WPS4919
P olicy R eseaRch W oRking P aPeR 4919
AIDS and Dualism
Ethiopia's Burden under Rational Expectations
Clive Bell
Anastasios Koukoumelis
The World Bank
Human Development Network-
Global HIV/AIDS Program
April 2009
Policy ReseaRch WoRking PaPeR 4919
Abstract
An AIDS epidemic threatens Ethiopia with a long alternative assumptions about mortality with and
wave of premature adult mortality, and thus with an without the epidemic. Although the epidemic does not
enduring setback to capital formation and economic bring about a catastrophic economic collapse, which is
growth. The authors develop a two-sector model hardly possible in view of Ethiopia's poverty and high
with three overlapping generations and intersectorally background adult mortality, it does cause a permanent,
mobile labor, in which young adults allocate resources downward displacement of the path of output per head,
under rational expectations. They calibrate the model amounting to 10 percent in 2100. An externally funded
to the demographic and economic data, and perform program to combat the disease is socially very profitable.
simulations for the period ending in 2100 under
This paper--a product of the Global HIV/AIDS Program, Human Development Network---is part of a larger effort in
the World Bank to assess the economic effects of HIV/AIDS, especially in Africa. Policy Research Working Papers are also
posted on the Web at http://econ.worldbank.org. The authors may be contacted at clive.bell@urz.uni-heidelberg.de or
anastasios.koukoumelis@urz.uni-heidelberg.de.
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 views of the International Bank for Reconstruction and Development/World Bank and
its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent.
Produced by the Research Support Team
AIDS and Dualism: Ethiopia's Burden under
Rational Expectations
Clive Bell and Anastasios Koukoumelis
First version: September, 2007
This version: March, 2009
Keywords: HIV/AIDS, Growth, Dualism, Ethiopia
JEL Classification: I10, I20, O11, O41
This paper a product of the Global HIV/AIDS Program, Human Development Network is part
of a larger effort in the World Bank to assess the economic effects of HIV/AIDS, especially in Africa.
An early version was presented at the Fifth Annual Meeting of the European Development Network
e
in Paris, 2007. We thank the participants, especially Alice M´nard, for valuable comments and
suggestions, and we are also much indebted to Hans Gersbach for extensive discussions of all aspects
of the study. Deepak Mishra provided a number of constructive comments on a recent version. We
absolve all of them of any responsibility for all remaining errors of analysis and opinion. The views
expressed in this paper do not necessarily reflect those of any of the institutions mentioned herein.
South Asia Institute, University of Heidelberg, Im Neuenheimer Feld 330, 69120 Heidelberg,
Germany. Tel.: +49-6221-548824, Fax.: +49-6221-545596. e-mail: clive.bell@urz.uni-heidelberg.de
South Asia Institute, University of Heidelberg, Im Neuenheimer Feld 330, 69120 Heidelberg,
Germany.
1
1 Introduction
Ethiopia is a so-called `next-wave' country where HIV/AIDS is concerned: populous,
in the early- to mid-stages of an epidemic, and with a government that some thought
to be rather tardy in recognizing the threat the disease poses (NIC, 2002). Some 1.2
million Ethiopians died of AIDS-related causes from the onset of the epidemic in the
early 1980s until the end of 2006, 111,000 alone in the latter year (MoH, 2006). In 2007
980,000 adults and children were living with HIV (UNAIDS, 2008). It is estimated that
in 2006 life expectancy at birth was lowered, purely as a result of AIDS, by 4.2 years
and 880,000 children had been orphaned by the disease.
A burgeoning AIDS epidemic is not Ethiopia's only burden. It is one of the world's
poorest countries, with low levels of physical and human capital: adults of working
age have, on average, about one year of schooling. The economy also exhibits strong
dualism, with clear differences between rural and urban families' living standards and
opportunities. High levels of fertility compound matters by increasing population pres-
sure on land, and so threaten to degrade the natural resource base. Both developments
are closely intertwined with stagnant agricultural productivity and internal migration
flows.1 These features of the Ethiopian economy must be incorporated into any anal-
ysis of its long-term development prospects, overshadowed as they are by the AIDS
epidemic.
Ethiopia merits attention as the object of study, not only for the above reasons in
themselves, but also for certain contrasts that it offers in relation to the much more de-
veloped economies of southern Africa, in which the epidemic is both mature and severe.
It can be argued that the wave of premature adult mortality that is now sweeping over
the latter is so great as to threaten a dramatic fall in productivity over the longer run.
Ethiopia's poverty leaves little scope for such a fall. Yet two questions remain. First,
1
See, inter alia, Hurni (1993), Grepperud (1996), and Shiferaw and Holden (1999).
2
does the AIDS epidemic pose a formidable obstacle to Ethiopia's attaining long-run
growth? Second, is it socially profitable to commit substantial resources to combating
the epidemic, even if its effects on the long-run rate of growth are modest? We seek to
answer both these questions, and in so doing to shed some light on the damage inflicted
by the pandemic on other parts of sub-Saharan Africa with similar characteristics.
To tackle this task, we employ an overlapping-generations (OLG) model of a dual
economy, in which the two sectors produce the same aggregate good and young adults
who grew up in villages can seek their fortunes in the towns. The model conforms to
the Ethiopian setting and experience in two ways: first, the growth of income depends
heavily on human capital formation,2 and second, the availability of land influences not
only intra-family allocations regarding education and child labor, but also population
movements.3 It draws on Bell, Devarajan and Gersbach (2006), whereby the extension
to two sectors and three generations greatly enlarges the set of variables over whose
future course agents must form their (rational) expectations. The resulting structure,
when calibrated to Ethiopia, is then used to simulate the path of the economy under
alternative demographic developments, namely, with and without the AIDS epidemic.
There is, of course, a substantial literature on the macro-economics of AIDS, but,
to our knowledge, not a single formal application to Ethiopia. Among a number of
notable studies, Cuddington (1993a) and Cuddington and Hancock (1994) use stan-
dard neoclassical (Solovian) models to simulate the impact of AIDS on macroeconomic
aggregates. The epidemic affects both the size and quality (health and accumulated
experience) of the labor force and the accumulation of physical capital (to the extent
that higher medical expenditures are financed out of domestic savings). In closely re-
lated frameworks, Over (1992) allows for infection to be concentrated among educated
2
`[T]he fact that the majority of Ethiopians remains uneducated explains in part the poor devel-
opment performance of the country over a prolonged period of time.' (Degefe et al., 2002: 218).
3
Abate (1995: 312) reports a statistically significant correlation between average land-holding and
urban population. Golini et al. (2001) and Mberu (2006) provide thorough analyses of internal
migration dynamics.
3
workers, and Cuddington (1993b) and Cuddington and Hancock (1995) incorporate
unemployment (workers who die from AIDS are replaced by the unemployed) and the
operation of dual labor markets (workers unable to secure employment in the formal
sector find lower-productivity employment in the informal sector). The macroeconomic
impact of AIDS depends, naturally enough, on the values of the parameters describing
the distribution of its prevalence among classes of workers, the speed of adjustment
in the labor market, and the proportion of health care costs that is financed out of
savings.
Cuddington, Hancock and Rogers (1994) add epidemiological elements to the one-
sector neoclassical growth model, and investigate the effects of health-sector policies
that either reduce the rate of transmission of HIV or increase the life expectancy of
AIDS patients. In this extended model, which allows for interactions between demo-
graphic and economic variables and the existence of multiple equilibria, cost-effective
policy interventions may propel the economy into a steady state in which the preva-
lence rate is lower or even zero. Finally, Young (2005) embeds a sub-model that
endogenizes labor force participation, fertility, and education decisions in a Solovian
framework with a constant savings rate. Higher mortality induces lower fertility, both
directly and in response to the resulting higher wages, and so keeps the population
below its current size for 70 years. This contraction of labor supply outweighs the
reduced accumulation of human capital among orphaned children, and individuals in
the surviving population (and succeeding generations) are, on average, correspondingly
better endowed with resources and enjoy higher levels of income.
A few country-specific studies are based on computable general equilibrium models
(Kambou, Devarajan and Over, 1992; Arndt and Lewis, 2001; and Arndt, 2003). Their
dynamic structure is similar to that of the standard neoclassical growth model, but
they go a step in the direction of realism by allowing some sectoral disaggregation of
GDP, and hence of disproportionate effects across sectors (those that use factors which
4
are more heavily affected by the epidemic tend to contract). As is well known, the
estimation and calibration of such models is very demanding of data, and numerous
assumptions about the constituent structural relationships are necessary.
Studies that employ OLG models often focus on how the epidemic influences hu-
man capital accumulation. In Bell, Bruhns and Gersbach (2006) and Bell, Devarajan
and Gersbach (2006), the accumulation of human capital is the force that generates
economic growth over the long run. Premature adult mortality destroys existing skills
and experience, and weakens the mechanism through which knowledge is transmit-
ted from parents to children. With reduced life-time resources, the level of schooling
may fall among affected families. Even when both parents survive, their expectations
concerning mortality rates among their own and their children's generation will still
influence decisions concerning current investments in schooling. The system normally
exhibits two or more equilibria, and if nothing is done to combat a sufficiently large
and sustained shock to mortality, the economy may fall into a poverty trap.
Corrigan, Glomm and Mendez (2004) concentrate on the fall in life expectancy, which
decreases the incentive to invest in all forms of capital, and the associated creation of
a large number of orphans. HIV-infected individuals face medical expenditures and do
not save during adulthood; uninfected individuals save a portion of their income for old-
age consumption. The level of human capital accumulation depends on whether parents
die prematurely due to HIV/AIDS, and on the amount of time that children spend
pursuing formal education rather than working. The macroeconomic consequences of
the epidemic are large, especially if the mortality shock persists.
The plan of the paper is as follows. Section 2 begins by setting out the basic approach,
followed by brief accounts of the sub-models of the rural and urban sectors, and of the
momentary equilibrium of the system, in which rural-urban migration under rational
expectations plays a central role. The section concludes with an examination of the
system's long-run behavior. The calibration of the system to the data for Ethiopia is
5
treated at some length in Section 3,4 thereby laying the groundwork for the results in
Section 4. The latter begins by setting out the sequences of the exogenous variables,
and then analyzes and compares the reference case, in which the epidemic runs its
projected course, with the counter-factual, in which it never breaks out. Turning to
policy, the question of what objectives should be pursued is addressed in Section 5,
the relation between public spending and mortality is estimated in Section 6, and
the social profitability of a particular policy program funded by aid is evaluated in
Section 7. Some concluding comments are drawn together in Section 8.
2 The model in outline
The basic economic unit is an extended family household, whose members pool their
resources.5 The following assumptions apply to urban and rural households alike. In-
dividuals live for up to three periods: childhood, adulthood and old age. Children may
divide their time between working and learning6 and all children within the family are
given the same schooling. On reaching adulthood, these individuals are homogeneous
in skills and preferences, and supply their labor completely inelastically. Those who
were born in villages also decide, at the very outset, whether they will stay or seek
their fortune in the towns, where some members of the extended family are already
established. Wherever young adults settle down, they raise children for altruistic as
well as self-interested reasons. A social rule determines the level of consumption in
the third period of life, with young adults making transfers out of the family's current
income to their parents. The old make no bequests, except of land.
4
The details of the algorithm for computing a rational expectations sequence are consigned to
Appendix B.
5
Schaffner (2004: 13), for example, reports that 89% of orphans live with their relatives.
6
According to the CSA (2001), in 2001, 49.0% of children aged 5 to 14 and 67.4% of those aged
15 to 17 were working. `All studies made on child labor in urban areas identified household poverty
as the major and primary cause of child labor'. (Tegenu, 2003: 28).
6
In what follows, we begin by describing demography and household behavior in the
rural and urban sectors of the economy, which produce the same aggregate good, but
are connected by the migration of young adults. This requires a full set of demographic
accounts for the entire economy, in which the profile of age-specific mortality rates plays
a leading role. Momentary equilibrium is brought about by a level of migration such
that young adults who have grown up in villages are indifferent between migrating to
the towns and staying on the family farm. The sequence of momentary equilibria is
such that agents' expectations are realized along the whole path.
2.1 The rural sector
The rural sector (labelled 1) is made up of a very large number of identical families.
Shortly after the beginning of each period, when the surviving children from the previ-
ous period have just attained adulthood, these young adults decide whether to migrate.
Only after choosing where to live do they form unions and have children, in rural areas
c
N1,t per union in period t. Death can come early to these adults, first, just after they
have had their children, and then once more later, just before reaching old age. Denote
ae al
these probabilities by q1,t and q1,t , respectively, so that the survival rate among all young
a ae al
rural adults, conditional on their surviving childhood, is (1 - q1,t ) (1 - q1,t )(1 - q1,t ).
Under the social rules, those young adults who survive the early phase are charged with
caring for all children, natural and adopted alike. Normalizing each household around
a c c ae
a surviving young `couple', N1,t = 2 t, let N1,t N1,t /(1 - q1,t ) denote the number
of children raised by such a couple in period t. The representative rural household is
o
completed by N1,t elders, all of whom die at the end of period t.
The population evolves as follows. Let all children born in villages stay there until
adulthood, and denote the probability that a child born in period t will survive into
c c c
adulthood in period t + 1 by (1 - q1,t ). Of the (1 - q1,t-1 )N1,t-1 children born to a
7
couple in period t - 1 who attain adulthood in period t, let Mta choose to migrate.
f
Denoting by N1,t the number of unions formed in the rural sector, just after the start
of period t, the above normalization to surviving couples implies that the number of
representative families (households) in the rural sector, F1,t , changes according to
ae f ae
c c
(1 - q1,t-1 )N1,t-1 - Mta f
F1,t = (1 - q1,t ) · N1,t = (1 - q1,t ) · · N1,t-1 . (1)
2
The number of elders per household in period t is thus given by
o al a
N1,t = (1 - q1,t-1 )N1,t-1 · (F1,t-1 /F1,t ). (2)
Each young adult possesses 1,t efficiency units of labor, which is a natural measure
of human capital. This is formed in a process whereby the benefits of formal education
are complemented by the quality of child-rearing, which depends on the human capital
of all the adults in the family.7 Let e1,t [0, 1] denote the fraction of childhood devoted
to formal education, the residual being allocated to work. Choosing the iso-elastic form
b1 · (e1,t )1 for the `educational technology', let a child's endowment of labor (measured
in efficiency units) on reaching adulthood in period t + 1 be given by
a o
1,t+1 = (1,t N1,t + 1,t-1 N1,t ) · b1 · (e1,t )1 + 1, (3)
where the product on the RHS implies that formal education and the quality of child-
rearing are indeed complements. It is plausible that < 1, that is, grandparents
contribute less, cet. par. , than parents to the children's potential capacity to build
human capital. Some education is required if 1,t+1 is to exceed unity, the human
capital of an uneducated young adult.8
7
For empirical evidence on the latter point, see Christiaensen and Alderman (2002) and Wolde-
hanna et al. (2005).
8
This is just a convenient normalization.
8
The aggregate good is produced by means of human capital and land. The higher
the level of rural-urban migration, the larger is the number of elders associated with
each rural young couple. The migrants' departure also relieves pressure on the land, for
those young adults who remain will have sole claim to the family's holding, t . Given
a fixed area available for cultivation, t evolves according to
t+1 = (F1,t /F1,t+1 ) · t . (4)
where the sequence {t }t=T is anchored to the observed value of 1 .
t=1
The resulting output is distributed among the members of the family as follows. Let
c1,t denote the consumption of a young adult, and let children and old adults consume
the constant fractions and , respectively, of c1,t , whereby the migrants have no
further claim on the household's resources. The parameters and define a social
sharing rule.
Subject to this rule, all allocative decisions within the family lie in the hands of
the young adults. Their choices determine not only all family members' current levels
of consumption, but also their children's level of human capital on reaching adult-
hood, and hence the family's consumption possibilities in the next period. It is thus
implied that, although the Ethiopian government is committed to the provision of
free and compulsory primary education for all children, school attendance in practice
depends on the household's income, its needs for child labor (which conflict with school-
ing), and parental awareness and appreciation of the benefits of education (Schaffner,
2004: 1). Those young adults who remain in the village choose an optimal allocation
(c0 , ep 0 | Mta ), which depends on the level of full income, the relative price of education,
1,t 1,t
the strength of their altruism towards their children, and their (subjective) estimates of
the levels of mortality that will afflict them later in young adult life and, in turn, their
children in adulthood. In choosing this optimum, they take the number of departing
9
migrants as parametrically given. The formal statement of their decision problem is
given in Appendix A.
By way of closing remarks, first, it is assumed that parents cannot borrow to finance
schooling (Schaffner, 2004: 17). Second, the only form of investment is education,
the costs of which are the children's potential contributions to current output. Indeed,
rural households in Ethiopia save little (Tengenu, 2003: 15) and cannot purchase formal
insurance against risk (Dercon, 2004). Third, since all households are assumed to be
identical, there is no trade in the services of labor and land.9
2.2 The urban sector
The number of children (per representative family) born in the urban sector (labelled
c c
2) in period t - 1 that survive into young adulthood in period t is (1 - q2,t-1 )N2,t-1 .
f
The total number of migrants arriving at the start of period t is Mta N1,t-1 . Hence,
the total number of young adults who form unions in urban areas at this juncture
f f
c c
is 1 - q2,t-1 N2,t-1 N2,t-1 + Mta N1,t-1 . Given the definition of Fi,t as the number of
surviving couples, we have
f f
ae
c c
1 - q2,t-1 N2,t-1 N2,t-1 + Mta N1,t-1
F2,t = (1 - q2,t )·
2
ae
(1 - q2,t ) c c Mta F1,t-1
= · (1 - q2,t-1 )N2,t-1 F2,t-1 + ae
. (5)
2 (1 - q1,t-1 )
The number of elders per urban household is, accordingly,
o al a
N2,t = (1 - q2,t-1 )N2,t-1 · (F2,t-1 /F2,t ). (6)
9
Land is state-owned and distributed equally among farmers on a strict usufruct basis by local
councils, though leasing is allowed (OPM, 2004: 3). Wage labor, whether in agriculture or otherwise,
remains relatively rare, and many off-farm activities (such as selling home-made products) are closely
linked to agricultural activities (Dercon, 2004: Sec. 5). Indeed, according to a survey conducted by
the Ministry of Labor in 1996, 44% of rural households reported non-agricultural sources of income,
but these sources accounted for only 10% of total income (Degefe and Nega, 2000: 179).
10
In effect, migrants join the established members of their extended kin groups, con-
tributing their labor and making the conventional claims on the common pot. They
are also assumed to adopt their urban counterparts' preferences, including those con-
cerning fertility, and make the same contributions to savings for the next period and
to the formation of human capital among the upcoming generation.
The urban sector differs from its rural counterpart in two other respects. First, as
there is some participation in secondary and higher education, one needs to distinguish
between children of primary school age (5-14 years) and youths aged 15 to 24. Extend-
cp
ing the notation of Section 2.1, let there be N2,t of the former, with net attendance
cp
ratio ep , and N2,t of the latter, with net attendance ratio es , where N2,t = N2,t + N2,t
2,t
cs
2,t
c cs
in order to make up the entire 20-year generation. Second, although families working
in the industrial and services sectors are landless, they may be endowed instead with
physical capital, which is accumulated through the savings of young adults. As in the
rural sector, the technology for producing the aggregate good exhibits constant returns
to scale, but now in efficiency units of labor and physical capital.
Analogously to eq. (3), the formation of human capital is assumed to be governed by
2
a o
(2,t + t 1,t )N2,t (2,t-1 + t-1 1,t-1 )N2,t ep + es
2,t 2,t
2,t+1 = + · b2 + 1 (7)
1 + t 1 + t-1 2
where
f f
Mta · (N1,t-1 /N2,t-1 )
t c ae c
(8)
(1 - q2,t-1 )(1 - q2,t-1 )N2,t-1
defines the ratio of newly arrived young migrants from the villages to the `native
young townies'. In effect, primary and post-primary education are treated as perfect
substitutes. This is certainly a strong assumption, but so long as ep is close to unity,
2,t
it is not particularly objectionable.
The level of physical capital in the next period, kt+1 , depends on savings in the
11
current period, st , it being assumed that the whole of the current capital stock lasts
for just the twenty years spanned by each new generation. Recalling that all young
adults are assumed to have their children before some of them are carried off in the early
ae
phase of adulthood (to be precise, the proportion q2,t ), and that the surviving children
themselves form couples at the start of the next period, the capital owned by each
representative urban household, normalized to a surviving couple, evolves according to
ae a
1 - q2,t 2N2,t st
kt+1 = ae
· c cp cs
. (9)
1 - q2,t+1 (1 - q2,t )(N2,t + N2,t )
The young adults' preferences are, in principle at least, defined over three goods: the
levels of consumption in young adulthood and old age, c2,t and c2,t+1 , respectively, and
the human capital attained by their school-age children on attaining full adulthood,
2,t+1 , which they may appreciate in both phases of their own lives. The latter quantity
depends heavily on the children's education, (ep , es ), and it produces two kinds of
2,t 2,t
pay-offs, namely, in the form of altruism and selfishly, inasmuch as an increase in 2,t+1
will also lead to an increase in c2,t+1 under the said social rules.
Although the pooling arrangement implicit in the extended family structure elimi-
nates the risk that orphaned children will be left to fend for themselves, 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 influence the levels of c2,t+1 and 2,t+1 .
Young adults in the current period must form expectations about the level of con-
sumption in old age should they survive into that phase of life. Forecasting consump-
tion in the next period is a formidable task, involving forecasting st+1 , ep , es
2,t+1 2,t+1 and
t+1 , all under rational expectations about the behavior of subsequent generations. Like
their rural counterparts, their task is to choose an optimal plan, albeit a complicated
12
one to compute: [c2,t , st , ep , es |ep , t , Et (st+1 , ep , es , t+1 )].
2,t 2,t 1,t 2,t+1 2,t+1
2.3 Momentary equilibrium
At the beginning of each period, the economy's endowments of productive resources
and their distribution among households are given. The two sectors are connected
by the flow of young rural migrants into the towns, the level of which all households
take as parametrically given when making their decisions concerning the allocation of
full income between investment and consumption. The central question in the deter-
mination of momentary equilibrium, therefore, is this: what governs the size of this
flow? In keeping with the general approach adopted here, we assume that on reaching
adulthood, members of the rising generation in villages decide whether to migrate to
the towns on the basis of the expected lifetime utilities, Et ui (i = 1, 2), offered by the
two locations. In each and every period, the endogenous variables of the system must
satisfy the condition
Et u1 (c0 , ep 0 ) + du = Et u2 (c0 , c0 , ep 0 , es 0 ),
1,t 1,t 2,t 2,t+1 2,t 2,t (10)
where du 0 represents the value of those net compensating advantages of rural life
not covered in Et u1 . Observe that agents' (rational) expectations concerning future
developments that have a bearing on current decisions appear in Et u2 . The sequence
of equilibria satisfying eq. (10) is anchored to the observed levels of investment in
education in the year 2000 (t = 1).
2.4 The system's long-run behavior
The dual economy formulated above constitutes a system of simultaneous, non-linear,
second-order difference equations, whose (exogenous) `forcing functions' are the se-
13
c t=T
quences of fertility, {Ni,t }t=1 , age-specific mortality, {qi,t }t=T , and total factor pro-
t=1
ductivity, {Ai,t }t=T , which are common knowledge to all economic agents. There is
t=1
no closed-form solution in the general case, and all the results presented in subse-
quent sections necessarily relate to transitional paths that are anchored to the initial
conditions in the period 1980-2000. Even in 2100, the last period of the sequences
that are computed under rational expectations, the economy is still far removed from
a state of steady growth in which the rural sector has shrunk to a mere appendage
and rural-urban migration has come to an end. In order to understand and interpret
these results, however, it is instructive to examine the steady-state path that will be
asymptotically attained.
The steady state in question is characterized by the said exogenous variables tak-
ing stationary values such that there is also a stationary population, and by parents
choosing a full education for their children. On reaching this state, the system becomes
tractable and linear, as we now demonstrate. Starting with the rural sector, and noting
that there is now no rural-urban migration, eq. (3) specializes to
al
1,t+1 = 2(1,t + (1 - q1 )1,t-1 ) · b1 + 1.10 (11)
With a stationary landholding and a Cobb-Douglas production technology, output per
family will grow at a steady rate equal to the rate of growth of A1,t plus the product
of the growth rate of 1,t and the elasticity of output with respect to inputs of labor
measured in efficiency units. Turning to the urban sector, eq. (7) specializes to
al
2,t+1 = 2(2,t + (1 - q2 )2,t-1 ) · b2 + 1. (12)
In a steady state, savings decisions will be such that kt grows at the same rate as 2,t ,
and with constant returns to scale in production, output per family will grow at the
10 al al
With a stationary population, qi,t = qi i, t.
14
sum of that rate and that of A2,t . The last step is to check that the urban sector is
indeed growing faster than the rural sector, thereby verifying that dualism has come
to an end. All this will be done in Section 4.
3 Calibration
Ethiopia's turbulent history in the second half of the 20th Century and the paucity
of relevant data pose formidable challenges to all forms of calibration and rule out
any that are heavily based on historical series.11 The method adopted here involves
anchoring the whole system to the observed values of key variables in the year 2000,
which is the starting point of all the simulations in the sections that follow.12 This is
a minimum requirement; but with a plethora of parameters to be determined, it is far
from sufficient. It is therefore supplemented by `guesstimates' based on what appear
to be reasonable values or regularities in cross-country data.
3.1 The rural sector
The values of the following variables in 2000 are observed: enrollments, average family
landholding, and all demographic magnitudes. In the absence of any estimates of the
years of completed schooling for this cohort, we have used USAID's (2005) estimate of
the net attendance ratio among children of primary school age, namely, ep = 0.243.
1,1
The average family landholding, 1 , is 1.02 hectares (OPM, 2004: 3). We define as
11
In particular, it is difficult to estimate the key relationship between changes in human capital
and economic performance. For example, enrollment rates increased by roughly 15% under both the
imperial regime, which was brought down by a coup in 1974, and the junta that followed (Negash,
1990), but GDP per capita increased under the first and decreased under the second. Similar difficulties
arise in evaluating the contribution of investment in physical capital. An examination of the data set
assembled by Nehru and Dhareshwar (1993) reveals that the growth rate of GDP per capita was
negatively correlated with the level of average gross domestic fixed investment.
12
The choice of base year naturally involves some compromises. Our choice was heavily influenced
by the availability of demographic, health and educational data. For example, 2000 is the only year
for which data on rural and urban attendance ratios are available.
15
`children' those persons falling between the ages of 5 and 24 years, whose number must
c ae c
be normalized to that of surviving young adults: recall that Ni,t (1 - qi,t )Ni,t . As
the main costs of schooling are the opportunity costs of this group's time and very
young children make limited claims on consumption, we take no account of the latter
in our calculations. Hence, the numbers of children, adults and elders in each family
are assigned according to the relative sizes of the cohorts 5-24, 25-44 and 45+, so
that the span of each generation is effectively 20 years. Recalling the normalization
a c o
N1,t = 2 t, we obtain N1,1 = 4.64 and N1,1 = 1.40 from CSA (2000). The premature
c a c
mortality rates are q1,1 = 0.062 and q1,1 = 0.206, where the latter implies N1,1 =
(1 - 0.1089) × 4.64 = 4.135.
We now turn to those parameters that must be set by guess and judgement. Begin-
ning with the production and consumption of the aggregate good, the values of 1,t are
not, of course, directly observable. In view of the very limited levels of education and
literacy of the rural population even up to the present, but with some progress over the
past generation, eqs. (3) and (7) suggest values of 1 in 1980 and 2000 that are close to
one: let 1,0 = 1.15 and 1,1 = 1.40. With these values as reference, we set the human
capital of a child at = 0.4.13 An adult's labor capacity is assumed to depreciate at
the rate = 0.2 per generation. The technology is taken to be Cobb-Douglas. The
initial value of the efficiency parameter can be chosen quite freely, since it does not
influence the steady-state behavior of 1,t ; hence, A1,1 = 1. Given the fact that, in
poor countries, the share of (imputed) rent in agricultural value added lies between
0.3 and 0.4, we choose 1 - 1 = 0.333, where 1 denotes the elasticity of output with
respect to inputs of labor measured in efficiency units. In the absence of data on the
consumption of children and elders relative to young adults, we postulate = = 0.6.
If parents choose to give their children a full primary education, consumption and
13
Cockburn (2002) claims that in rural Ethiopia "the marginal productivity of children is roughly
one-third to one-half that of male adults".
16
income can grow so long as the transmission of knowledge across generations, as repre-
sented by the parameter b1 , is sufficiently strong, even if A1,t grows slowly and the size
of the holding falls. In this connection, grandparents are assumed to be less capable
than parents in imparting knowledge: let = 0.75. When schooling is incomplete, the
parameter 1 comes into play. We choose b1 = 0.35 and 1 = 0.4.
We are not aware of any micro-economic studies of rural household's preferences
over current consumption and children's educational attainment. We assume that the
preferences are Stone-Geary, with `taste' parameters and 1 - , respectively. In view
of the fact that these households are very poor and that investment in education is not
wholly altruistic, but also produces a pay-off for the parents in old age, we set = 0.8.
The `subsistence' parameter c1,min is a flexible component of the algorithm that yields
the rational expectations path of the economy.14 Finally, ep
1,min is set to an arbitrarily
small value, namely 0.001.
3.2 The urban sector
The starting values of education are ep = 0.740 and es = 0.502 (USAID, 2005).
2,1 2,1
Where the magnitude of k1 is concerned, recall that the household's savings in period
t form the entire investment fund for the next period. Since capital, so defined, is a
stock, the capital-output ratio has the dimension of time, the unit of which in this
framework is 20 years.15 Hence, the usual magnitude of 3 (years) translates here into
0.15. The value of k1 actually chosen, namely, 1.0, satisfies the requirement that
the path of kt thereafter be tolerably smooth. It yields, along with the initial values
of other variables and various parameters, the rather modest capital-output ratio of
14
The resulting value is reported in the notes to Table 2.
15
Marquetti (2004) provides estimates of the net fixed standardized capital stock for the period 1963-
2000, based on the sum of depreciated past aggregate investment(the perpetual inventory method),
including both gross residential capital formation and changes in stocks. He assumes that equipment
and structures represent 20 and 80 per cent, respectively, of gross capital formation, and that both
categories of asset have the same life of 14 years.
17
a
2.57 (= 20 × 1.00/6.964) years (see Table 2). With N2,t = 2 t, CSA (2000) yields
c o
N2,1 = 4.15 and N2,1 = 0.990. The adult premature mortality rate is set with that in
the rural sector, where HIV-prevalence rates are much lower, so as to yield the estimated
a cp cs c
national average. We obtain q2,1 = 0.299 (we assume N2,t = N2,t = N2,t /2 t).
We retain the values of , and in the rural sector. Given the much higher
educational attainments of town-dwellers, the initial value of 2,t should be pitched
well above its rural counterpart.16 In the light of the prevailing differences in the
quality of schooling and the attainment in vocational and higher education, it seems
safe to conclude that in the year 2000, the level of human capital among adults in
the cities was at least twice that among those in the country: we set 2,0 = 2.3 and
2,1 = 2.8. We are no longer at liberty to choose the efficiency parameter A2,1 as we
please; for there are (crude) estimates of the productivity differential between the two
sectors, and these must be largely respected. This consideration yields A2,1 = 2.0.
Since inputs of labor fully reflect human capital, we choose 2 = 0.667.
Turning to the preferences of young urban adults, let these be Stone-Geary once
more, with parameters j (j = 1, 2, 3, 4), which relate to current and future con-
sumption and the two levels of education, respectively. The parameters 3 and 4
are endogenously determined so that the optimal values of primary and secondary
education in period 1 are equal to 0.740 and 0.502, respectively. The remaining i
parameters are obtained from the normalization 1 + 2 + 3 + 4 = 1 and the rate
of pure impatience in consumption, 1 - 2 /1 . We are unaware of any attempts to
estimate this rate for Ethiopia, but for the U.S., Fullerton and Rogers (1993) find that
it lies between 0.25% and 0.5% per annum. With this as a guide, we set 2 /1 = 0.85.
Finally, ep s p
2,min = e2,min = e1,min , and the subsistence parameter c2,min is assumed to
about 17% larger than its rural counterpart.17
16
Adult illiteracy rates in urban and rural Ethiopia in 1999 were 28.0% and 77.9%, respectively,
(ILO, 2005).
17
According to the 2000 Welfare Monitoring Survey (World Bank, 2003), the poorest 20% of house-
18
3.3 Population
c ae
Knowing the size of the rural population in the year 2000, P1,1 , as well as N1,1 /(1-q1,1 ),
a o
N1,1 and N1,1 , we can obtain F1,1 from
c a o
P1,t = N1,t + N1,t + N1,t F1,t . (13)
Using eq. (2), we then obtain F1,0 .18
Similarly, for the urban population, we have
c a o
P2,t = N2,t + N2,t + N2,t F2,t (14)
which yields F2,1 ; F2,0 then follows from eq. (6). For any sequence {Mta }t=T , it is
t=1
then possible to calculate all the demographic variables Fi,t and Pi,t (i = 1, 2) in all
subsequent periods. Note that the mortality rates in the early and late phases of
ae al
adulthood, qi,t and qi,t , are assumed to be equal for all i and t.
holds in both sectors spent `almost the same amount' (Birr 49 compared to Birr 42).
18 c a
Values of qi,0 and qi,0 are also required. These are calculated using the following mortality rates
from World Bank (2003):
Adult female Adult male Infant Under-5
1980 400.98 490.97 143.00 213.00
2000 535.00 594.00 117.00 174.00
a
where qi,t is the average over the two sexes. Child mortality in 1980 is proportional to child mortality in
c c
2000, that is, q1,0 = q2,0 = (213/174)·0.062 = 0.0759. Similarly, for premature adult mortality in 1980,
a a
assuming that the sectoral mortality profiles do not differ: q1,0 = q2,0 = [(400.98 + 490.97)/(535.00 +
594.00)] · 0.200 = 0.158. Our data indicate fewer children in period 0 than in period 1, but the World
Bank (2003) reports total fertility rates of 6.60 in 1980 and 5.65 in 2000. Note that Pi,t corresponds
to the population aged 5 or more years.
19
4 The results
Given the initial conditions, the system evolves under the influence of demographic
developments and (disembodied) technical progress, which are treated as exogenous.
c
Recalling Section 2.4, these are the sequences of fertility, {Ni,t }t=T , age-specific mortal-
t=0
ity, {qi,t }t=T , and total factor productivity, {Ai,t }t=T . In order to establish the effects
t=0 t=1
of the epidemic, we need these sequences for two disease environments: first, there is
the counter-factual (denoted by D = 0), in which the epidemic never appears; second,
there is its actual and projected course (denoted by D = 1).
We begin with demography. Although there are projections for the overall Ethiopian
population by age group till 2050 (UN, 2004), no such projections exist for the two
sectors separately. To produce them, we assume that there will be a complete demo-
c
graphic transition to stationary fertility levels in five periods. To be precise, Ni,t is
assumed to decline linearly from its current value (t = 1) to 2.041 at the start of period
a
5 (see Table 1). The series for premature mortality among adults, q1,t (D = 1), was
estimated on the following basis. (i) Drawing on the experience from the epidemic else-
where in Africa, Loewenson and Whiteside (1997) conclude that: the spread of HIV
becomes exponential when about 2% of the sexually active population are infected;
the ensuing period until the peak is reached is comparatively short, probably 10 years
or less; and the epidemic may then become endemic, with perhaps about 10% of the
sexually active population being HIV-positive at any one time. (ii) The (estimated)
upswing in prevalence rates in Ethiopia's rural sector (see Figure 1) will probably gain
momentum;19 for urbanization is commonly associated with return flows, including
those associated with the agricultural peak periods, all of which intensify the spread
of HIV/AIDS (Degefe and Nega, 2000: 74). A notable feature of D = 1 is that mor-
a
tality reaches its peak before 2020. The counterfactual series for qi,t (D = 0) has been
19
This is consistent with Hladik et al. (2006). Their analysis suggests an `expanding HIV epidemic
in rural and all Ethiopia, but a possible decline in some urban areas'.
20
interpolated from the W.H.O. life tables for Ethiopia for the year 2000.
Where the growth of total factor productivity (TFP) is concerned, we proceed as
follows. The growth rate of A1,t , denoted by gA1,t , is set at a level such that the resulting
rates of income and consumption growth in agriculture, together with that of rural-
urban migration, remain plausible. If, at first glance, the annual rate of 1% (i.e., 22%
over 20 years) looks rather low, it should be recalled that the accumulation of human
capital will also contribute to the growth of individual productivity. In the urban sector,
we choose, parsimoniously, gA2,t = 0. This apparently startling assumption does not
rule out the growth of individual productivity, even if physical capital per head stays
constant. For, as shown below, the parameters of the educational technology allow
unbounded growth of individual human capital when all children are fully educated,
and the production technology is Cobb-Douglas.
There remains the (exogenous) quantity du in eq. (10). Since (expected) `utils' are
not observable and the purist position that du = 0 is a useful benchmark, even when
many factors have been omitted from the model, we sought results with du as close to
zero as the algorithm would permit when given a day or two in order to converge. With
du = 0.5, convergence became very slow indeed, and there we left it provisionally.
As it turns out, the resulting value of c1,1 would have to be about 17% larger in order
that a young rural adult be indifferent between the two locations, which seems quite
modest and so justifies the decision to proceed with du = 0.5.
As a final preliminary, we calculate the rate of growth in the steady-state setting of
Section 2.4. With the parameter values adopted above, eq. (11) becomes, for D = 1,
1,t+1 = 2(1,t + 0.75(1 - 0.05)1,t-1 ) · 0.35 + 1,
whose relevant characteristic solution is 1,t = 1,0 (1.139)t . Hence, the sector grows
asymptotically at the rate 0.315 (= 0.222 + 0.667 × 0.139) per generation, or 1.38% p.a.
21
For the urban sector, eq. (12) yields 2,t = 2,0 (1.368)t , or 1.58% p.a., which is also the
whole economy's asymptotic rate, and which is finally approached from above.
4.1 The projected course with AIDS
Table 2 presents the results for the reference case D = 1. In the rural sector, the human
capital of a young adult grows, on average, at about 32% per period over the whole
time horizon, which is over twice the asymptotic rate. This outcome is the immediate
result of two developments. First, there is the rise in educational attainments early on,
with full primary education attained by period 3. Second, after heavy out-migration
in period 2, the number of elders per family, all of whom contribute to the capacity of
their grandchildren to build human capital, reaches a minimum in period 3, and then
increases substantially.
Land per family decreases until period 5, the consequence of a fixed overall cultivable
area, high fertility in the early periods and insufficient relief from outmigration. The
a c c
upturn is then quite sharp. With Ni,t = 2 t and (1 - q1,1 )N1,1 = 3.879, i.e., with the
size of the cohort of rural young adults expected to almost double over the period 2000-
2020, then under any reasonable assumptions about the growth of TFP, agriculture will
fail to generate much of a livelihood for them, and spurred on by a narrowing of the
gap between urban and rural mortality rates, outmigration is indeed heavy in period
2. With the relief offered by this alternative, consumption per young adult grows
throughout, increasing by about 22% in the first period, and then accelerating to 55%
in the fifth, yielding an average rate over the whole horizon of about 39% per period,
thus exceeding the asymptotic rate of 31.5%.
Turning to developments in the towns, despite migration throughout, only 60% of
the population is urban in period 6.20 Even so, this internal `brain drain' of young
20
In reality, the rate of urbanization is affected not only by the number of young adults who move
from their villages to the cities, but also by population growth in some rural settlements, which
22
adults has its drawbacks, not only for rural families, but also for urban ones. For there
are substantial differences in the levels of young adults' human capital across the two
sectors, so that the arrival of migrants reduces the average level of human capital in
urban areas and slows down the rate of growth of 2,t . In any event, ep and es both
2,t 2,t
reach unity in period 3, so that thereafter, it is only the contribution of the family's
adults to the children's capacity to build human capital that affects the evolution of
2,t . As t peaks in period 2, the influence of the term (1,t t + 1,t-1 t-1 ) / (1 + t )
weakens in later periods, and the growth rate of 2,t pick ups strongly: the average
growth rate of 2,t over the whole horizon is about 23% per period, well below its
asymptotic rate. Thus, while the level of urban young adults' human capital in the
cities is always higher than their rural cousins', the ratio thereof falls from 2.0 in
period 1 to 1.44 in period 6.
A salient feature of the urban sector is the use of physical capital in production.
Premature adult mortality has an adverse impact on the savings of young adults, and
yet st drops by 20% between periods 1 and 2, before starting to accelerate, attaining
almost two and a half times its initial level in period 6. Other factors are also at work,
of course; so that a comparison with the counter-factual D = 0 is needed to settle
matters (see Section 4.2 below). The consumption of a young adult increases six-fold.
A brief comment on the main national aggregates is also required. From Table 2 it
is seen that the total population of those aged 5 and older increases by a factor of 4.3
over the time horizon from 2000 to 2100. If this seems rather staggering, the reader is
reminded that the Ethiopian population doubled over the past 25 years alone. Given
the country's very youthful population, only a much faster demographic transition or
a sharp reduction in completed fertility due to the HIV/AIDS epidemic will impose a
much less numerous population in the steady-state.
The other index of central importance is GDP per head. Let Y1,t y1,t F1,t and
gradually acquire the organizational characteristics and judicial status of towns.
23
Y2,t y2,t F2,t denote the levels of aggregate income in the rural and urban sectors,
respectively. Their sum, Yt Y1,t + Y2,t , is aggregate income (GDP) when the two
sectors are combined, and yt Yt /Pt is income per capita in the overall economy. The
latter increases by 50% in the first period, decelerates to about 30% in the next period,
in part under the delayed effects of high mortality in period 1, and then picks up again
to 55-60% in the final three again, in part, as a result of the decrease in premature
adult mortality. In this connection, it may be helpful to express yt , which relates to
an indivisible 20-year period, as a more familiar annual quantity. The level of GDP
per head in the year 2000 was $725 (Heston et al., 2006). Scaling in the same way,
y6 then corresponds to $5428 in 2100, which implies an average annual growth rate of
2.01% over the entire horizon. This would be a decided improvement over Ethiopia's
performance in the last century, but it is rather modest by the standards set by many
Asian countries.
4.2 The counterfactual: no outbreak
Table 3 presents the results for the benchmark D = 0, which differs from D = 1 only
a
in the series {qi,t }, with the economy's initial total endowments being unaffected. The
lower levels of premature adult mortality encourage investment in both physical and
human capital in both sectors, and so lead to somewhat faster growth. With the urban
sector as the main motor of the economy, migration is higher throughout, the rate
being notably greater in period 1, when mortality is highest under D = 1. The end
result is that in period 6, about 65% of the population is urban.
Going into the essential details, full education is reached in the same periods as under
D = 1, but e1,t , ep and es are higher in the interim. The general course of kt is also
2,t 2,t
similar to, but lies above that under D = 1. The courses of the fixed factor land, t , are
virtually identical up to period 4, but begin to diverge quite markedly thereafter. The
24
upshot is that the consumption enjoyed by a young rural adult in period 6 is a mere
8.5% higher than under D = 1. The consumption enjoyed by a young urban adult,
whose level grows six-fold over the whole horizon, is but 5.0% higher than under D = 1.
GDP per head in 2100 is about 10% higher in the counter-factual. This difference seems
rather small, but it is permanent, as can be seen by examining the difference equations
(3) and (7), in which divergences in the transitional dynamics produce cumulative level
effects.21 The trajectories of all the main variables under D = 0 and D = 1 are depicted
for comparison in Figure 2. What is striking is that the effects of the epidemic on the
levels of per capita endowments and GDP are still growing in 2100, even though the
epidemic has weakened to an endemic state by period 5. Also striking is the finding
that the epidemic has virtually no effect on the size of the total population, even at
the close of the horizon in 2100. The reasons for this are, first, the assumption that
while the epidemic kills off young adults,22 it has no effect on net reproduction in each
sector, and second, that urban fertility is lower, so that faster urbanization under the
counterfactual leads to lower fertility in aggregate.
5 Policy objectives
What objectives commend themselves in the present setting, where the population
is growing rapidly, but is also beginning to experience a wave of premature adult
mortality? In each period, the level of human capital attained by the rising generation
of young adults has a powerful influence on both output and the formation of human
capital, not only in the present, but also in the future provided they and their offspring
do not die prematurely in adulthood. Mortality on the scale that now rules in Ethiopia
is also strongly undesirable in itself on purely ethical grounds, a consideration that
should appear in any evaluation of social welfare quite independently of the fact that
21
The characteristic roots are the same if the mortality rates are likewise.
22
Recall that Pt is defined not to include those young adults who die at the start of period t.
25
high levels of premature adult mortality undermine the process through which human
capital is formed and also hinder the accumulation of physical capital. In particular,
it can certainly be argued that the level of social welfare ought to be higher, cet. par.,
when any cohort is larger and none is smaller.
In Section 4, we concentrated heavily on the stream of losses in GDP per head,
which, by definition, accords no weight whatsoever to changes in the sizes of cohorts
or, for that matter, to those in the size of the whole population. It turns out, however,
that the assumptions concerning fertility and the timing of premature mortality leave
but little scope for the population to vary much in the face of the epidemic. Net
reproduction in the villages and towns, respectively, is independent of premature adult
mortality, so that, in each period, the level of fertility in the aggregate depends only
on the distribution of the population of young adults between town and country, and
hence on rural-urban migration, whose level is endogenous. In the long run, after the
epidemic has subsided and fertility has reached stationary levels, it is quite possible
that the population will be smaller in the counterfactual without the epidemic; for
urban fertility is lower than rural fertility in the transition and there is more migration.
Granted these assumptions, therefore, the use of GDP per head as the basic argument
of the social welfare function is relatively innocuous provided, of course, the use of
measures based on current aggregate output is admissible in this context. As it turns
out, the size of the population does enter into the calculations, for reasons that will
become clear in the sections that follow.
6 Public spending and mortality
The next step is to introduce public policy explicitly aimed at altering the course
of mortality in the presence of the epidemic. In order to produce the corresponding
variants of the benchmark D = 1, it is necessary to establish the relationship between
26
the level of spending on measures to contain the epidemic and treat those infected, and
premature mortality. We draw on the procedure and estimates in Bell, Devarajan and
Gersbach (2006), a brief sketch of which will suffice here.
It is argued that the efficacy of spending depends directly on the number of families,
Ft , that make up the whole society. Aggregate spending is then written as Ft · Gt ,
where Gt is the level of spending per family in period t. Let such aggregate expendi-
tures 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), (15)
where the superscript `a' to qt can now be suppressed without ambiguity and 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 . We also assume, and this seems sensible,
that the effects of these expenditures last for only the period in which they are made.
Our definition of the benchmark D = 1 implies that qt (0; D = 1) should yield the
estimates in Table 1. 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, again as set out in Table 1. The four-
parameter logistic,
1
qt (Gt ; D = 1) = dt - , (16)
at + ct e-bt Gt
possesses an asymptote and sufficient curvature. With four parameters to be deter-
mined, the above conditions must be augmented by two others.
One way of proceeding is to pose the question: what is the marginal effect of ef-
ficient spending on qt in high- and low-prevalence environments, respectively? That
is to say, we need estimates of the derivatives of qt (Gt ; D = 1) at Gt = 0 and some
value of Gt that corresponds to heavy spending, when the scope for using cheap in-
terventions has been exhausted. Here, we draw on the costs of preventing a case of
27
AIDS or saving a disability-adjusted-life-year (hereinafter, a DALY), as estimated by
Marseille, Hofmann and Kahn (2002, Table 1) using various methods. These authors
put the average annual cost per DALY saved of a diverse bundle of HIV-prevention
interventions (e.g., targeting sex workers and condom promotion, voluntary counseling
and testing, ensuring a safe blood supply) at $12.50, whereby it may be remarked that,
for these particular interventions, the assumption that Gt produces a pure public good
does not seem to be wide of the mark.
Following the purposive and determined implementation of the full battery of pre-
ventive measures, the remaining intervention is to treat the infected. Opportunistic
infections can be treated in the later stages of the disease, and the onset of full-blown
AIDS can be delayed for some years through the controlled use of anti-retroviral ther-
apies. Such measures will do little to reduce qt as strictly defined here, but by keeping
infected individuals healthier and extending life somewhat, they will raise lifetime in-
come and improve the parental care enjoyed by children in the affected families. In
the context of the model, therefore, it seems perfectly defensible to interpret these
gains in longevity as a reduction in qt . The estimated average annual cost of the rec-
ommended first-line HAART regimens is about $360 per person (WHO, 2005). It is
assumed here that the latter is the cost of the efficient, marginal form of intervention
when determined and extensive efforts at prevention are already being undertaken.
With an additional assumption about how quickly diminishing returns to preventive
measures set in as the prevalence rate falls (see the note to Table 4), the above con-
ditions yield the values of the parameters at , bt , ct and dt . The years in which such a
program might be undertaken fall into the decades beginning in 2000 and closing in
2040 (see below). The resulting values of the parameters are set out in Table 4, and
the associated functions qt (Gt ; D = 1) are plotted in Figure 3.
28
7 The policy program
The instruments potentially at the government's disposal are: (i) the set of all measures
designed to contain the disease and to treat those who become infected; (ii) measures
aimed at supporting the victims' families, especially in the form of consumption and
school-attendance subsidies: and (iii) the taxes or better still outside aid needed to
finance the spending under items (i) and (ii). The introduction of any of these measures
brings us at once to private decisions, for these respond to the program of taxes and
spending that the government chooses. Specifying all this in detail is a complicated
matter (for a full account, see Bell, Devarajan and Gersbach [2006]), and no attempt
will be made to implement the full procedure here.
We adopt, instead, a simpler approach, which not only does away with the compli-
cations that arise when the magnitudes of fiscal variables are endogenous, but is also
quite plausible in the light of the international community's willingness to shoulder a
large part of the costs of combating the epidemic in poor countries. The proposal is
that donors would commit themselves to making outright grants to the government of
Ethiopia for this purpose. It would be unrealistic to expect indefinite budgetary sup-
port for such programs. Given the course of the epidemic as expressed by qt (D = 1),
the worst should be over by 2040 (the start of period 3), so that one can entertain
the possibility of a stream of outright grants that ends in 2039. Let this stream be
denoted by B = (B1 , B2 ). In each period, the said amount is allocated to measures
designed to combat the epidemic, as summarized in the function qt (Gt ; D = 1), where
Gt = Bt /Ft , (t = 1, 2). The model is then run with the corresponding exogenous
mortality profile {qt (Gt ; D = 1)}t=7 .
t=1
The next step is to name sums of money. We suppose, perhaps rather optimistically,
that a spending program of $100 million a year in real terms will commence in 2008
and continue until 2040. Since the model's unit time period is 20 years, this translates
29
into two sub-programs. The first translates, somewhat roughly, into an outlay of $60
million a year, starting in 2000 and ending in 2019: equivalently, about $4.20 is spent
annually per family, whereby some allowance has been made for population growth over
that interval.23 The second runs from 2020 until 2040. Expenditure per family falls
to $4.03 a year for that period, again making allowance for population growth. These
two elements constitute the spending program B. The evolution of the main variables
under this program is set out in Table 5. As intuition would suggest, the reductions
in mortality lead to a slightly more rapid accumulation of human and physical capital
than under D = 1, but leave the population size virtually unchanged, as slightly higher
levels of migration induce lower aggregate fertility.
In order to assess the social profitability of this particular program B, we need an
index of welfare. A very conservative choice is confined to the resulting change in GDP
per head in each period relative to the reference case D = 1 without any intervention
(see Table 2), a choice that pays no heed to the premature deaths warded off by B. The
absolute values of GDP per head and the series of corresponding differences are reported
in Table 6, in both absolute and proportional terms. The next step is to make these
differences in 20-year per capita incomes commensurable with the aggregate program B.
This involves, first, multiplying each by the size of the corresponding population, and
second, converting the resulting sums into annual flows of constant dollars. Recalling
that GDP per head in 2000 was $725 annually (Heston et al., 2006), y1 (D = 1) = 0.488
implies an `exchange rate' of 29,710 in our 20-year unit period.
A social discount rate, r, is also needed. The real, long-term annual yield offered
by good quality paper in international capital markets is about 4%, so this is the
minimum real rate that public investment should yield. As a further check, results
are also reported for the more stringent standard of 5% annually; but our preferred
23
In the year 2000, there were about 28.5 million Ethiopians between the ages of 15 and 44 (UN,
2004), the groups which make up the overwhelming majority of the sexually active population. These
numbers have been normalized to yield the numbers of families as defined in Section 2.1.
30
rate is the lower one. These rates yield the two columns of aggregate present values
reported in Table 6. Adding up each column, we obtain the aggregate gross benefits
of the program, as implied by our choice of welfare measure. The present value of the
aggregate costs is obtained by discounting the stream of annual (aggregate) outlays
under B = (B1 , B2 ). Table 7 presents the corresponding findings using full income
per head.24 The resulting ratio of benefits to costs ranges from 3.5 to 1 to 5.3 to 1,
depending on the discount rate and, to a much lesser extent, the choice of income
measure employed. These levels of social profitability seem high indeed, especially
when one considers that the positive net benefits accruing after period 7 have been
left out. As argued above, the said level effects are permanent, and if, conservatively,
the differences in period 6 continued indefinitely, the associated discounted stream of
benefits would contribute about another 0.7 units when r = 0.04.
Are there reasons to doubt the robustness of this finding? In view of the discussion in
Section 6, the most compelling point to be made against them is that the effectiveness of
spending on prevention when prevalence rates are relatively high and levels of spending
are rather low has been over-estimated. Suppose, therefore, that the (absolute) slopes
of the qt (·) functions at zero levels of spending were only one half those derived in
Section 6. Since the changes in qt under the program are small anyway, it follows
from the approximately linear responses of the whole system in any sufficiently small
neighborhood that the benefits in any period would be only half as large. That is to
say, the social benefit-cost ratio would then range from at least 1.75 to 1 up to 2.65 to
1 or more. On this basis, we can still conclude with some confidence that the program
analyzed in this chapter would yield a large social return for each dollar spent.
A second important consideration is whether we have adopted an unduly pessimistic
view of Ethiopia's growth prospects over the long run. We have therefore experimented
by increasing the "transmission" parameter b2 from 0.45 to 0.505, which raises the
24
Full income, yt , is equal to (y1,t |ep =0 F1,t + y2,t |ep =es =0 F2,t )/Pt t 1.
1,t 2,t 2,t
31
asymptotic rate of growth of GDP per capita from 1.58% p.a. to 2.0% p.a. (see eq. (12))
and yields an average rate of 2.18% over the 100-year horizon.25 The program to combat
AIDS becomes even more profitable: the benefit-cost ratio with respect to GDP per
capita (full income), assuming a 5% discount rate, is now 5.11 (4.90). The reason is
as follows. Faster urban growth induces more migration, with unchanged mortality.
The program to combat AIDS yields sharper reductions in mortality per $ spent in the
towns, where the prevalence rates are much higher, than in the villages. Hence, the
pay-off to this form of intervention rises as the underlying rate of growth of the urban
sector increases. If one wants to go further and make the case that an asymptotic rate
of, say, 2.5% p.a. is attainable, this would simply reinforce our finding.
8 Conclusions
Taking a long-term view, the effects of the AIDS epidemic in Ethiopia estimated here
might, at first glance, seem rather small. There is no threat of an economic collapse,
and the size of the population, which increases somewhat more than fourfold over the
century 2000 to 2100, is scarcely affected though many meet an early death along the
way. At 2.01% over the entire horizon, the average annual rate of growth of output per
head would be a decided improvement over Ethiopa's performance in the last century,
however modest that may appear in comparison with East Asia or even some fully
developed countries. Under the counter-factual, in which there is no outbreak, output
per head at the end of the horizon is, moreover, only 10% higher.
Yet such a view savors of complacency. First, the threat of an economic collapse,
using output per head as yardstick, cannot be that imminent when the economy is
already mired in poverty and backwardness. Second, the said difference in output
per head is permanent, a consequence of the weaker inter-generational transmission
25
Results for a range of values of b2 are depicted in Figure 4.
32
of human capital during the transistional phase. Third, premature adult mortality
would be high in Ethiopia even if there had been no outbreak, so that AIDS effectively
`competes' with other mortal afflictions. Fourth, the projected course of the epidemic
is much less severe than that now ravaging southern Africa. All this stands in stark
contrast to South Africa in particular, a middle-income country which enjoyed rather
low adult mortality in 1990, when the adult prevalence rate was about 1%, but is now
experiencing a great wave of mortality, with the prevalence rate in ante-natal clinics
already exceeding 25%. Fifth, in following the common practice of measuring national
economic well-being in terms of GDP per capita admittedly with misgivings we
have neglected the fact that premature death is also something to be avoided in itself,
not only for the infected individuals themselves, but also on account of its consequences
for their dependent children, towards whom all parents presumably have feelings of love
and altruism.
33
Table 1: Exogenous demographic variables.
a
qi,t
t c
qi,t ~c
N1,t ~c
N2,t
D = D D=0
1 0.062 4.135 3.474 0.220 0.150
2 0.052 3.612 3.116 0.180 0.120
3 0.041 3.088 2.757 0.140 0.090
4 0.031 2.564 2.399 0.120 0.060
5 0.020 2.041 2.041 0.100 0.040
6 0.020 2.041 2.041 0.100 0.040
Note
D = D is an hypothetical scenario with an
epidemic and equal mortality rates across sectors.
D = 1 denotes the actual case with epidemic.
34
Table 2: The evolution of the economy and demographic variables, D = 1.
t 1,t t c1,t ep
1,t y1,t 2,t kt c2,t st ep
2,t es
2,t y2,t
yt
yt
1 1.400 1.020 0.572 0.243 3.219 2.800 1.000 0.783 1.492 0.740 0.502 6.964 0.488 0.512
2 1.796 0.697 0.700 0.706 3.566 3.147 1.448 1.106 1.197 0.932 0.632 7.200 0.733 0.804
3 2.483 0.421 0.899 1.000 4.177 3.173 1.391 1.459 1.298 1.000 1.000 9.075 0.956 1.033
4 3.260 0.338 1.305 1.000 5.907 4.633 1.773 2.100 1.675 1.000 1.000 12.124 1.471 1.547
5 4.260 0.327 2.004 1.000 8.801 5.972 2.615 3.092 2.383 1.000 1.000 17.328 2.357 2.431
6 5.553 0.407 3.111 1.000 14.640 7.972 4.521 4.537 3.551 1.000 1.000 26.458 3.652 3.735
c a a c c o o
t qi,t q1,t q2,t N1,t N2,t N1,t N2,t Mta t F1,t F2,t P1,t P2,t Pt
35
0 0.076 0.174 0.174 3.840 0.423 4.51 0.65
1 0.062 0.206 0.299 4.640 4.150 1.400 0.990 0.391 0.846 5.86 1.19 47.09 8.47 55.56
2 0.052 0.162 0.215 3.946 3.516 1.217 0.394 1.028 1.463 8.58 5.04 61.43 29.78 91.21
3 0.041 0.126 0.162 3.304 3.012 1.106 1.055 0.183 0.102 14.19 8.47 90.96 51.39 142.35
4 0.031 0.104 0.139 2.709 2.586 1.499 1.043 0.499 0.310 17.69 14.87 109.83 83.69 193.52
5 0.020 0.100 0.100 2.151 2.151 1.836 1.286 0.427 0.214 18.25 21.45 109.24 116.64 225.89
6 0.020 0.100 0.100 2.151 2.151 2.360 1.626 0.392 0.167 14.67 25.03 95.53 144.60 240.13
1
The rural sector: A1,1 = 1.000, gA1,t = 0.222, 1,0 = 1.150, = 0.800, c1,min = 0.500, ep
1,min = 0.001, b1 = 0.350, 1 = 0.400.
2
The urban sector: A2,1 = 2.000, gA2,t = 0.000, 2,0 = 2.300, 1 = 0.539, 2 = 0.458, 3 = 0.001, 4 = 0.001, c2,min = 0.584,
ep s
2,min = e2,min = 0.001, b2 = 0.450, 2 = 0.400.
du = 0.500, i = 0.667, = 0.40, = 0.20, = 0.60, = 0.60, = 0.75. yt grows at 1.98% per annum. Fi,t , Pi,t and Pt in
millions.
Table 3: The evolution of the economy and demographic variables, D = 0.
t 1,t t c1,t ep
1,t y1,t 2,t kt c2,t st ep
2,t es
2,t y2,t
yt
yt
1 1.400 1.000 0.573 0.264 3.171 2.800 0.947 0.794 1.409 0.723 0.491 6.671 0.491 0.515
2 1.823 0.702 0.707 0.782 3.599 3.053 1.567 1.135 1.233 0.968 0.657 7.291 0.761 0.836
3 2.588 0.429 0.930 1.000 4.307 3.181 1.540 1.516 1.400 1.000 1.000 9.485 1.014 1.091
4 3.359 0.362 1.375 1.000 6.259 4.686 1.987 2.192 1.808 1.000 1.000 12.614 1.603 1.682
5 4.446 0.362 2.140 1.000 9.447 5.983 2.984 3.239 2.600 1.000 1.000 18.357 2.586 2.665
6 5.824 0.468 3.376 1.000 16.096 8.112 5.095 4.765 3.886 1.000 1.000 28.096 4.016 4.102
c a c c o o
t qi,t qi,t N1,t N2,t N1,t N2,t Mta t F1,t F2,t P1,t P2,t Pt
36
0 0.076 0.174 3.840 0.423 4.51 0.65
1 0.062 0.150 4.485 3.768 1.400 0.990 0.519 1.124 5.97 1.25 47.09 8.47 55.56
2 0.052 0.120 3.850 3.321 1.295 0.431 1.079 1.578 8.51 5.36 60.77 30.81 91.58
3 0.041 0.090 3.237 2.891 1.145 1.128 0.200 0.108 13.94 8.91 88.97 53.62 142.60
4 0.031 0.060 2.645 2.475 1.611 1.034 0.631 0.373 16.51 16.44 103.31 90.56 193.87
5 0.020 0.040 2.083 2.083 1.941 1.354 0.508 0.220 16.49 23.55 99.37 128.04 227.41
6 0.020 0.040 2.083 2.083 2.529 1.693 0.451 0.158 12.78 27.27 84.50 157.49 241.98
1
The rural sector: A1,1 = 1.000, gA1,t = 0.222, 1,0 = 1.150, = 0.800, c1,min = 0.500, ep
1,min = 0.001, b1 = 0.350, 1 = 0.400.
2
The urban sector: A2,1 = 2.000, gA2,t = 0.000, 2,0 = 2.300, 1 = 0.539, 2 = 0.458, 3 = 0.001, 4 = 0.001, c2,min = 0.584,
ep s
2,min = e2,min = 0.001, b2 = 0.450, 2 = 0.400.
du = 0.500, i = 0.667, = 0.40, = 0.20, = 0.60, = 0.60, = 0.75. yt grows at 2.06% per annum. Fi,t , Pi,t and Pt in
millions.
Table 4: Parameters of the functions qt (Gt ).
rural sector urban sector
m=3
t=1 t=2 t=1 t=2
at 3.939 5.213 1.472 2.323
bt 0.078 0.107 0.029 0.048
ct 1.109 1.467 0.414 0.654
dt 0.404 0.312 0.829 0.550
t 16.398 11.970 43.888 26.857
Note
t denotes the level of spending that would
restore D = 0 if there were no diminishing re-
turns to preventive measures; m is the multiple
of t at which HAART becomes cost-effective.
37
Table 5: The evolution of the economy and demographic variables under program B.
t 1,t t c1,t ep
1,t y1,t 2,t kt c2,t st ep
2,t es
2,t y2,t
yt
yt
1 1.400 1.020 0.573 0.251 3.211 2.800 1.000 0.786 1.477 0.737 0.500 6.931 0.489 0.513
2 1.806 0.693 0.701 0.717 3.564 3.127 1.450 1.111 1.198 0.934 0.634 7.208 0.734 0.804
3 2.500 0.423 0.904 1.000 4.211 3.180 1.416 1.469 1.309 1.000 1.000 9.152 0.961 1.037
4 3.284 0.338 1.312 1.000 5.933 4.646 1.788 2.111 1.685 1.000 1.000 12.194 1.476 1.551
5 4.279 0.328 2.012 1.000 8.838 5.993 2.630 3.106 2.392 1.000 1.000 17.395 2.362 2.436
6 5.578 0.408 3.121 1.000 14.683 7.992 4.539 4.551 3.563 1.000 1.000 26.538 3.658 3.741
c a a c c o o
t qi,t q1,t q2,t N1,t N2,t N1,t N2,t Mta t F1,t F2,t P1,t P2,t Pt
38
0 0.076 0.174 0.174 3.840 0.423 4.51 0.65
1 0.062 0.193 0.285 4.603 4.110 1.400 0.990 0.415 0.899 5.88 1.19 47.09 8.47 55.56
2 0.052 0.150 0.201 3.917 3.486 1.221 0.402 1.012 1.442 8.66 5.02 61.79 29.54 91.33
3 0.041 0.126 0.162 3.304 3.012 1.126 1.066 0.192 0.109 14.18 8.42 91.16 51.15 142.31
4 0.031 0.104 0.139 2.709 2.586 1.494 1.046 0.490 0.306 17.74 14.73 110.04 82.95 192.98
5 0.020 0.100 0.100 2.151 2.151 1.836 1.283 0.427 0.217 18.29 21.30 109.53 115.74 225.28
6 0.020 0.100 0.100 2.151 2.151 2.357 1.625 0.390 0.167 14.73 24.87 95.85 143.64 239.49
1
The rural sector: A1,1 = 1.000, gA1,t = 0.222, 1,0 = 1.150, = 0.800, c1,min = 0.500, ep
1,min = 0.001, b1 = 0.350, 1 = 0.400.
2
The urban sector: A2,1 = 2.000, gA2,t = 0.000, 2,0 = 2.300, 1 = 0.539, 2 = 0.458, 3 = 0.001, 4 = 0.001, c2,min = 0.584,
ep s
2,min = e2,min = 0.001, b2 = 0.450, 2 = 0.400.
du = 0.500, i = 0.667, = 0.40, = 0.20, = 0.60, = 0.60, = 0.75. yt grows at 1.98% per annum. Fi,t , Pi,t and Pt in
millions.
Table 6: GDP per head with and without the programme B.
present value1
t y (B) y (D = 1) difference % change
r = 0.04 r = 0.05
1 0.48893 0.48801 0.00092 0.19 1.523 1.523
2 0.73379 0.73313 0.00066 0.09 0.817 0.674
3 0.96072 0.95633 0.00439 0.46 3.870 2.640
4 1.47587 1.47142 0.00445 0.30 2.434 1.371
5 2.36242 2.35664 0.00578 0.25 1.685 0.784
6 3.65845 3.65225 0.00620 0.17 0.876 0.337
Benefit-cost ratio: 5.30 3.75
1
Present values are reported in billions of dollars.
Note
yt series from Tables 5 and 2.
Table 7: Full income with and without the programme B.
present value1
t y (B) y (D = 1) difference % change
r = 0.04 r = 0.05
1 0.51331 0.51220 0.00111 0.22 1.741 1.741
2 0.80387 0.80352 0.00035 0.04 0.416 0.344
3 1.03734 1.03307 0.00428 0.41 3.591 2.449
4 1.55136 1.54712 0.00424 0.27 2.207 1.243
5 2.43638 2.43074 0.00564 0.23 1.565 0.728
6 3.74082 3.73477 0.00605 0.16 0.814 0.313
Benefit-cost ratio: 4.89 3.49
1
Present values are reported in billions of dollars.
Note
yt series from Tables 5 and 2.
39
16
urban
national
rural
14
12
10
8
6
4
2
0
1990 1994 1998 2002 2006 2010
Time
Figure 1: Estimated and projected adult HIV prevalence rates in Ethiopia, 1990-2010
(MoH, 2006).
40
rural human capital rural land rural consumption
(in logarithms)
6
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
1.0
5
0.5
4
3
0.0
2
-0.5
1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6
urban human capital urban physical capital urban consumption
(in logarithms)
1.5
8
5
7
4
1.0
6
3
0.5
5
2
4
0.0
3
1
1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6
migration rural population share overall income per capita
(per 1000 persons) (in logarithms)
70
0.8
1.0
60
0.7
50
0.5
0.6
40
0.0
30
0.5
-0.5
20
0.4
-1.0
10
1 2 3 4 5 6 1 2 3 4 5 6 0 1 2 3 4 5 6
Figure 2: D = 1 (solid line) and D = 0 (dashed line).
41
rural sector urban sector
0.30
t =1 t =1
0.20
t =2 t =2
0.25
0.18
mortality
mortality
0.16
0.20
0.14
0.15
0.12
0 10 20 30 40 50 0 10 20 30 40 50
spending spending
Figure 3: Values of the functions qt (Gt ).
42
GDP per capita GDP per capita
r = 0.04 r = 0.05
7
7
6
6
5
5
4
4
0.45 0.47 0.49 0.51 0.45 0.47 0.49 0.51
full income full income
r = 0.04 r = 0.05
7
7
6
6
5
5
4
4
0.45 0.47 0.49 0.51 0.45 0.47 0.49 0.51
Figure 4: Benefit-cost ratios as a function of b2 .
43
A Microfoundations
We lay out the microfoundations of the model in greater detail, beginning with rural
households.
Children are less productive than uneducated adults; they provide at most (0, 1)
efficiency units of labor. In a setting with three overlapping generations, each period is
taken to span 20 years. Thus, with at most 10 years of primary education as the only
option in rural areas, the labor actually supplied by the household in period t is
(2 - ep )N1,t
1,t
c
a o
ep N1,t
1,t
c
+ 1,t N1,t + (1 - ) 1,t-1 N1,t 1,t - ,
2 2
where 1,t is the family's total endowment of labor in efficiency units and (> 0)
denotes the depreciation rate of labor `capacity' across the last two periods of life.
The technology for producing the aggregate good is assumed to be Cobb-Douglas in
form, with constant returns to scale in labor and land. The level of output produced
by a family in period t is thus given by
1
ep N1,t
1,t
c
y1,t = A1,t 1,t - 1-1 ,
t
2
where A1,t and t denote the level of technical efficiency and the size of the family's agri-
cultural land holding, respectively. Under the social sharing rule, the upper boundary
of the rural household's budget set in the space of c1,t , ep becomes
1,t
1
ep N1,t
1,t
c
A1,t 1,t - 1-1 - N1,t + N1,t + N1,t c1,t = 0 .
t
c a o
(17)
2
The parents' preferences over current consumption and the level of education attained
by their children are influenced by the expectations that they form in period t about
the premature mortality that will afflict their children as adults in period t + 1. Let
44
these preferences take the Stone-Geary form:
Et u1 (c1,t , ep ) = ln (c1,t - c1,min ) + (1 - ) (1 - q1,t+1 ) ln (ep + ep
1,t
a
1,t 1,min ) ,
a
where (1 - q1,t+1 ) is the (point) expected survival rate in the next period, as formed in
the minds of the young adults when they make their decisions in period t, c1,min > 0
is an endogenously determined subsistence level of consumption (see Appendix B),
e1,min > 0 is likewise defined with respect to education, and is a `taste' parameter.
The presence of inter-generational altruism, in the sense that parents place some value
on their children's education, ensures that the process of human capital accumulation
may get underway and so lead to sustained growth. Observe, however, that c1,min > 0
and e1,min > 0 imply that whereas consumption is a necessary good, education is a
luxury. Hence, there is the possibility that a household is so poor that the children
will not be sent to school, that is, the parents' altruism will not actually be operative.
A poverty trap may, therefore, arise.
The family's decision problem is
max Et u1 s.t. (17), (c1,t , ep ) 0, ep 1 ,
1,t 1,t (18)
(c1,t , ep |Mt
1,t
a
)
where it should be noted once more that they take this decision after the migrants have
departed for the towns.
Turning to the urban sector, the production technology is also Cobb-Douglas. Anal-
ogously to eq. (17), the upper boundary of the urban household's budget constraint
is
2
(2 - ep - es )(N2,t + N2,t ) (2,t + t 1,t )N2,t (1 - )(2,t-1 + t-1 1,t-1 )N2,t
2,t 2,t
cp cs a o
A2,t + + (kt )1-2
2 (1 + t ) (1 + t-1 )
cp cs a o a
- (N2,t + N2,t ) + N2,t + N2,t c2,t + N2,t st = 0, (19)
45
where st denotes the level of savings per young adult. These savings are available to
the family as physical capital in the next period.
Let the preferences of a young adult at time t be represented as follows:
Et u2 (c2,t , c2,t+1 , ep , es ) = 1 ln(c2,t - c2,min ) + 2 (1 - q2,t )Et [ln(c2,t+1 - c2,min )] +
2,t 2,t
al
3 (1 - q2,t+1 ) ln(ep + ep
a
2,t
a s s
2,min ) + 4 (1 - q2,t+1 ) ln(e2,t + e2,min ) .
where the third and fourth terms on the RHS represent altruism towards her children
in the form of the levels of their primary and secondary education. It should be
noted, first, that the `pay-offs' in the event that the parent should die prematurely
al
(with probability q2,t ), or that the children, in their turn, should die prematurely in
a
adulthood (with probability q2,t+1 ), have been normalized to zero, and second, that
c2,t+1 is also, in general, 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.
With this scheme of expectations, the household's decision problem at time t is
max Et u2 s.t. (19), (c2,t , st , ep , es ) 0, (ep , es ) 1.
2,t 2,t 2,t 2,t
[c2,t ,st ,ep ,es |ep ,t ,Et (st+1 ,ep
2,t 2,t 1,t
s
2,t+1 ,e2,t+1 ,t+1 )]
(20)
B Algorithm
Where the rational expectations sequence is concerned, inspection of problems (18)
and (20) reveals that we need the forecasts Et st+1 , Et ep , Et es
2,t+1 2,t+1 (from which, for
any given migration series, optimal or otherwise, we may obtain Et c2,t+1 from eq. (19))
a
and Et t+1 (which, from eq. (8), depends on Et Mt+1 ). In addition, when t = 1 the
solution must yield the observed levels of educational attainment. To summarize, the
46
following must hold:
a0
· Et (Mt+1 ) = Mt+1 (where Mta 0 is the value that satisfies eq. (10) in period t + 1),
a
· Et st+1 = st+1 , Et (ep ) = ep 0 , Et (es ) = es 0 ,
2,t+1 2,t+1 2,t+1 2,t+1
· ep 0 = 0.243, ep 0 = 0.740 and es 0 = 0.502.
1,1 2,1 2,1
The algorithm set out below describes our estimation approach for D = 1 (i.e., the
scenario wherein all conditions specified above are satisfied). It proceeds in five stages.
In the first, households are assumed to form stationary expectations where future
levels of savings and education are concerned. Given the resulting values, a rational
expectations path with respect to migration flows is computed. In the following three
stages, these restrictions to stationary expectations are gradually lifted, and a full
rational expectations path in migration and urban consumption is sought.
The utility function parameters 3 and 4 are gradually adjusted in stages 1 to 4
so as to satisfy the boundary conditions at t = 1 at the end of each stage. If, during
this adjustment process, either 3 or 4 becomes very small, the system of first-order
conditions in the two sectors and eq. (10) often fails to converge to a solution. For
¯ ¯
this reason, floor values are specified for these parameters, namely, 3 and 4 , and
whenever either 3 or 4 becomes smaller than the corresponding floor value, all i
are restored to their starting/provisional values and the process starts again with a
lower value of du . The following table gives the parameters' starting and floor values
used to produce the results reported in Table 2. Stage 5 replicates stage 4 (i.e., the
du 1 2 3 4 ¯
3 ¯
4
2.50 0.5360 0.4556 0.0047 0.0037 0.00425 0.00325
solution of the full rational expectations system) for increasingly smaller values of du .26
26
No floor values are specified at this stage (which explains why the adopted 3 and 4 values are
¯ ¯
smaller than 3 and 4 ).
47
¯
The utility differential decreases at this final stage up to some user-specified du value
(which equals 0.50 in the paper, but could also be zero).
Another complication arrises from the fact that we need to know the population
structures under D = D and D = 0 in order to calculate the mortality profile under
D = 1.27 But the population structure depends on (among other factors) migration,
and the solutions of problems (18) and (20) under D = D and D = 0 require values of
certain parameters (e.g., j (j = 1, 2, 3, 4) and ci,min (i = 1, 2)), which we obtain from
the analysis of the factual scenario.
To overcome this difficulty, we employed an iterative process, where, using a provi-
a
sional qi,t (D = 1) series, we estimate all the parameters under D = 1. We then use
these parameters to find all demographic variables under D = D and D = 0, and
a
employ the latter to recalculate qi,t (D = 1) using eqs. (21) to (24)). This process is
a
repeated till the difference between the provisional and recalculated qi,t (D = 1) series
becomes very small (in the present case, smaller than 1/105 ).
27
In order to obtain a separate premature adult mortality series for each sector, we solve the following
system of equations:
P1,t (D = D ) q1,t (D = D ) + P2,t (D = D ) q2,t (D = D )
a a
qt (D = D ) =
a
(21)
P1,t (D = D ) + P2,t (D = D )
q2,t (D = D ) - q2,t (D = 0)
a a
Rt = a a (22)
q1,t (D = D ) - q1,t (D = 0)
a a
a
P1,t (D = 0) q1,t (D = 0) + P2,t (D = 0) q2,t (D = 0)
qt (D = 0) = (23)
P1,t (D = 0) + P2,t (D = 0)
a a
q1,t (D = 0) = m · q2,t (D = 0) (24)
Eq. (21) states that the economy-wide premature adult mortality rate is a weighted average of the
premature adult mortality rates in the two sectors, with weights equal to the respective population
sizes (the estimation results when D = D are available from the authors upon request). Eq. (22)
relates the excess mortality due to the epidemic in the urban sector to the excess mortality due
to the epidemic in the rural sector. We assume that Rt changes linearly from 3 in period 1 to 2
in period 4 (thereafter the two sectors have the same premature adult mortality rates). Eq. (23)
replicates eq. (21) for the No-AIDS scenario. Finally, eq. (24) specifies the relationship between
premature adult mortality in the two sectors in the absence of an epidemic. We assume that m = 1.1,
that is, premature adult mortality would be 10% higher in the villages than in the cities, which, in
view of the existing differences in the provision of medical care, seems plausible.
48
1: Start with a set of provisional values for du , j (j = 1, 2, 3, 4), j (j = 3, 4) and
¯
{Mta }t=0 +2 .28
t=T
2: for stage = 1 to 4 do
3: repeat
4: repeat
5: Find c1,min such that ep 0 = 0.24329 and set c2,min = 1.167 × c2,min .
1,1
6: for t = 1 to T + 1 do
7: Solve the first order conditions in the two sectors and eq. (10) imposing
the restrictions:
Et (st+1 ) = s0 , Et (ep ) = ep 0 and Et (es ) = es 0 if stage = 1.
t 2,t+1 2,t 2,t+1 2,t
Et (ep ) = ep 0 and Et (es ) = es 0 if stage = 2.
2,t+1 2,t 2,t+1 2,t
Et (es ) = es 0 if stage = 3.
2,t+1 2,t
8: Use the obtained values of c0 , ep 0 , c0 , s0 , ep 0 , es 0 , Mta 0 and eqs. (3), (4),
1,t 1,t 2,t t 2,t 2,t
(7) and (9) to calculate the households' endowments in the next period.
9: end for
10: Calculate the demographic variables with {Mta }t=0 +2 set equal to t=T
a 0 t=T +2 a t=T +2
{Mt }t=0 (or a linear combination of {Mt }t=0 and {Mta 0 }t=0 +2 ). t=T
11: until {Mta }t=0 +2 = {Mta 0 }t=0 +2 . // rational expectations in migration
t=T t=T
p0
12: if e2,1 = 0.740 or es 0 = 0.502 then
2,1
13: ¯
if 3 > 3 and 4 > 4 then ¯
14: adjust the urban utility function i parameters.30
15: else
16: reduce the value of du .
17: end if
18: end if
19: until ep 0 = 0.740 and es 0 = 0.502.
2,1 2,1
20: end for
21: ¯
while du du do // stage = 5
22: repeat
23: repeat
24: Replicate line 5.
25: for t = 1 to T + 1 do
26: Solve the first-order conditions in the two sectors and eq. (10).
27: Replicate line 8.
28: end for
29: Replicate line 10.
30: until {Mta }t=0 +2 = {Mta 0 }t=0 +2 .
t=T t=T
p0 s0
31: if e2,1 = 0.740 or e2,1 = 0.502 then
32: adjust the urban utility function i parameters.
33: end if
34: until ep 0 = 0.740 and es 0 = 0.502.
2,1 2,1
35: reduce the value of du .
36: end while
49
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