear interpolation
is used to arrive at the estimates for all intervening years within each decade. All three
programmes pass the test with flying colours whatever be the choice of index; but the
two that involve the promotion of health have a benefit-cost ratio that is roughly four-
to five-fold that involving educational subsidies alone. This striking improvement is
due not only to the saving of lives, but also to the fact that the resulting reduction in
expected mortality provides an additional potent incentive to invest in education.
43
7 Concluding discussion
From about 1980 onwards, Kenya has run into increasing difficulties in three separate,
but related spheres, all of which pose serious threats to its citizens' long-term prosperity
and well-being. The first is a significant weakening of the mechanism through which
human capital is transmitted and accumulated from one generation to the next
so much so, that in the absence of improvements in the primary school system and
significant levels of secondary and higher education, the level of output per head may
well approach a ceiling not very many times higher than that ruling at present. The
second is a fall in the productivity of human capital, measured in terms of aggregate
output, a fall that surely has much to do with failings in policy-making and governance
during the past two decades.
The third is the outbreak of the AIDS epidemic, whose effects on mortality and
morbidity became evident by the early 1990s. A useful summary measure of the epi-
demic's economic effects is that, in the absence of any public intervention, it will delay
by about a decade the attainment of the levels of human capital in 2040 under the
counterfactual in which there is no outbreak.
The damage caused by these developments is not simply additive; rather they interact
in mutually reinforcing ways. Conversely, measures designed to remedy failings in one
sphere will relieve the problems in the other two, in part at least. Indeed, the results
for a combined programme of measures in the spheres of health and education reveal
a definite synergy among them. It is for this reason that any programme to combat
the spread of the epidemic, to treat its victims and to support needy families must be
complemented by reforms in the educational system and in economic policy in general.
All three elements play a central role in determining the expected returns to investment
in human capital, and hence the rate of growth over the long haul.
Where future work is concerned, we think it important to introduce the effects of
44
learning on the job where the young are concerned. Translating freshly derived edu-
cational attainment into output of goods and services is not accomplished overnight;
rather it can be likened to the maturing of wine. The waiting involved will reduce the
earnings, and hence the opportunity costs, of primary school graduates; but it will also
delay the stream of expected returns from all education. What the net effect turns out
to be is a matter of more than academic interest.
References
[1] Azam, J.-P., and Daubr´ee, C. (1997), Bypassing the State: Economic Growth in
Kenya, 1964-90, Development Centre, O.E.C.D., Paris.
[2] Altig, D. Auerbach, A.J., Kotlikoff, L.J., Smetters, K.A., Walliser, J. (2001), "Sim-
ulating Fundamental Tax Reform in the United States", American Economic Review,
91(3): 574-595.
[3] Appleton, S., Bigsten, A. and Manda, D.K. (1999), "Educational expansion and
economic decline: returns to education in Kenya, 1978-1995". CSAE Working Paper
Series, Working Paper 90, Oxford.
[4] Bell, C., Gersbach, H., Bruhns, R., and V¨olker, D. (2004), "Economic Growth,
Human Capital and Population in Kenya in the Time of AIDS: A Long-run Analysis
in Historical Perspective", mimeo, University of Heidelberg.
[5] Bell, C., Devarajan, S., and Gersbach, H. (2006), `The Long-run Economic Costs
of AIDS: A Model with an Application to South Africa', The World Bank Economic
Review, forthcoming.
[6] Corrigan, P., Glomm, G., and M´endez, F. (2004), "AIDS, Human Capital and
Growth", Indiana University, Bloomington, IN.
45
[7] Corrigan, P., Glomm, G., and M´endez, F. (2005), "AIDS Crisis and Growth",
Journal of Development Economics, 77(1): 107 124.
[8] Evans, D. and Miguel, E. (2005), "Orphans and Schooling in Africa: A Longitudinal
Analysis", Harvard University, Cambridge MA.
[9] Evenson, R.E., and Mwabu, G., (1995), "Household Composition and Expenditures
on Human Capital Formation in Kenya", Economic Growth Center, Discussion Pa-
per No. 731, Yale University.
[10] Fullerton, D., and Rogers, D. (1993), Who Bears the Lifetime Tax Burden?, Brook-
ings Institution Press, Washington, DC.
[11] Heston, A., Summers, R., and Aten, A., (2002), Penn World Tables Version 6.1,
Center for International Comparisons at the University of Pennsylvania (CICUP),
October.
[12] Kalemli-Ozcan, S. (2006), `AIDS, Reversal of the Demographic Transition and
Economic Development: Evidence from Africa', University of Houston, mimeo.
[13] Kenya, Central Bureau of Statistics, Kenya Population Census, various years,
Government Printer, Nairobi.
[14] Kenya, Central Bureau of Statistics (1996), Welfare Monitoring Survey, 1994,
Government Printer, Nairobi.
[15] Manda, D.K. (2002), "Globalization and the Labour Market in Kenya", Glob-
alization, Production and Poverty Discussion Paper No. 6, Overseas Development
Group, University of East Anglia.
[16] O'Connell, S.A., and Ndulu, B.J. (2000). "Africa's Growth Experience A Focus
on Sources of Growth", processed, Swarthmore College.
46
[17] Sheffield, J. R. (1973), Education in Kenya An Historical Study, Teachers College
Press, Columbia University.
[18] Thias, H., and Carnoy, M., (1972), Cost-Benefit Analysis in Education: A Case
Study of Kenya, World Bank Staff Occasional Paper No. 14, IBRD, Washington,
D.C.
[19] United Nations (2004), World Population Prospects, Online Database:
http://esa.un.org/unpp/
[20] Wambugu, A. (2003), "Essays on earnings and human capital in Kenya", School
of Economics and Commercial Law, Goeteborg University.
[21] Yamano, T., and Jaynes, T.S. (2005), "Working-Age Adult Mortality and Primary
School Attendance in Rural Kenya", Economic Development and Cultural Change,
: 616 653.
[22] Young, A. (2005), "The Gift of the Dying: The Tragedy of AIDS and the Welfare
of Future African Generations", Quarterly Journal of Economics, 120(2): 423 466.
47