ï»¿ WPS6169
Policy Research Working Paper 6169
The Benefits of Indiaâ€™s Rural Roads Program
in the Spheres of Goods, Education
and Health
Joint Estimation and Decomposition
Clive Bell
The World Bank
South Asia Region
Transport Unit
August 2012
Policy Research Working Paper 6169
Abstract
All-weather rural roads usually improve not only villagersâ€™ of human capital in the presence of both morbidity
terms of trade, but also their educational attainments and mortality. Based on survey evidence from upland
and health. Obtaining empirical estimates of the benefits Orissa in India and Bangladesh, as well as elements of
generated by the first is straightforward, not so those more usual forms of calibration, the model yields a ratio
generated by the others. The object of this paper is to of commercial to non-commercial benefits of about
estimate the relative sizes of their respective contributions two-to-one in the first generation, falling to three-to-
to total benefits in connection with the all-India rural four in the second. This is broadly consistent with the
roads program Pradhan Mantri Gram Sadak Yojana, valuations expressed by respondents in the Orissa survey,
using an overlapping generations model featuring the who ranked the latter benefits at least on a par with the
production and consumption of goods and the formation former.
This paper is a product of the Trasnport Unit, South Asia Region. It is part of a larger effort by the World Bank to provide
open access to its research and make a contribution to development policy discussions around the world. Policy Research
Working Papers are also posted on the Web at http://econ.worldbank.org. The author may be contacted at clive.bell@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
The Beneï¬?ts of Indiaâ€™s Rural Roads Program in the
Spheres of Goods, Education and Health: Joint
Estimation and Decomposition
Clive Bellâˆ—
(Appendix by Jochen Laps)
First version: January, 2010. This version: August, 2012
Abstract
All-weather rural roads usually improve not only villagersâ€™ terms of trade, but
also their educational attainments and health. Obtaining empirical estimates of
the beneï¬?ts generated by the ï¬?rst is straightforward, not so those generated by
the others. The object of this paper is to estimate the relative sizes of their
respective contributions to total beneï¬?ts in connection with the all-India ru-
ral roads program Pradhan Mantri Gram Sadak Yojana, using an overlapping
generations model featuring the production and consumption of goods and the
formation of human capital in the presence of both morbidity and mortality.
Based on survey evidence from upland Orissa in India and Bangladesh, as well
as elements of more usual forms of calibration, the model yields a ratio of com-
mercial to non-commercial beneï¬?ts of about two-to-one in the ï¬?rst generation,
falling to three-to-four in the second. This is broadly consistent with the val-
uations expressed by respondents in the Orissa survey, who ranked the latter
beneï¬?ts at least on a par with the former.
Keywords: Rural roads, transport costs, education, health, India
JEL Classiï¬?cation: E62, O19, O12
Sector Board: Transport
âˆ—
University of Heidelberg, INF 330, D-69120 Germany. This paper was prepared as part of the
documentation for a World Bank loan proposal in connection with PMGSY. The views expressed
herein are not, however, necessarily those of the World Bank or its Board.
1
1 Introduction
A new all-weather rural road will bring various beneï¬?ts to the villagers along its route.
As producers, they will enjoy higher net prices for their marketed surpluses; as con-
sumers, they will pay less for urban goods. If there is no school in the village itself,
those children who already attend one elsewhere will spend less time traveling to and
fro, and those who did not attend earlier may do so now. If there is a village school, it
is the teachers themselves who may appear more regularly. Much the same holds for
medical treatment. Children and those adults with chronic ailments are more likely to
make regular visits to the clinic; and in an emergency, those in need of medical atten-
tion will be able to reach the clinic sooner, which may make the diï¬€erence between life
and death.
Measuring the roadâ€™s eï¬€ects on these movements of goods and people is, in principle
at least, relatively straightforward. Valuing the resulting beneï¬?ts is another matter
altogether. For the new road aï¬€ects not just the decisions of what to produce and
consume in the sphere of what might be called â€˜textbook goodsâ€™, but also those having
to do with the formation and maintenance of human capital, including life itself. These
decisions are not, moreover, readily separable, which calls for their analysis within a
uniï¬?ed framework. Yet more is at stake than consistency and rigor. It will be argued
that valuing the beneï¬?ts that arise in connection with more favorable prices of goods,
improved educational attainment and lower morbidity involves a common (money)
metric which is directly related to eï¬€ects that are fairly readily measurable â€“ provided
families decide not to educate their children fully. In contrast, the beneï¬?ts of reduced
mortality, even if such reductions can be measured with some conï¬?dence, do not ï¬?t
into this convenient scheme of things. The same will hold, moreover, if children are
fully educated. How, then, are these beneï¬?ts to be estimated in practice? A uniï¬?ed
framework, in which decisions in the spheres of production, consumption and education
are made in a particular environment of morbidity and mortality, provides one way of
answering this question. For one can set up the model so that, counter-factually, the
road lowers transport costs but nothing else; or, at the other extreme, everything else
but transport costs. Granted that the model can be numerically calibrated, one can
use the equivalent variation (EV) for each setting, relative to the benchmark of â€˜no-
roadâ€™, to establish the size of the beneï¬?t arising from lower transport costs relative to
those arising from other eï¬€ects. Since estimating the money-metric beneï¬?ts of lower
transport costs is a relatively straightforward task and does not involve the apparatus
of a uniï¬?ed model, combining such an estimate with the corresponding contribution
2
to the total beneï¬?t, as yielded by the model, will then yield an estimate of the total
beneï¬?t that has a substantial empirical anchoring.
The object of this paper is to estimate the relative size of the contribution of reduced
transport costs, in connection with the all-India rural roads program known as Pradhan
Mantri Gram Sadak Yojana (PMGSY), a very large scheme that will eventually cover
some 170,000 habitations, take at least 20 years to implement and cost in excess of
US$40 billion (World Bank, 2010). The uniï¬?ed framework employed for this purpose
is the overlapping generations (OLG) model of Bell, Bruhns and Gersbach (2006), sub-
stantially extended so as to cover transport costs and morbidity. Important elements
of the empirical basis for this application are supplied by an analysis of a survey of
some 240 households drawn from 30 villages lying in the upland â€“ and poor â€“ region
of Orissa (Bell and van Dillen, 2012) and two large surveys in Bangladesh (Khandker
et al., 2009). There is also an indirect check, in the form of completely independent
estimates of the so-called value of a statistical life provided by Lui et al. (1997) and
Simon et al. (1999).
The plan of the paper is as follows. The model is set out and analyzed in Section 2,
the essential diï¬ƒculty with valuing the beneï¬?ts arising from eï¬€ects other than those of
lower transport costs being addressed in Section 2.3. The numerical set-up follows in
Section 3, which is divided up into subsections dealing with functional forms, parameter
values and calibration under perfect foresight. This is the basis for the exact welfare
analysis in Section 4, treating in sequence the benchmark of â€˜no-roadâ€™, the world with
the road and the contribution of lower transport costs to the whole resulting beneï¬?t.
The conclusions are drawn together in Section 5.
2 The Model
The basis is the model of Bell, Bruhns and Gersbach (2006), which deals with human
capital formation and growth when there is premature adult mortality. To summarize,
an extended family comprises three overlapping generations, with all surviving adults
caring for all related children in each period, which stretches over a generation.1 The
young adults are assumed to decide how current resources are to be allocated between
consumption and the childrenâ€™s education. The level of current resources available to
1
This arrangement is admittedly a rather idealized description of the social structure even in Kenya,
let alone in rural India, but the pooling of the risks of premature mortality among the adults greatly
simpliï¬?es the analysis.
3
the family is largely determined by the level of the parentsâ€™ human capital and their
survival rate, but the children themselves can also work instead of attending school.
At the end of each period, some young adults die prematurely, all old adults die, and
the children are on the point of becoming young adults.
How much of childhood, if any, is spent at school depends not only on the familyâ€™s
available resources, but also on three further factors. First, there is the parentsâ€™ desire
to provide for their old age and their childrenâ€™s future, motives which express themselves
in the parentsâ€™ willingness to forego some current consumption in favour of investment
in their childrenâ€™s schooling, and hence of the childrenâ€™s human capital when they attain
adulthood in the next period. Second, there is the eï¬ƒciency with which schooling is
transformed into human capital, which arguably depends on the quality of the school
system and child-rearing within the family, whereby the latter ought to improve with
the parentsâ€™ human capital. Third, the returns to the investment in any child will be
eï¬€ectively destroyed if that child dies prematurely in adulthood. This implies that
the expected returns to education depend on parentsâ€™ (subjective) assessments of the
probability that their children will meet an untimely death.
For present purposes, we need to extend this framework in two ways. First, the
transportation of goods and persons must be brought into the picture. Instead of the
aggregate consumption good in Bell, Bruhns and Gersbach (2006), there are now two
consumption goods, one of which the household produces; the other is an â€˜urbanâ€™ good,
which the household can obtain only through exchange. The resulting trade necessar-
ily involves transportation. The same holds for education and health, insofar as the
children and their teachers must travel to school and the sick to a clinic, which may lie
some distance oï¬€ and, in the absence of an all-weather road, be inaccessible at times.
Second, we introduce morbidity, which, as formulated below, reduces individualsâ€™ ca-
pacities to go about their daily business.
2.1 Human capital and output
We begin by introducing some notation.
Nta : the number of individuals in the age-group a (= 1, 2, 3) in period t,
Î»a
t : the human capital possessed by an adult in age-group a (= 2, 3),
Î³ : the human capital of a school-age child,
Î±t : the output produced by a unit of human capital input in year t,
4
et : the proportion of their school-age years actually spent in school by the cohort of
children (a = 1) in period t.
Human capital is formed through a process that involves the adultsâ€™ human capital
and the educational technology. The human capital attained by a child on becoming
an adult in period t + 1 depends, in general, on the numbers and human capital of
the adults, the level of schooling that child received, and the number of siblings of
school-going age, who were presumably competing for the adultsâ€™ attention, care and
support â€“ all in the previous period. Formally,
t+1 = Î¦(et , Î»t , Nt ),
Î»2 (1)
where Î»t = (Î»2 3 1 2 3
t , Î»t ) and Nt = (Nt , Nt , Nt ) and Î¦ is increasing in all its arguments,
except for Nt1 .
Simplifying with the application in mind, let Î¦ be multiplicatively separable in: (i)
the educational technology, which involves only et ; (ii) the average level of the adultsâ€™
human capital; and (iii) the degree of competition among siblings. This implies that
formal education and the adultsâ€™ human capital are complements in producing the
childrenâ€™s human capital, which is intuitively plausible.2 Parents in rural India have
most of their children when they are in their twenties, and in their thirties, they are
busy rearing them to adulthood. Normalizing the structure to a representative couple
within the extended family, these assumptions yield the following specialization of (1):
2(Nt2 Î»2 3 3
t + Nt Î»t ) Nt1
t+1 = ft (et ) Â·
Î»2 Â· Ïˆ + 1, (2)
Nt2 + Nt3 Nt2 + Nt3
where ft (Â·) represents the educational technology, whose eï¬ƒciency may vary with time,
and the function Ïˆ (Â·) the eï¬€ects of competition among siblings for their parentsâ€™ time
and attention. These functions are assumed to have the following properties: ft (Â·) is
continuous and increasing âˆ€et âˆˆ [0, 1), with ft (0) = 0; and Ïˆ (Â·) is continuous and de-
creasing in the number of children per adult, and goes to zero as that number becomes
arbitrarily large. The assumption ft (0) = 0 implies that a child who receives no school-
ing will attain only some basic level of human capital, which, without loss of generality,
may be normalized to unity â€“ hence the â€˜1â€™ on the RHS of (2). The assumption that
Ïˆ (Â·) is a decreasing function implies that, cet. par., an increase in mortality among
parents that outweighs any reduction in fertility will hinder the formation of human
2
Becker, Murphy and Tamura (1990) and Ehrlich and Lui (1991) pioneered the approach based on
the direct transmission of potential productivity from parent to child.
5
capital among their children. Let there be no depreciation of human capital.
The diï¬€erence equation (2) governs the systemâ€™s dynamics. A brief remark will
suï¬ƒce on the asymptotic behaviour of Î»2t when there is full education. Observe that
under stationary technological and demographic conditions, (2) may be written as
Î»2 2 2
t+1 = 2f (et )(a2 Î»t + a3 Î»tâˆ’1 ) + 1,
where a2 and a3 are constants and Î»2 tâˆ’1 = Î»t . Suppose et = 1 âˆ€t, so that the relevant
3
characteristic root is a2 [1+ (1+2a3 /(a2
2 f (1)))]f (1). Then, starting from a suï¬ƒciently
large value of the parentsâ€™ combined human capital, when they would choose et = 1,
unbounded growth of Î»2 t is possible if a2 [1 + 2 f (1)))]f (1) â‰¥ 1, and the
(1 + 2a3 /(a2
inter-generational growth rate then approaches
g âˆ— â‰¡ a2 [1 + 2 f (1)))]f (1) âˆ’ 1,
(1 + 2a3 /(a2 (3)
from above. If, however, 1 > 2(a2 + a3 )f (1), Î»2
t will approach the stationary value
1/[1 âˆ’ 2(a2 + a3 )f (1)]. It should be remarked that if et is restricted below unity, to
Â¯t say, then the latter value will replace unity as the argument of f (Â·) in the above
e
expressions. This will be important when analysing the long-run eï¬€ects of morbidity,
travelling to school and involuntary absences in Sections 3 and 4.
The household produces a single consumption good (good 1) solely by means of labor,
measured in eï¬ƒciency units, under constant returns to scale. A natural normalization is
that a healthy adult who possesses human capital in the amount Î»a t is endowed with Î»t
a
eï¬ƒciency units of labor, which he or she is assumed to supply completely inelastically.
To accomodate sickness and injury, denote the fraction of each period that a surviving
adult spends in disability by da
t (a = 2, 3). Children, too, suï¬€er ailments, which reduce
t âˆ’ (1 + Ï„ )et )Î³
the eï¬€ective time left for schooling and work. Each child supplies (1 âˆ’ d1
eï¬ƒciency units of labor when it spends et units of time in school and, unavoidably, Ï„ et
units of time travelling to and from school, whereby Î³ âˆˆ (0, 1), i.e., a full-time working
child is less productive than an uneducated adult. The household therefore produces
t âˆ’ (1 + Ï„ ))et )Î³ + (1 âˆ’ dt )Nt Î»t + (1 âˆ’ dt )Nt Î»t
y1t = Î±t Nt1 (1 âˆ’ d1 2 2 2 3 3 3
(4)
units of good 1 in period t.
6
2.2 The familyâ€™s preferences and decisions
Some additional notation is needed.
xit : the consumption of good i (= 1, 2) by each young adult,
Î² : the proportion of a young adultâ€™s consumption received by each child,
Ï?: the proportion of a young adultâ€™s consumption received by each old adult,
Ïƒt : the direct costs per child of each unit of full-time schooling,
nt : the number of children born to a representative couple who survive to school age
in period t,
x qa : the probability that an individual aged a will die before reaching the age of
a + x,
qt : the probability that a young adult in period t will die before reaching the third
phase of life.3
The parameters Î² and Ï? are viewed as binding by all concerned through a social norm
and they are enforced by appropriate social sanctions.
The extended familyâ€™s expenditure-income identity involves, in principle, outlays on
the two consumption goods, health care and education, where the latter include both
the direct expenditures and the opportunity costs of the pupilsâ€™ time. In what follows,
we resort to the drastic simpliï¬?cation that there are no expenditures on getting to
the clinic and being treated: mortality and morbidity rates are set exogenously, at
levels that depend on the availability or otherwise of an all-weather road. This is not
merely simpliï¬?cation in the interests of making the analysis more tractable. For the
relationship between choosing treatment and the experience of disability over the whole
stretch of a generation is not only diï¬ƒcult to model, but is also not well-established
empirically: for example, better access to a clinic may induce timelier and heavier
outlays on treating an acute ailment, and so save outlays on undoing even more damage
later, should the condition go untreated at the outset.
It will be convenient to normalize the budget identity by the number of young adults.
The said identity may then be written as:
Pt (Î², Ï?) Â· pt Â· xt + Qt (Î±t , Î³, Ïƒt , Ï„ ) Â· et â‰¡ p1t Î±t Â· (Î›t + Nt1 (1 âˆ’ d1 2
t )Î³ )/Nt , (5)
3
In the present structure, this statistic corresponds to 20 q20 , the probability that an individual will
die before 40, conditional on surviving until 20.
7
where the household faces the price vector pt for the two consumption goods and
t )Î»t + Nt (1 âˆ’ dt )Î»t
Î›t â‰¡ Nt2 (1 âˆ’ d2 2 3 3 3
(6)
is the adultsâ€™ aggregate supply of eï¬ƒciency units of labor. The term
Pt (Î², Ï?) â‰¡ (1 + (Ï?Nt3 + Î²Nt1 )/Nt2 ) (7)
expresses the eï¬€ect of the familyâ€™s demographic structure on the â€˜priceâ€™ of the con-
sumption bundle xt relative to education. Analogously, the â€˜priceâ€™ of a unit of full
education (et = 1) is
Qt â‰¡ (p1t Î±t (1 + Ï„ )Î³ + Ïƒt )Nt1 /Nt2 . (8)
The RHS of (5) is the level of (normalized) full income in period t. In this connection,
observe that Nt1 , Nt2 and Nt3 are related as follows:
Nt1 = nt Nt2 /2, (9)
Nt2 = ntâˆ’1 Nt2
âˆ’1 / 2 , (10)
Nt3 = (1 âˆ’ qtâˆ’1 )Nt2
âˆ’1 . (11)
The young adultsâ€™ preferences are, in principle at least, deï¬?ned over the following: the
levels of consumption in young adulthood and old age, xt and Ï?xt+1 , respectively, and
the human capital attained by their school-age children on attaining full adulthood
(Î» 2
t+1 ), which they may appreciate in both phases of their own lives. Investment in
the childrenâ€™s education therefore produces two kinds of pay-oï¬€s, one altruistic, as
expressed by the value directly placed on Î»2 t+1 , and the other selï¬?sh, inasmuch as an
2
increase in Î»t+1 will also lead to an increase in Ï?xt+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. The incidence of morbidity is uncertain at the
level of the individual, though if the extended family is large enough, its realized levels
of morbidity will diï¬€er little from the population rates. There is also uncertainty about
future demographic developments, which will inï¬‚uence the realized level of Ï?xt+1 .
At the start of period t, young adults choose like partners. Each couple produces nt
8
children, and then draws up a plan for current consumption and investment in the chil-
drenâ€™s education based on the familyâ€™s resources and expectations about its membersâ€™
state of health and other relevant variables in the coming and future periods. Given
such assortative mating, the pair will agree wholly on what is to be done. Appealing to
the law of large numbers in order to rid the system of any uncertainty about the real-
ized levels of morbidity in period t, so that full income in that period is non-stochastic,
let the preferences of a young adult at time t be represented as follows:
Et U = b1 u(xt ) + b2 (1 âˆ’ qt )Et [u(Ï?xt+1 )] + Et [(1 âˆ’ qt+1 )] Â· nt Ï†(Î»2
t+1 ), (12)
where goods 1 and 2 are private goods in consumption, but the childrenâ€™s attainment
of human capital is a public one within the union. It should be noted, ï¬?rst, that
no account has been taken of the pain and suï¬€ering associated with morbidity, even
though its level may change exogenously; second, that the â€˜pay-oï¬€sâ€™ in the event that
the parent should die prematurely (with probability qt ), or that any of the children, in
their turn, should die prematurely in adulthood (each with probability qt+1 ), have been
normalized to zero; and third, that conditional on surviving into old age at t + 1, the
associated level of consumption, Ï?xt+1 , is also a random variable viewed at time t, for
its level depends on a whole variety of future economic and demographic developments.
Finally, observe also that the parentsâ€™ altruistic motive makes itself felt only when they
themselves are young and actually make the sacriï¬?ces, whereby Î»2 t+1 is non-stochastic
by virtue of et being non-stochastic.
These adults take all features of the environment in periods t and t + 1 as paramet-
rically given. It will be helpful to distinguish between what they know and what they
must forecast. At the time of decision, the current endowment and environment are
described by the vector
Zt â‰¡ (Nt , nt , Î»t , Pt , Qt , pt , Ï„, Î±t , qt , dt ). (13)
This is assumed to be known.4 What is unknown are the (future) realizations of xt+1
and qt+1 . Under the social norm expressed by Ï?, the parents at t must form expectations
about how their surviving children will allocate full income in period t + 1, a decision
that depends, not only on Zt+1 , which will have been revealed at that time, but also
on all future constellations thereafter, to the extent that these inï¬‚uence (xt+1 , et+1 ).
4
It can be argued that there is uncertainty about qt at the point of decision at time t, the (indi-
visible) unit period being rather long. This possibility is addressed below.
9
For simplicity, all individualsâ€™ forecasts of all elements of the future environment are
assumed to be point estimates, so that whilst there is uncertainty about an individualâ€™s
personal fate, there is none about the future mortality proï¬?le itself or future fertility.
Indeed, we go farther down this path, and assume not only that all individuals share
t=âˆž
the same forecasts of {Zt+1 }t =1 , but also that these forecasts are unerring: that is to
say, there is perfect foresight about everything â€“ with the vital exception of whether
a particular individual will die prematurely. Under this assumption, Ï?xt+1 becomes
non-stochastic, conditional on surviving into old age at time t + 1, so that (12) may be
written
t+1 ) + (1 âˆ’ qt+1 ) Â· nt Ï†(Î»t+1 (et )),
Et U = b1 u(xt ) + b2 (1 âˆ’ qt )u(Ï? x0 2
(14)
where the superscript â€˜0â€™ denotes the optimal choice of those making decisions at time
t + 1, which their parents forecast unerringly at t. The functions u and Ï† are assumed
to be strictly concave.
A young adultâ€™s decision problem therefore takes the following form:
max Et U s.t. xt â‰¥ 0, et âˆˆ [0, (1 âˆ’ d1
t )/(1 + Ï„ )], (2), (5). (15)
(xt , et |{Zt+t }t =âˆž )
t =0
If ft1 (Â·) is concave, Et U (Â·) will be strictly concave in (xt , et ). Hence, problem (15) has
a unique solution and the ï¬?rst-order necessary conditions are also suï¬ƒcient. Observe
that the optimum always involves xt > 0. The corner solution in which the children are
not educated at all can also be ruled out when f (Â·) satisï¬?es the lower Inada condition,
since âˆ‚Î»2
t+1 /âˆ‚et is then unbounded at et = 0; but this condition will not be imposed.
2.3 Comparative statics
The associated Lagrangian is, omitting the non-negativity constraints for brevity,
L = b1 u(xt ) + b2 (1 âˆ’ qt )u(Ï? x0
t+1 ) + (1 âˆ’ qt+1 ) Â· nt Ï†(Î»t+1 (et ))
2
t )Î³ )/Nt âˆ’ Pt (Î², Ï?) Â· pt Â· xt âˆ’ Qt (Î±t , Î³, Ïƒt ) Â· et . (16)
+ Î¼ p1t Î±t Â· (Î›t + Nt1 (1 âˆ’ d1 2
10
The Envelope Theorem yields the following results, all of which accord with elementary
intuition. Where the movement of goods and school children is concerned, we have
âˆ‚Et U 0
t )Î³ )/Nt âˆ’ Pt (Î², Ï?) Â· x1t âˆ’ (Î±t (1 + Ï„ )Î³Nt /Nt )et > 0,
= Î¼ Î±t Â· (Î›t + Nt1 (1 âˆ’ d1 2 1 2
âˆ‚p1t
(17)
and
âˆ‚Et U 0
= âˆ’Î¼Pt (Î², Ï?) Â· x2t < 0, (18)
âˆ‚p2t
by virtue of the fact that the household is a net seller of good 1 and a net buyer of
good 2, and et â‰¤ (1 âˆ’ d1t )/(1 + Ï„ ). An increase in the travel-time to school is likewise
t )/(1 + Ï„ ) â‰¡ e
damaging. Denoting by Î½ the multiplier associated with et â‰¤ (1 âˆ’ d1 Â¯t ,
âˆ‚Et U 0
= âˆ’Î¼p1t Î±t Î³ (Nt1 /Nt2 )et âˆ’ Î½ (1 âˆ’ d1 2
t )/(1 + Ï„ ) . (19)
âˆ‚Ï„
Morbidity reduces not only the familyâ€™s productive endowments, pain and suï¬€ering
Â¯t :
having been ruled out by assumption, but also e
âˆ‚Et U 0
= âˆ’Î¼p1t Î±t Î³ Â· (Nt1 /Nt2 ) âˆ’ Î½/(1 + Ï„ ) , (20)
âˆ‚d1t
âˆ‚Et U 0
2
= âˆ’Î¼p1t Î±t Â· Î»2
t , (21)
âˆ‚dt
âˆ‚Et U 0
= âˆ’Î¼p1t Î±t Â· (Nt3 /Nt2 )Î»3
t . (22)
âˆ‚d3t
Turning at last to mortality, we begin by recalling (11); so that qt exerts an inï¬‚uence
on Et U , not only directly, but also on the feasible set in period t + 1, and hence on
x0
t+1 . We have
âˆ‚Et U 0 âˆ‚ x0
= âˆ’b2 u(Ï? x0
t+1 ) + b2 (1 âˆ’ qt ) âˆ‡ u ( Ï? x 0
t+1 ) Â· Ï? t+1
; (23)
âˆ‚qt âˆ‚qt
and
âˆ‚Et U 0 âˆ‚ x0
= b2 (1 âˆ’ qt ) Â· âˆ‡u(Ï? x0
t+1 ) Â· Ï? t+1
âˆ’ nt Ï†(Î»2
t+1 (et )), (24)
âˆ‚qt+1 âˆ‚qt+1
which reï¬‚ects the fact that an increase in premature mortality among the children on
reaching adulthood will also aï¬€ect the parentsâ€™ consumption in old age, should they
survive to enjoy it. In this connection, note that an increase in qt+1 may well induce
an increase in x0t+1 , since higher premature mortality in any period makes old-age
provision for the next less attractive.
11
3 Setting up the System Numerically
The beneï¬?ts ï¬‚owing from an all-weather road stem from more favorable prices facing
the household as producer and consumer, from reduced time for the children to go to
and from school, and from lower morbidity and mortality due to timelier treatment.
Under the above assumptions, it is seen from (17) - (22) that, with the exception of re-
duced mortality, suï¬ƒciently small changes in each of these features of the â€˜environmentâ€™
yield beneï¬?ts that, in money-metric utility, are equal to the gains or savings calculated
at the allocation ruling before the said change and valued at the corresponding oppor-
tunity cost.5 Inspection of (23) and (24), however, reveals that there is no such ready
simpliï¬?cation where mortality is concerned; for the sub-utility functions u and Ï† appear
0 t=âˆž
t+1 , in the perfect-foresight sequence {xt }t=0 . In
explicitly, as does the next bundle, x0
order to obtain some feel for the size of the value placed on reduced mortality relative
to that of other beneï¬?ts, a resort to some numerical examples is unavoidable. This
task involves the construction of the whole perfect-foresight sequence.
3.1 Functional forms
There is no hope of estimating more than a tiny part of this system econometrically;
and even â€˜calibrationâ€™ for the system as a whole is ruled out for want of suitable data.
The approach, therefore, is to choose functional forms that are both tractable and
plausible, if only through common usage in other contexts, and then constellations of
associated parameter values such that certain key magnitudes correspond to what are
called the â€˜stylized factsâ€™.6
1. Technologies. Let ft (et ) = zt et âˆ€t, where zt represents an inter-generational trans-
mission factor, which reï¬‚ects the quality of both child-rearing and the school system.
This limiting form is certainly the simplest, and it causes no technical problems in view
of the assumption that Ï† is strictly concave (see below). The absence of diminishing
returns does not seem especially odd when one reï¬‚ects on the need for children to spend
some years in school before they have mastered the three Râ€™s, which form the basis of
all other acquired abilities involving literacy. The great majority of school-children in
5
Observe that if Î½ = 0, each of these expressions is scaled by the Lagrange multiplier Î¼. As it turns
out, this simpliï¬?cation is lost after period 1, when Î½ > 0; so that the whole apparatus is also needed
in connection with chnages in trip-time and morbidity.
6
As Solow once wrote in the original connection with the character of growth in industrialized
countries in the decades following WWII, they are certainly stylized, but whether they are facts is
another matter.
12
Indiaâ€™s villages now receive some education, moreover, so that this form of f (et ) can
also be thought of as applying over the relevant range up to a full education. Turning
to competition among siblings, rather little is known about its eï¬€ects on human capital
formation, so we adopt the agnostic position that Ïˆ = 1 âˆ€ Nt . With these choices, (2)
specializes to
Nt2 Î»2 3 3
t + Nt Î»t
t+1 = 2zt Â· et Â·
Î»2 + 1. (25)
Nt2 + Nt3
2. Preferences. In the macroeconomics literature, especially the empirical kind, the
logarithm tends to hold sway. There is usually, however, an aggregate consumption
1âˆ’a
1t Â· x2t (0 <
good. For present purposes, therefore, form the Cobb-Douglas aggregate xa
a < 1), which is homogeneous of degree one in xt . Applying the logarithm to this index
of consumption, we obtain
u(xt ) = a ln x1t + (1 âˆ’ a) ln x2t âˆ€ t.
There is much less guidance to be had about Ï†(Â·). In their study of Kenya over the his-
torical period 1950-1990, Bell, Bruhns and Gersbach (2006) employed the logarithmic
form for u, but the data resisted their attempts to impose this on Ï†. More curvature
was needed, and successful calibration was achieved with the iso-elastic form
âˆ’Î·
t+1 ) = 1 âˆ’ (Î»t+1 ) /Î· ,
Ï† (Î» 2 2
whereby the value of Î· lay in the range 0.35 â€“ 0.65, with a clustering around 0.5. This
form will be adopted here, too. The associated values of Î· will provide a useful point
of departure.
3.2 Parameters and the values of exogenous variables
We need some starting values, namely, for period t = 1. In view of Indiaâ€™s demographic
history and the prevailing state of aï¬€airs in rural areas, let N1 = (3, 2, 0.75), with
n1 = 3.5. There is much illiteracy among the old, but less among their children, who
are todayâ€™s parents. Rising productivity over the past generation also suggests that Î»2
1
1 . Hence, let Î»1 = (1.7, 1.2).
is substantially larger than Î»2
Given the numerous parameters, and corresponding degree of under-identiï¬?cation,
there is no call for great precision everywhere. Let Î³ = 0.65 and the social norms
demand Î² = 0.6 and Ï? = 0.8. Households are still rather poor, so their taste for good
13
1 should be at least as strong as that for good 2: accordingly, let a = 0.5. Without
loss of generality, set the prices of both goods in the town at unity in all periods. A
survey of 30 villages in upland Orissa for the year 2009-10 (Bell and van Dillen, 2012)
yields the ï¬?nding that in the absence of an all-weather road, the unit transport costs
for paddy, the main crop, were a bit less than 0.1, but those for other commercial crops
somewhat higher. Applying the same to fertilizers, seeds and other goods bought in,
households in such villages then face the price vector pt = (0.9, 1.1) âˆ€ t. As for the trip
to school, the great majority of Indiaâ€™s villages have a primary one of their own; but
only a small minority have a secondary or high school, and in the absence of an all-
weather road, the daily round-trip time can be rather long. The averages for primary
and secondary school pupils in the Orissa villages lacking such a road were 23 and 76
minutes, respectively. Allowing, say, twelve hours for sleeping, eating and bathing at
home, and taking into account the higher opportunity costs of older childrenâ€™s time,
let Ï„ = 0.08. The direct costs of state schooling are surely modest: recalling (8),
let Ïƒt be 0.15 times the opportunity cost factor p1t Î±t (1 + Ï„ )Î³ , whereby Î±t has yet to
be determined. For the moment, we also defer discussion of the inter-temporal taste
parameters b1 and b2 .
Coming by estimates of premature adult mortality is a far easier task than that of
morbidity. In a setting of three overlapping generations, each generation corresponds
to about 20 years, the age at which full adulthood is attained. The rate qt therefore
corresponds to 20 q20 , the probability that an individual will die before reaching 40,
conditional on reaching 20. For India in 2005, WHO (2007) gives 20 q20 = 0.065 and
30 q20 = 0.123. Something closer to the latter is suited to our present purposes, ï¬?rst, to
allow for some mortality in the ï¬?rst part of old age, which the rigid time structure of
the model rules out, and secondly, to reï¬‚ect higher mortality among rural middle-aged
adults than the all-India average. Hence, let q1 = 0.125. If history and international
experience are any guides, this is sure to fall over the coming generation, PMGSY or no.
It does not seem too much to hope that India will do as well then as China does now,
so let q2 = 0.053. Where morbidity and disability are concerned, school-age children
typically suï¬€er less sickness than their parents, who, in turn, are in better health
than their aged parents. The ï¬?ndings from the Orissa survey conï¬?rm as much, the
average number of days of reported sickness being 5.6 a year, though the contribution
of chronic ailments may not have been properly covered. It is also possible that sickness
was interpreted as being too unwell to go about the usual daily business at all, as
opposed to being in a state of diminished capacity to do so. Thus, the vector d1 =
(0.02, 0.04, 0.08) represents a rather speculative stab at an estimate for period 1. Since
14
it is easier to ward oï¬€ premature death than morbidity, the associated improvement to
d2 = (0.015, 0.03, 0.06) in period 2 is a bit less dramatic than that in mortality.
3.3 Setting up the sequence under perfect foresight
The system must be set up in such a way that it satisï¬?es two requirements. First,
the household must choose a plan that is in keeping with what we observe in the
present. The key variable here is the level of investment in education, e1 . Children
in Indiaâ€™s rural areas typically start school at 6 years of age and complete about 6
years of schooling on average. Hence, with up to 12 years of schooling available, and
noting that morbidity reduces the endowment of productive time, the model must be
1 = (6/12)(1 âˆ’ d1 ). A further adjustment is needed for the
set up so as to yield (1 + Ï„ )e0 1
number of days lost due to bad weather, especially in the monsoon. The Orissa survey
yields an estimated 8.6 and 9.4 days a year, on average, for primary- and secondary-
school children, respectively, involuntary absences that were mainly attributable to
their teachersâ€™ failure to arrive (Bell and van Dillen, 2011). Assuming a school year
1 = (1 âˆ’ 9/180)(6/12)(1 âˆ’ d1 ) =
of 180 days, the required condition becomes (1 + Ï„ )e0 1
(19/40)(1 âˆ’ d1 1
1 ), where Ï„ = 0.08 and d1 = 0.02.
Secondly, the whole sequence must be anchored to some plausible conï¬?guration in
the future. In this connection, there is much talk of meeting the so-called Millennium
Development Goals by 2015, and suchlike. Let us suppose, therefore, that a full ed-
ucation for all is attainable within one complete generation, where the deï¬?nition of a
â€˜full educationâ€™ must make a full allowance for the claims on a childâ€™s time made by
illness, traveling to school and other involuntary absences. That is to say, the model
must be set up such that parents in period 2 do choose e0 Â¯2 = 0.95(1 âˆ’ d1
2 = e 2 )/(1 + Ï„ ).
If the general environment described by Zt does not deteriorate thereafter, it will then
follow that e0
t = e t )/(1 + Ï„ ) âˆ€t â‰¥ 3. For once et = 0.95(1 âˆ’ dt )/(1 + Ï„ )
Â¯t = 0.95(1 âˆ’ d1 0 1
is attained, the young adults in that period can then be certain that all future gen-
erations will continue this policy, with its corresponding eï¬€ects on their consumption
in old age, as formulated in (15). The desired anchoring of the system will have been
accomplished.
The simplest way of ensuring all this where Zt is concerned is to impose stationarity
from period 2 onwards. On the assumption that fertility will fall to replacement levels
by 2030, and allowing for premature mortality among adults, let Nt = (2, 2, 1.5) âˆ€t â‰¥ 2.
Also stationary are prices, tastes and costs, a road being built, if at all, at the start
15
of period 1 (see below). Recalling the discussion of mortality and morbidity in Section
3.2, we also have qt = 0.053 and dt = (0.015, 0.03, 0.06) âˆ€t â‰¥ 2.
The ï¬?nal step is to choose the productivity parameters zt and Î± and the intertemporal
2 = 0.95(1 âˆ’ 0.015)/1.08 = 0.866 with as little
taste parameters b1 and b2 so as to yield e0
to spare as possible. With the exception of zt , this must be accomplished by trial and
error as part of the process of computation. The only prior restriction is that with pure
impatience for consumption, b1 > b2 . Since premature mortality already appears in
connection with preferences over dated consumption, the pure discount rate arguably
should not greatly exceed 15 percent per generation of 20 years.
One can arrive at an appropriate value of z by imposing the assumption that indi-
vidual productivity, Î»t , will grow without limit if, after some point, all generations are
fully educated. As noted in Section 2.1, by choosing a suï¬ƒciently productive technology
f (et ) and otherwise setting up the system so that e0 Â¯t âˆ€t â‰¥ 2, where e0
t = e Â¯t is suf-
t = e
ï¬?ciently close to 1, we ensure that Î»2t will indeed grow without bound. Recall from (3)
that the relevant characteristic root when the upper limit on et is unity is 1+ g âˆ— . Reduc-
Â¯ = 0.866 yields the root a2 [1 + (1 + 2a3 /a2
ing this upper limit to e 2 f (0.866))]f (0.866),
where f (et ) = zt et , a2 = 2/3.5 and a3 = 1.5/3.5. Let zt = 0.85 âˆ€t, which implies that
output per head will grow at the rate of 41.3 percent per generation, or 1.74 percent a
year. This seems defensible for an horizon many generations oï¬€.
4 A New Road: Exact Welfare Measures
Imagine two islands, each inhabited by a representative extended family. The ï¬?rst is
characterized by the constellation of numerical values set out in Section 3. The second
is identical, except for the happy event that a road is provided free of charge at the
very start of period 1. As a result, unit transport costs, travel-times, morbidity and
mortality all become lower than those on island 1. In this more benign environment, it
t > 0.866 âˆ€t â‰¥ 2, but also highly likely that e1 will be higher
is not only certain that e0 0
âˆž
than on island 1. The latter being so, the path {Î»2 t }t=2 will lie everywhere above its
counterpart on island 1. The two paths will, moreover, exhibit diï¬€erent asymptotic
rates of growth, even though they share the common value of zt . This â€˜growth-eï¬€ectâ€™
stems from lower morbidity, reduced round-trip times to school and fewer involuntary
Â¯t closer to unity â€“ as long as the road does
absences, all of which bring the upper limit e
not fall into utter disrepair. If it were to do so at the end of period 1, for example,
t = 0.866 âˆ€t â‰¥ 2 would still be ensured thereafter, under the hypothesis that it holds
e0
16
on island 1, which will have no road in any period. Even in the face of such neglect,
therefore, there would remain a â€˜level-eï¬€ectâ€™.
Island 1, therefore, provides the benchmark for the following thought-experiment.
At the very start of period 1, the inhabitants of island 2 are given the choice between
having the road and making do with the conditions ruling on island 1, but receiving a
lump-sum payment instead. If the latter sum is such that they are indiï¬€erent between
these two alternatives, we will have found the equivalent variation (EV) corresponding
to the provision of the road, as assessed by the young adults in period 1 on both islands.
The road will, of course, yield further beneï¬?ts in later periods, even if it falls into utter
disrepair at the end of period 1. For the level-eï¬€ect alluded to above will come into
play, and to the associated increase in the familyâ€™s full income there will correspond an
EV for that generation of young adults. In what follows, however, we will conï¬?ne our
attention to the EV for young adults in periods 1 and 2, leaving the task of estimating
the whole sequence thereof to another paper.
4.1 The benchmark: No road
Recalling Section 3.3, the task here is to choose Î±, b1 and b2 so that e0
2 = 0.866 is
âˆž
t }t=1 can be derived on
barely attained. It is easily seen that the whole sequence {Î»2
the hypothesis that e0t = 0.866 âˆ€t â‰¥ 2 without any reference to Î±, b1 and b2 . The
parameter values, the initial conditions and (25) yield
(19/40)(1 âˆ’ 0.02) 2 Ã— 1.7 + 0.75 Ã— 1.2
2 = 2 Ã— 0.85 Â·
Î»2 Â· + 1 = 2.1457 .
1.08 2 + 0.75
Continued recursion using {Nt }âˆž
t=2 yields
2 Ã— 2.1457 + 1.5 Ã— 1.7
3 = 2 Ã— 0.85 Ã— 0.866 Â·
Î»2 + 1 = 3.8791 ,
2 + 1.5
2 Ã— 3.8791 + 1.5 Ã— 2.1457
4 = 2 Ã— 0.85 Ã— 0.866 Â·
Î»2 + 1 = 5.6195 ,
2 + 1.5
and so forth.
Having thus determined that Î»2 0
2 = 2.1457, the hypothesis e2 = 0.866 leaves the
household in period 1 with almost all the information needed to derive p2 Â· x0
2 from
0
(5), whereupon x2 would follow from the assumption that the sub-utility function u
is (transformed) Cobb-Douglas. The missing element is the value of Î±. Moving back
to calibration, therefore, it follows that by hazarding a guess at Î±, we are then left to
17
ï¬?nd a pair (b1 , b2 ), with b2 â‰ˆ 0.85b1 , such that the solution to problem (15) in period
1 indeed involves e0 1 = (19/20)(6/12)(1 âˆ’ 0.02)/1.08, whereby Î± may be varied until
the desired result is obtained. Mindful of the need to fulï¬?ll the hypothesis e0
2 = 0.866,
but not too comfortably, some experimenting yielded Î± = 5, albeit with quite strong
impatience (b2 = 0.8b1 = 21.8047) and modest curvature of the sub-utility function
Ï† (Î· = 0.1).
The resulting sequence of the main endogenous variables when there is no road is
set out in the upper panel of Table 1. Three generations on, in some 60 years, a
young adult is over three times more productive and enjoys a level of real consumption
likewise three times higher than his or her great-grandparents were in young adulthood
in period 1.
Table 1: The sequences of the main variables, with and without the road
Period 1 2 3 4
No Road x1t 2.8177 3.6598 5.9275 9.2346
x2t 2.3054 2.9943 4.8495 7.5556
e0
t 0.4310 0.8664 0.8664 0.8664
Î»2
t 1.7000 2.1457 3.8791 5.6195
Road: Variant 1 x1t 2.7605 3.8022 6.4384 10.4149
x2t 2.4976 3.4401 5.8252 9.4230
e0
t 0.4907 0.9366 0.9366 0.9366
Î»2
t 1.7000 2.3044 4.2567 6.4454
Road: Variant 2 x1t 2.7537 3.8173 6.4631 10.4478
x2t 2.4914 3.4538 5.8476 9.4527
e0
t 0.4959 0.9366 0.9366 0.9366
Î»2
t 1.7000 2.3182 4.2692 6.4661
Variant 1: q1 = 0.125, qt = 0.053 âˆ€t â‰¥ 2. Variant 2: q1 = 0.101625, qt = 0.0451 âˆ€t â‰¥ 2.
18
4.2 Life with the road
We need to specify how the road improves the householdâ€™s environment in period 1,
as described by the vector Z1 . Starting with unit transport costs for goods, Bell and
van Dillen (2012) estimate that these are reduced by about 5 percent of the net price
received for commodities marketed. Khandker et al. (2009) obtain a similar estimate
for rural roads in Bangladesh, once an allowance is made for greater volumes marketed.
They also estimate that the farm-gate price of fertilizers declined by 5 percent. With a
new road, therefore, let p1 = (0.95, 1.05). Where schooling is concerned, Bell and van
Dillenâ€™s (2012) analysis of the Orissa survey yields substantial savings in travel-time for
secondary-school pupils (just over 30 minutes a day) and far fewer days of involuntary
absences for all grades (about 2 instead of about 9 for primary and 14 for secondary).
These estimates imply Ï„ = 0.04 and e t )/(1+0.04) = 0.9508(1 âˆ’ dt ).
Â¯t = (1 âˆ’ 2/180)(1 âˆ’ d1 1
It is not to be expected that the provision of an all-weather road will have such strong
eï¬€ects on morbidity, at least in the very short run; and Bell and van Dillen (2012) ï¬?nd
none in the Orissa villages. Households did, however, respond by switching away from
local healers, quacks and primary clinics to hospitals, which certainly suggests that
all-weather roads are valued in this connection. To be quite conservative, therefore, let
there be no improvement in dt âˆ€t. The ï¬?ndings where mortality is concerned are mixed.
An analysis of individual mortality in the sample households revealed no eï¬€ects; but
the respondents in the village â€˜focus groupâ€™ interviews were adamant that mortality
had dropped sharply, as the acutely sick and injured could be treated in good time.
According to their claims, in the group of nine such villages, the average number of
such deaths had dropped from 3.25 to 0.75 annually. Even allowing for â€˜unavoidableâ€™
deaths, these claims are rather startling; for the crude mortality rate of about 10 per
1000 in the sample drawn from all 30 villages implies just 10 deaths annually in a
typical village of 1000 souls. Since qt is assumed to fall from 0.125 in period 1 to 0.053
in period 2 even with no road, let the provision of the road reduce q1 , not by the implied
25 percent or so, but rather by 10 percent, to 0.1125.
Turning to future periods, let the road be perfectly durable, thereby maintaining
these more favourable prices of goods, as well as travel-times to, and involuntary ab-
sences from, school indeï¬?nitely. As argued in Section 3.2, however, morbidity and
mortality will surely fall over the next generation even in the absence of a road.
Staying on the conservative side, let the road continue to have no eï¬€ect on morbid-
ity in period 2, but let the 10 percent reduction in mortality continue to hold, so
that q2 = 0.0477. Thereafter, all these rates remain stationary. The conï¬?guration
19
(q1 = 0.1125, qt = 0.0477 âˆ€t â‰¥ 2) is denoted by (mortality) Variant 1. In view of the
more striking reductions reported in the survey, we also examine Variant 2, in which
there is a 15 percent reduction in q : (q1 = 0.101625, qt = 0.0451 âˆ€t â‰¥ 2).
The next step is to compute the sequence of perfect foresight equilibria that will arise
under these more favorable sequences of {Zt }âˆžt=1 . The values of the main variables for
the ï¬?rst four periods are reported in the lower two panels of Table 1. It is seen that,
three generations on in Variant 1, the levels of real productivity and consumption7 are
about 14.7 and 18.6 percent higher, respectively, than on island 1. With the lower
levels of mortality in Variant 2, the corresponding ï¬?gures are 15.1 and 19.0 percent,
respectively.
4.3 The beneï¬?ts generated by the road
The EV for young adults in period 1 is obtained by ï¬?nding lump-sum payments such
that they would attain the same level of expected utility were they confronted instead
with the less favorable environment ruling on island 1. Since most will survive into
old age and, in this model, there are no ï¬?nancial instruments beyond pooling and the
social norm expressed by Ï?, it is desirable to smooth the payments. For simplicity, let
2
there be one payment in period 1, T1 , which each young adult will enjoy for sure, and
another of equal size in period 2, conditional on the individual surviving into old age.
The level of the familyâ€™s (normalized) full income in period 1 is therefore augmented
2
by the amount T1 , and its allocation remains subject to the social norms expressed by
Î² and Ï?. In keeping with the assumption that altruism disappears with the onset of
old age, let surviving individuals keep the contingent payment in period 2 wholly for
themselves. Under the social norms, these receipts have no eï¬€ect on their childrenâ€™s
decisions as adults in period 2, and thus introduce no further complications into the
2
computation of the whole sequence. The task, therefore, is to ï¬?nd a T1 such that
young adults attain E1 U 0 with the road in period 1, given that they correctly forecast
Ï? x0 2
2 when solving problem (15) with (normalized) full income augmented by T1 and so
enjoy purchasing power p Â· Ï? x0 2
2 + T1 in old age, conditional on their surviving to enjoy
it. Given Cobb-Douglas preferences, the sum p Â· Ï? x0 2
2 + T1 will be spent in proportions
a and 1 âˆ’ a on goods 1 and 2, respectively, which completes the formulation of their
2
decision problem, and so permits the joint computation of T1 and the whole sequence.
7
Form and compare the Cobb-Douglas aggregates (x1t x2t )0.5 .
20
The payments in question for Variants 1 and 2 are reported in the column so labeled
in Table 2. To put them in perspective, recall that a young adult in period 1 can
produce 5 Ã— 1.7 = 8.5 units of good 1, with a farmgate price of 0.9 in the absence
2
of the road. In Variant 1, therefore, T1 is 7.7 percent of the value of this output. In
the still healthier environment of Variant 2, it rises to 8.3 percent thereof. Here, it
should be recalled that those who survive into old age will receive the same payment
2
once more on reaching that phase of life, and that in view of the fact that T1 reï¬‚ects
both intertemporal substitution and pure impatience, there is no reason to discount
the second (contingent) payment when summing up over both periods to yield the total
beneï¬?t received by each adult who is young in period 1.
Table 2: The EV and its decomposition, t = 1
2 2
E1 U1 T1 E1 U1 (Î”p) T1 (Î”p) 2
E1 U1 (Î”pc ) T1 (Î”pc )
No Road 82.8544
Road: V1 86.1470 0.5868 85.0189 0.3994 83.8044 0.2044
(68.1%) (34.8%)
Road: V2 86.4477 0.6379 85.0189 0.3994 84.0982 0.2510
(62.6%) (39.3%)
Later generations also beneï¬?t, not only because the road is assumed to be perfectly
durable, but also because it promotes human capital formation in period 1, and so
confers higher productivity on adults in all subsequent periods. In Variants 1 and 2,
2
T2 is 14.7 and 15.7 percent, respectively, of the value of a young adultâ€™s output in the
absence of a road (5 Ã— 2.1457 Ã— 0.9). These beneï¬?ts, and those accruing to subsequent
generations, must be discounted back to period 1 at the appropriate rate; for those
receiving them arrive progressively later on the scene under conditions of improving
living standards. Even so, their contribution to the discounted sum of all beneï¬?ts will
be large: with an inter-generational discount rate of 100 percent (almost 4 percent per
2 2
annum), for example, the present value of T2 comfortably exceeds T1 .
21
Table 3: The EV and its decomposition, t = 2
2 2
E2 U T2 E2 U2 (Î”p) T2 (Î”p) 2
E2 U2 (Î”pc ) T2 (Î”pc )
No Road 86.0689
Road: V1 91.7588 1.4220 88.6707 0.6309 89.3451 0.7991
(44.4%) (56.2%)
Road: V2 92.1203 1.5182 88.6707 0.6309 89.7038 0.8896
(41.6%) (58.6%)
4.4 Decomposing the contributions
The EV measures the combined beneï¬?ts generated by favourable changes in transport
costs, the time available for schooling, morbidity and mortality. As established in
Section 2.3, the ï¬?rst lends itself fairly readily to empirical estimation through observa-
tion of householdsâ€™ actual behavior, but the last poses severe problems, as do changes
Â¯ and child morbidity when parents decide to educate their children fully. If the
in e
parametrization of the model be accepted, however, we can proceed by answering the
question, how big is the contribution of the reduction in transport costs to the EV
in comparison with that of the rest taken together? For given that the former can
be measured independently and empirically, the model will then yield a rough, but
defensible estimate of the latter.
To recap from Sections 4.1 and 4.2, the road causes the constellation
pt = (0.9, 1.1) âˆ€t, Ï„ = 0.08, d1 = (0.02, 0.04, 0.08), dt = (0.015, 0.03, 0.06) âˆ€t â‰¥ 2,
q1 = 0.125, qt = 0.053 âˆ€t â‰¥ 2, e
Â¯ = 0.866
to become, in Variant 1,
pt = (0.95, 1.05) âˆ€t, Ï„ = 0.04, d1 = (0.02, 0.04, 0.08), dt = (0.015, 0.03, 0.06) âˆ€t â‰¥ 2,
q1 = 0.1125, qt = 0.0477 âˆ€t â‰¥ 2, e
Â¯ = 0.9366.
The desired decomposition involves the changes in pt . Here, the problem of path de-
22
pendence cannot be avoided; for the EV corresponding to the change in the whole
constellation is not, except by mere ï¬‚uke, additively separable in the component parts.
We therefore proceed as follows. Suppose, at one extreme, the road were to aï¬€ect only
2
pt . As before, we can calculate the value of T1 that corresponds to this hypothetical
change in the original constellation, and then express it as a proportion of its coun-
2
terpart under the complete change. The columns labelled E1 U1 (Î”p) and T1 (Î”p) in
Table 2 report the corresponding levels of expected utility and the said payments. In
Variant 1, the value of the reduction in transport costs by itself makes up 68.1 percent
2
of T1 , falling to 62.6 percent in Variant 2.
At the other extreme, one can calculate the payment that corresponds to all the
2
changes in the original constellation except pt , which is denoted by T1 (Î”pc ). The
2
residual left after subtracting this from the value of T1 corresponding to the complete
change is then attributable to the change in pt plus any interaction eï¬€ects between
changes in transport costs and all other changes. In Variant 1, the corresponding share
is 65.2 per cent, just 2.9 percentage points smaller than that when the reduction in
transport costs stands alone. The share in Variant 2 is likewise lower, at 60.7 percent.
In period 2, these shares fall quite sharply, ranging between 41.4 and 44.4 percent.
By way of an independent check on these ï¬?ndings, one can compare the above es-
timates with those yielded by a completely diï¬€erent approach. The one that comes
immediately to mind involves the derivation of the so-called Value of a Statistical Life
(VSL) from variation in wage rates across occupations or industries and the associated
variation, if any, in fatality rates, an approach vigorously promoted by Viscusi (for a
survey, see Viscusi [1993]). Pursuing this line, Simon et al. (1999) employ data from
Indian manufacturing ï¬?rms and arrive at the rather startling estimate that the VSL
is 20 to 48 times larger than the present value of lifetime foregone earnings. As the
authors note, this greatly exceeds the estimate of 7 to 8 obtained by Liu et al. (1997)
for Taiwan, China in the 1980s.
Be that as it may, what is the ratio implied by the approach adopted here? Consider
2
Variant 1 in period 1. The equivalent sum T1 (Î”q ), which is deï¬?ned in relation to
changes in mortality alone, stems from a reduction in q1 of 0.0125 (= 0.1 Ã— 0.125).
Since the said sum is paid twice, the implied VSL is 19.264 (= 2 Ã— 0.1204/0.0125),
which is 2.5 times the value of output in old age of a surviving young adult, namely,
7.65 (= 5 Ã— 1.7 Ã— 0.9). The implied VSL in period 2 is 82.566 (= 2 Ã— 0.2188/0.0053), as
against lost output in the value of 9.656 (= 5 Ã— 2.1457 Ã— 0.9), a ratio of just over 8.5 to
one. In the present framework, however, Tt2 (Î”q ) relates not to the risk that a young
23
adult will fail to survive into old age and so not produce any output in that phase of
life, but rather that he or she will not enjoy her claim on the common pot in old age,
whose value is Ï? Â· pt+1 xt+1 . Young adults in period 1 face the risk of losing 5.270 in
period 2, which in relation to the VSL implies a ratio of 3.66 to one. Their counterparts
in period 2 face the possible loss of 8.535 in period 3, which implies a ratio of 9.67 to
one. All in all, these ratios for period 1 do seem to be rather on the low side, which
suggests that the beneï¬?ts stemming from the posited reductions in mortality may be
somewhat higher than those implicitly reported in Tables 2 and 3.
5 Conclusions
Providing backward rural areas with all-weather roads promotes not only production
and trade in what can be called the â€˜commercialâ€™ sphere of life, but also the forma-
tion of human capital and health in the â€˜non-commercialâ€™ one. The main object of
this paper has been to develop a method to value their respective contributions to
the total beneï¬?t generated by rural roads. For given that obtaining independent es-
timates of the beneï¬?ts, deï¬?ned as willingness to pay, in the commercial sphere of
production and trade is relatively straightforward, the respective shares of the â€˜com-
mercialâ€™ and â€™non-commercialâ€™ spheres in the total beneï¬?ts, as derived in the uniï¬?ed
OLG-framework employed here, then yield an estimate of the willingness to pay in
the non-commercial sphere that is anchored to the independent estimate of that in
the commercial sphere. Based on survey evidence from a backward region of Orissa in
east India and Bangladesh, as well as elements of more usual forms of calibration, the
model yields a ratio of commercial to non-commercial beneï¬?ts of about two-to-one in
the ï¬?rst generation, falling to three-to-four in the second. This is broadly consistent
with the valuations expressed by respondents in the Orissa survey, who ranked the
latter beneï¬?ts at least on a par with the former.
24
Appendix
by Jochen Lapsâˆ—
This appendix describes the solution methods used to produce the results in Tables
1, 2 and 3. To that end, it is divided into two sections. The ï¬?rst summarizes the
parameters and exogenous variables with and without the road. The second deals with
the calibration of the model, the calculation of the sequences reported in Table 1 and
the equivalent variation of Tables 2 and 3. The computer package used is MATLAB. The
code is available upon request.
A.1 Parameters and Variables
The road causes the householdâ€™s demographic environment and its expectations con-
cerning the latter to change as reported in Table A.1. The vectors d1 = (0.02, 0.04, 0.08)
and dt (0.015, 0.03, 0.06) âˆ€t â‰¥ 2 are unaï¬€ected by the road.
Table A.1: Demographic Environment
Period 1 tâ‰¥2
2
No Road (qt , Et qt+1 ) (0.125, 0.053) (0.053, 0.053)
2
Road (V1) (qt , Et qt+1 ) (0.1125, 0.0477) (0.0477, 0.0477)
2
Road (V2) (qt , Et qt+1 ) (0.10625, 0.0451) (0.0451, 0.0451)
Table A.2 summarizes all remaining parameter values, including prices, time for
travelling to and from school, and the eï¬ƒciency factor in the educational technology.
Some of these variables change with the road, though not across scenarios.
A.2 Algorithm
The anchoring of the system ï¬‚eshed out in Section 2 is achieved by appropriate choices
for the productivity parameter Î± and the intertemporal taste parameters b1 and b2 .
âˆ—
University of Heidelberg
25
Table A.2: Parameter Values
Î² Ï? Î· Î³ a z p Ï„
No Road (0.9, 1.1) 0.08
0.6 0.8 0.1 0.65 0.5 0.85
Road (0.95,1.05) 0.04
Algorithm 1 calibrates the model such that, without the road,
Â¯2 = (19/20) Â· (1 âˆ’ d1
1 = (19/40)(1 âˆ’ d1 ) and e2 = e
(1 + Ï„ )e0 2 )/(1 + Ï„ ), t â‰¥ 2.
1 0
(A.1)
That is to say, a young agent in period t correctly anticipates that the next generationâ€™s
optimal choice involves full education in period t + 1. This requirement has to be
imposed in the maximization problem of the young agent in period t+1. If indeed e0 t+1 =
(19/20)Â·(1âˆ’d1t+1 )/(1+Ï„ ), and if the general environment does not deteriorate thereafter,
then the second part of equation (A.1) follows immediately. As the road encourages
education, it is also clear that, with the road, agents will choose full education from
period t + 1 onwards. Note that the maximal education level is higher with the road
Algorithm 1 Calibration
1: for it = 1 : maxit do maxit: prespeciï¬?ed # of iterations
2: Given Î±, b1/b2, and some inititial interval [b1 , b1 ], solve the young adultâ€™s deci-
sion problem
1 âˆ’ e1,it ) < tol ) then
if (abs(e0 0
3: e0
1 given in (A.1)
4: tol = 1e âˆ’ 4: prespeciï¬?ed tolerance level
5: A solution found.
6: Go to Algorithm 2
7: else
8: Adjust the taste parameter b1 , using the fact that e0 is decreasing with b1 .
9: end if
10: end for
because of fewer days of involuntary absences and a smaller Ï„ . Algorithm 1 searches
for the taste parameter b1 . Given Î±, the ratio b1 /b2 , and an initial guess for the interval
in which we expect the b1 satisfying (A.1) to be located, Algorithm 1 solves the young
adultâ€™s decision problem (15). As e0 1,it is continuous and monotone in b1 , there are
root-ï¬?nding methods available. I use a simple version of the derivative-free regula falsi
method, which combines elements of bisection and secant methods, to solve for the
26
1 âˆ’ e1,it . This results in
root of the function e0 0
Î± = 5, b1 = 27.2266 and b2 = 0.8 Â· b1 = 21.8047.
Algorithm 2 generates the sequences reported in Table 1. Before doing so, special
attention is paid to the second period. If, without the road, e0
2 is either smaller than,
or unreasonably higher than its maximal level 0.95(1 âˆ’ d1
2 )/(1 + Ï„ ) = 0.866, then one
goes back to Algorithm 1 with appropriately adjusted parameter values. Here z is a
promising candidate, as the eï¬ƒciency factor directly aï¬€ects individual productivity,
Î»t . Recall that for all values of z larger than 0.577, there is unbounded growth of Î»2
t
at the rate given in equation (3). Note, however, that a value for z that is too low
results in a non-monotone sequence for expected utility, at least without the road. The
reason is that an ineï¬ƒcient eductional technology is compatible with a young adult
choosing extensive education for his oï¬€spring only if the weight n on the altruistic
term in Et Ut is relatively high, given Ï†(Â·). Hence, for this model speciï¬?cation, the
taste parameters b1 and b2 must fall dramatically with a decline in z in order to satisfy
equation (A.1). If n is indeed relatively high at the outset, however, its decline along
the presumed demographic transition outweighs the growth in human capital, at least
in period t = 2. In order to avoid such behaviour, we set z = 0.85. The parameter
constellation that yields an e0
2 consistent with (A.1) triggers the calculation of the
sequence.
To complete the picture, Table A.3 displays the sequences of Et Ut for all scenarios.
Table A.3: The sequences of Et Ut , with and without the road
Period 1 2 3 4
No Road 82.8544 86.0689 109.1654 129.8546
Road V1 86.1470 91.7588 117.0121 139.7707
V2 86.4477 92.1203 117.3874 140.1497
Algorithm 3 takes as inputs the sequences for Et Ut with and without the road and
solves for the equivalent variation in terms of the familyâ€™s (normalized) full income.
A young adult in period t enjoys the lump-sum payment Tt2 for sure, while a second
payment of Tt2 in period t + 1 is conditional on surviving into old age. Note that a sole
payment received as a young adult results in an equivalent variation that is greater
27
Algorithm 2 The Sequences under Perfect Foresight
1: for time = 1 : T do T : # of periods considered
2: Î±, b1 ,b2 , given.
3: if e0
t,it > eÂ¯t then
4: Solve the young adultâ€™s decision problem in current and future periods with
e0
t = Â¯
et t â‰¥ 2.
,
5: else
6: Return to Algorithm 1 using other parameter values
7: end if
8: end for
Algorithm 3 Calculating the EV
1: for time = 1 : T do T : # of periods considered
2: for it = 1 : maxit do
3: Given Î±, b1 and b2 and some initial guess for the transfer Tt2 , solve the young
adultâ€™s decision problem.
4: if (abs(Et Utroad âˆ’ Et Ut,it
no road
) < tol) then tol = 1e âˆ’ 4
5: A solution found.
6: else
7: Adjust the Transfer Tt2 , using the fact that Et Utno road increases with Tt2 .
8: end if
9: end for
10: end for
than 2 Â· Tt2 , a reï¬‚ection of the desire to smooth out consumption over the life cycle.
Algorithm 3 ï¬?nds the transfer Tt2 that yields Et Ut0 with the road in period t when
solving the young adultâ€™s decision problem without the road but with (normalized) full
income augmented by Tt2 , given a correct forecast of his purchasing power in old age,
pÏ?x0 2 2
t+1 + Tt . As expected utility is continuous and strictly increasing in Tt the regula
falsi method is again appropriate. The system is much simpliï¬?ed by the assumption
that altruism disappears in old age, because the transfer received in this stage of life will
not aï¬€ect the individualâ€™s childrenâ€™s future decisions as adults. Note that the transfer
in period t = 1 alters the current optimal decision and hence the future environment
such that the system exhibits path dependency with respect to the transfer payments.
In order to ï¬?nd the equivalent variation for periods later periods, t â‰¥ 2, one then has to
reset the Et Ut â€“sequence to its values without the road and without transfer payments.
Algorithms 2 and 3 are applied to all scenarios and all decompositions of the combined
beneï¬?ts stemming from the road.
28
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29