Journal of Economic Growth, 1: 449-486 (December 1996)
9 1996KluwerAcademicPublishers,Boston.
The Economics of Poverty Traps
Part One: Complete Markets
COSTAS AZARIADIS
University of California, Los Angeles
This paper lists theoretical reasons why neoclassical models of one-sector growth imply that nations with identical economic structures need not converge to the same steady state or balanced growth path, and outlines the
empirical significance and policy implications of conditional nonconvergence. We survey poverty traps in both
convex and nonconvex economies with complete market structures. Among the potential causes of traps are subsistence consumption; distorted international trade in intermediate inputs; demographic transitions when fertility
is endogenous; technological complementaritiesin the production of consumption goods, financial intermediation
services, manufactures, or human capital; coordination failures among voters; various restrictions on borrowing;
indivisibilities in human capital formation or child rearing; and monopolistic competition in product or factor
markets.
Keywords: nonconvergence,persistence, history
JEL classification: D91, J13, J24, 011,041
1.
Introduction
Economists have spent a great deal of time and resources in the last ten years to understand
why the very richest and the very poorest countries do not converge in output per worker. The
data are unequivocal on this point: the world distribution of income is thinning in its middle
r a n k s ) For example, among the countries included in the columns o f the income distribution
histograms reported by Barro and Sala-i-Martin (1995), and drawn in Figure 1, only 26%
o f the total number o f 118 countries belonged to the extreme 8 columns (specifically, 17%
are in the four richest and 9% in the four poorest income categories) in 1960. By 1990,
however, 35% o f 129 countries were in these tail positions, with approximately 23% of
listed nations ranking in the top four income classes and 12% ranking in the bottom four.
Quah (1996) confirms these findings using a more refined statistical methodology.
Our profession finds nonconvergence interesting because well-known turnpike results
from optimum growth theory and related descriptions of the development process lead us
to expect convergence; in addition, many o f us share a gut instinct that increased resource
mobility and faster diffusion o f technology will enable the poor to follow in the footsteps
o f the rich, narrowing existing gaps in per capita GDP. Figure 2(a) sums up this view o f
growth as a smoothly converging dynamical system that applies universally to all nations,
both developing and developed.
F r o m formal tests and anecdotal evidence, we have identified regional groups for which
the convergence hypothesis seems to be at work (States of the Union, Japanese prefectures,
450
AZARIADIS
12-2,
8
6J
onllllll
5.5
5.9
6.3
.6.7. . 7.1. . 7.5. . .7.9.
8.3
8.7
9.1
(a)
12~
10 L
8
5.5
6
6.5
7
7.5
8
8.5
9
9.5
(b)
Figure 1. Worldincome distribution. (a) Histogram of log (real per capita GDP) in 1960. (b) Histogram of log
(real per capita GDP) in 1990. From R, Barro & X. Sala-i-Martin, Economic Growth.
O E C D countries, East Asian tigers), and others for which it does not (regions of Italy,
Africa, some petroleum exporters, parts of Latin America). To explain nonconvergent
growth, a number of researchers have modified the standard models of accumulation to
obtain nonergodie equilibrium growth paths that contain several attractors, e.g., steady
POVERTY TRAPS
451
states, balanced growth paths, or asymptotic distributions of world income. We call the
lowest of these attractors a "poverty trap" or a "low-level development trap."
The theoretical modifications needed to extract nonergodic behavior from otherwise standard models of economic growth range from such mild assumptions as subsistence consumption and credit market imperfections to quite substantial postulates like large external
increasing returns in producing consumption goods or financial intermediation services, or
else highly demand-elastic cost markups.
As of this writing, the evidence against reasonably rapid convergence among nations
seems to be much more solid than the evidence for any one of the individual mechanisms
that may prevent convergence to a high-level steady state. It seems useful at this stage
of our knowledge to make up a thorough list of the economic mechanisms that encourage
poverty traps as a prelude to evaluating the empirical relevance of each mechanism, and
to assessing some of the ways and means of escaping from aggregate states of persistent
national poverty.
This is, in rough outline, the ground I propose to cover in the remainder of this paper.
I place particular emphasis on listing the technologies, preferences, market structures,
fertility patterns, and public policies that preserve and augment initial inequality in per
capita income among otherwise identical national economies. Section 2 starts with an
examination of conditional convergence (that is, of persistent income inequality supported
by national differences in exogenous economic fundamentals like technology, fiscal policy,
and market structure). From that point forward, nations are regarded as fundamentally
identical. Section 3 surveys poverty traps for convex economies with complete markets,
and Section 4 nonconvex economies with complete markets} A summary with extensions
takes up Section 5.
2.
Alternative Explanations of Nonconvergence
Social scientists try to understand persistent differences in income at any level of aggregation
by deciding whether or not the incomes in question accrue to groups of fundamentally
identical earning potential. Are some families or nations persistently poorer than others
because they are innately disadvantaged in acquiring income-producing skills? Or are the
poor merely the victims of the circumstances in which they are initially placed by chance,
environment, or history? If the answer to the last question is yes, can a small or temporary
improvement in the opportunities of persistently poor groups result in a large or permanent
betterment of their lifetime income?
These are hard questions to answer with assurance at all levels--national, regional, or
family--because the data at our disposal do not permit us to disentangle easily the effect on earnings of innate characteristics from that of circumstantial position. Continued uncertainty over the factors responsible for persistent income inequality means that
researchers have to start with a working hypothesis about the ultimate causes of this inequality. For instance, the empirical research exemplified in the writings of Barro (1991),
Barro and Sala-i-Martin (1995), and of Mankiw et al. (1992) is based on the premise
that countries are inherently different in one or more fundamental aggregate features
(e.g., time preference, technology, demography, market structure, or economic policy)
452
AZARIADIS
and may follow distinct development paths converging to distinct steady states, as shown in
Figure 2(b).
An alternative working hypothesis, articulated in work by Galor and Ryder (1989), Murphy, Shleifer, and Vishny (1989), Azariadis and Drazen (1990), Boldrin (1992a), Durlauf
(1993a), Smith and his collaborators (various dates), and many others, 3 takes the growth
process of nations to be fundamentally the same except for differences in history, e.g., in the
circumstances from which the growth process begins. These are chiefly the starting stocks
of human and physical capital, and the state of technology. Figure 2(c) illustrates how a
history-dependent growth model explains the phenomenon of convergence clubs. Membership in these international clubs, one for the poor and one for the rich, in Figure 2(c), is
defined by reference to the threshold level fr of initial capital, which serves as an admission
ticket for the club of wealthy nations. Development paths quality for membership in this
club if and only if they start above f~2A third working hypothesis----exemplified in research by Krugman (1991), Matsuyama
(1991), Gali (1994), Klenow (1993), and in the work surveyed by Matsuyama (1995), and
Benhabib and Gali (1995)--proposes multiple equilibrium growth paths as an explanation
for asymptotic differences in the standard of living among nations that are similar both in
economic structure and in starting conditions. This approach pays particular attention to
how the community's expectations influence phenomena like overtaking and rank reversals,
that is, to the reasons why some nations gain ground in the world income distribution at the
expense of other, apparently similar nations.
Figure 2(d) depicts the mechanics of overtaking for two economies starting with different
stocks of material wealth. Economy A converges to the low steady state kl from a high
initial capital stock ka while economy B, starting with a low initial stock kff, ends up in the
higher steady state f~3,overtaking economy A in the process. The upper and lower branches
of the policy correspondence in Figure 2(d) represent the rapid growth path of economy B
and the slow one of economy A.
Each approach explains persistent income inequality in a distinct manner, with the fundamental view stressing inherent differences, the historical view emphasizing historical
accidents compounded by the inner workers of economic development, and the multiple
equilibrium paths view focusing on the role of beliefs. All three approaches are complementary, and each is likely to contribute to our understanding of why nations continue to
differ in income levels or growth rates. However, I shall present them in sharp contrast for
the sake of theoretical clarity, beginning with the fundamental approach in this section, focusing on the historical view in Sections 3 and 4, and dealing with the multiple equilibrium
approach in Azariadis (1996).
The fundamental approach holds that the asymptotic distribution of per capita world
income would become degenerate, except for random shocks, or institutional frictions in
the transmission of technical progress and in the movement of factors of production across
international borders. These frictions enable national differences in technology, institutions,
and policy to sustain permanent differentials in international standards of living. We call
this argument conditional convergence.
To understand conditional convergence, let us think of the Earth as consisting of a number
of completely isolated national economies indexed by i = 1 . . . . . H and growing along
POVERTYTRAPS
453
kt+l
45*
(a)
kt~1I
45~
k"/ ..............................................................................
k'L~
country
kt
(h)
Figure 2. Alternative growth paths. (a) convergence (b) conditional convergence.
distinct balanced growth paths. Suppose next that each nation's growth process is well
described by the accumulation decisions of a standard one-sector optimum growth model,
that is, by a single infinitely-lived household with rate of time preference pi > O, a smooth,
well-behaved intensive-form production function ft': ~ + -+ R+, which represents output
(net of depreciation) per unit of labor services and a capital income tax rate z-i E [0, 1].
Taxes are also levied on labor income at the rate 0 i ~ [0, 1], and all tax revenue is completely
454
AZARIADIS
kt+l
~3
45"
.
~
aL
k2 1...................
kt
(c)
kt§
45*
ko a
ko A
[~3
kt
(d)
Figure 2. Alternative growth paths. (c) History-drivenpovertytrap. (d) Expectations-drivenpovertytrap.
wasted. Labor services are inelastically supplied, and their efficiency grows exogenously
at the rate gi > O.
Along a balanced growth path, each economy grows at the rate gi, with per capita income
at time t being
POVERTY TRAPS
455
and each nation having constant capital intensity ki, which solves the equation
(1 - r ; ) f / ( k i )
= pi
(2)
Let hi be the inverse function of the marginal product schedule ft. Then solving (1) for k i
and substituting into (2) we obtain
),~= (1 -1- gi)t
fi [hi
( 1- - - ~ ) ] P i
(3)
Equation 3 says that international differences in long-run income growth rates are directly
and uniquely attributable to corresponding differences in rates of labor-augmenting technical
change. However, persistent differences in the level of income may have a fairly long list of
root causes: high patience or a low rate of time preference p, a low rate of capital income
taxation r, and high initial total factor productivity f will permanently favor any country i
over another, less-fortunate country j. However, if an econometrician can correct for the
variables (p, f, g, r), say, by treating economic structure as a fixed effect, then international
long-term income patterns should be identical.
This is the (oversimplified) crux of the conditional convergence argument. When we
take into account international mobility in technology and financial capital, the list of the
variables on which convergence is conditioned shrinks. For instance, if capital flows freely
across borders, all after-tax interest rates, net of depreciation, will equal the rate of time
preference of the most patient nation. Long-run income disparities will now reflect differences in technology (total factor productivity and technical progress) and in the taxation of
capital. Once we allow some scope for the diffusion of knowledge, say, by allowing any one
country to adopt the best available inventions and innovations at some fixed cost, technology
becomes a less important explanation of international differentials in growth performance.
As countries advance on their growth paths, they have more to gain by adopting advanced
technologies and will rationally choose to pay the fixed cost of innovation.
Mankiw, Romer, and Weil (1992) estimate the conditional convergence hypothesis in a
descriptive growth model and report that it explains about 60% of international variation
in per capita GNP. Two problems with this approach derive from strong assumptions researchers make about structural heterogeneity in the sample and about strict exogeneity in
the fundamental parameters of the growth model.
Heterogeneity in the sample is problematic because any pattern of international income
distribution is consistent with a sufficiently diverse sample of countries. More precisely,
any finite collection of income data will be matched reasonably well by a large enough space
of fundamental characteristics, that is, one with enough degrees of freedom in structural
parameters. For example, empirical work on growth often uses dummy variables to isolate
political and social influences on economic development.
To keep the number of explanatory variables small, we may choose to regard the parameter
vector of each country to be stationary for a period of time sufficiently long to qualify
economic structure as a fixed effect. By removing the effects of this fixed economic structure
on growth performance, we are again in a position to make international comparisons of
convergence in the standard of living.
456
AZARIADIS
Ag
1.0
4
0.5
0.0
-0.5
OO 9
9
9
-1.0
..
Ak
-1.5
-1.5
-1.0
0.0
-0.5
0.5
(a)
1.0
Ag
9
0.5
0.0
9
lio-O
|
9
-0.5
9
' ~-~
9
-1.0
-1.5
-1.5
Ak
-1.0
-0.5
0.0
0.5
(b)
Figure 3. Growth rates and capital intensity. (a) Output per worker. (b) Income per capita.
Benhabib and Gali (1995) report the results of two tests of conditional convergence
based on splitting into two successive decades a sample of observations from 1965 to
1985. Conditional convergence implies that nations that increased (decreased) their capital
intensity, measured by changes in the capital-to-output ratio, between 1965 and 1975 should
experience a slowdown (speedup) in their growth rate over the subsequent decade 19751985. Figures 3(a) and 3(b) are scatter plots of the change in the decadal growth rate
of output per worker in panel (a), and the growth rate of per capita income4 in panel (b)
against the change in the capital-output ratio during the preceding decade. More than half
of the countries sampled violate the maintained hypothesis of negative correlation between
growth rates and capital-output ratios, calling into question the power of the conditional
POVERTY TRAPS
457
convergence approach to explain observed development patterns in samples that include
the least developed countries.
A deeper theoretical concern with tests of conditional convergence is the presumed exogeneity of parameters that both economic theory and political science tend to regard as
endogenous. Examples are the speed of technology diffusion among nations, the mobility
of factor services and goods across borders, and various fiscal parameters. Are we to think
of these as set by nature or by collections of individuals acting on purpose? If we choose
the former, as much empirical work on growth does, how will we interpret the finding
that differences in capital per worker are partly explained by differences in rates of capital
income taxation s or, equivalently, by probabilities of expropriation?
One possibility is to work out the political aspects of taxation and economic development
in an effort to understand why a sequence of governments or electorates would adopt policies
that are individually rational or group-rational, but prove collectively harmful for long-run
growth. 6 The approach I take in the rest of this survey is a shortcut: let us assume that all
countries have identical structural characteristics (tastes, technology, institutions, policies)
but that random historical events have placed some countries at a disadvantage relative to
others. In particular, will there be an ergodic distribution of world per-capita income as
countries overcome initial differences in stocks of physical or human capital? The key issue
is whether, in this idealized setting, there are economic mechanisms that will enable some
of the initially underendowed countries to surmount their disadvantage and join the club of
wealthy nations, while others lag permanently behind the leaders in income level or growth
rate.
3.
Convex Economies with Complete Markets
This section surveys standard neoclassical growth models--with convex preferences, convex technology, and complete markets--whose equilibrium behavior accords qualitatively
with Figure 2(c). In that figure an unstable steady state 1c2separates two asymptotically stable steady states, km and ~:3- We call kl a poverty trap because it is the asymptotic destination
of any economy whose initial capital stock is in interval (0,/~2).
The phase diagram in Figure 2(c) is not consistent with convex one-sector models of
optimum growth with a constant utility discount rate since the positive steady state of these
models, sometimes called the modified golden rule, is unique. This follows immediately
from the observation that the equation
f ' ( k ) = p q-8
(4)
has one solution for any constant returns technology. Here again p > 0 is the representative
household's rate of time preference, 3 ~ [0, 1] is the depreciation rate, and f : •+ --+ R+
is a standard intensive-form production function gross of depreciation.
Overlapping generations economies save out of current or accumulated wages, and admit
poverty traps whenever, in the neighborhood of an unstable steady state like/c2, aggregate
458
AZARI~X~IS
investment is sensitive to changes in the capital-labor ratio. Sensitivity of this sort may
come from three sources:
1. Low elasticity of substitution between capital and labor, which implies that a small
change in capital intensity is associated with a large change in the wage-rental ratio
and, hence, in the wage bill and the flow of savings.
2. The saving rate itself is very sensitive to income at low income levels, as it may be for
an economy near its subsistence consumption level.
3. A demographic transition occurs in a critical range of income in which fertility drops
rapidly as per-capita national income improves.
All three mechanisms are at work in a modified version of the simplest overlapping
generations model without national debt due to Diamond (1965) in which generations
consist of identical, two-period lived households. Equilibria of this model will turn out to
satisfy a first-order difference equation of the general form
(5)
n(kt)kt+l = z[R(kt+l), w(kt)]
in which k is the capital-labor ratio; f is the intensive form of a standard neoclassical
production function that exhibits constant returns; w ( k ) = f ( k ) - k f ' ( k ) is the wage rate
as a function of the capital-labor ratio; R ( k ) = 1 - 8 + f ( k ) is the gross yield on loans at
t; n(kt) is the gross fertility rate at t, which we allow to depend on economic conditions;
and z ( R , y) is aggregate saving per worker when R is the yield on loans, y is wage income
in youth, and zero is wage income in old age.
To derive the dynamical system in equation 5 from first principles, we introduce endogenous fertility starting with the utility function of the representative household in generation
t. Preferences are defined over three consumption goods in the vector c t = (c~, ct2, nt+l),
which lists youthful consumption, old-age consumption, and the number of children each
household chooses to have. For the sake of concreteness and tractability, let us suppose
that the lifetime utility associated with the vector c t is
vt=log(ctl--xl)-1-~61og(ctg--X~)+~logn,+l
ifc] > x t l a n d c t 2 > x~
(6)
In this expression we assume 15 > 0, y > 0 and (x~, x~) is a vector of subsistence consumption levels in youth and old age, which is endogenous to generation t, being generated
by a as-yet-to-be-described process of habit formation. 7
Children are a pure consumption good whose marginal cost involves both material resources and time diverted away from gainful labor market activity. Households possess one
unit of time that they divide into labor supply and parenting; let L ( n ) c [0, 1] be the labor
supply of a household with n > 0 children. The endogenous fertility literature 8 typically
asserts that households maximize a lifetime utility function like (6) subject to a budget
constraint of the general form
c] 4- ct2/Rt+l < y ( w , , n,+l),
(7)
POVERTYTRAPS
459
where Rt+I is the gross yield on loans payable at t + 1; y: R~_ --+ N+ is the labor income of
a household earning wages w per unit time with n offspring. The function y is increasing
in its first argument, decreasing in the second argument, and satisfies
yCw, O ) = w
(8)
Setting aside integer constraints associated with the choice of family size and ignoring
altogether complications like infant mortality, twins, and the like, we obtain the following
first-order conditions to the consumer problem defined by equations 6 and 7:
(1 + fl)(c] - x~) = wt =-- y ( w t , nt+,) - x[
- x~/Rt+
1
(9a)
(1 "t- fl)(ct2 -- Xt2) = fl Rt+lO)t
(9b)
(1 + f l ) y n ( w t , nt+l) + yo)t = 0
(9c)
where Yn is the marginal cost of a child expressed in terms of forgone labor income. From
equation 9a it is easy to see that household saving is
y ( w t , n t + l ) - - c ~ -- 1 +fl f l [ y ( w t , n t + l ) - - x ~ l - t -
1
l+flRt+l
(lo)
Equilibria in this economy are sequences (kt, nt) that conform to the following requirements:
1. the asset market clearing condition
L(nt+2)nt+lkt+l = y ( w t , nt+l) -- c~;
(11)
2. equations 9c and 10
3. the factor demand equations
wt = f ( k t ) - k t f ' ( k t ) ,
Rt = 1 - (~ + f ' ( k t ) ;
(12)
4. a description of how the subsistence consumption vector (x[, x~) evolves over time
5.
initial conditions (kl, nl) for t = 1
The RHS of equation 11 is saving per household in generation t; the LHS is the product
of capital intensity and labor supply by offspring of generation t. There are nt+l of these
offspring, each one with nt+2 children of its own, and with a labor supply equal to L(nt+2).
Within this general framework, we analyze four particular types of poverty traps stemming from impatience, technological complementarity, distorted international trade, and
demographic factors.
460
AZARIADIS
3.1. Impatience Traps
The first type of trap is familiar from earlier work by Kuznets (1966), Beals and Koopmans
(1969), and Magill and Nishimura (1984); it occurs when the saving rate is sensitive to
changes in income at low levels of consumption. The key idea is due to Irving Fisher (1965):
consumption early in life reduces the degree of impatience for subsequent consumption. It
is shown in the indifference diagram of Figure 4(a) where, along any ray through the origin,
the household becomes more willing to defer current consumption as it becomes richer.
A special case in point is an isoelastic utility function with subsistence consumption,
along the lines of equation 6. In particular, we assume y = 0, y ( w , n ) = w for all n,
technology parameters 3 = 1 and f ( k ) = k ~, 0 < ot < 1, and a subsistence consumption
vector (x~, x~) = (0, x t k t + l ) , where xt is a share-of-GNP parameter defined by
xt =
0 if kt < [r
5 if k t > / c
(13a)
with 5 E [0, min(1, o m / ( 1 - s))], and/c a critical value of the capital stock.
With this specification fertility becomes exogenous, assumed to be constant at nt = n > 0
for all t; labor income at t is simply w t = (1 - ot)k~; and equation 11 reduces to
s(1 - a )
kt+ 1 = n - (1 - s ) x t / o t k at ,
(13b)
with x t defined in (13a), and s being the saving propensity
s =-- fl/(1 +
i~)
(13c)
Note that the fertility parameter n in the denominator of equation 13b equals one p l u s the
rate of population growth.
Figure 4(b), the phase diagram of the dynamical system consisting of equations (13a)
and (13b), demonstrates that two stable steady states, such as/r and/r may exist if the
critical value/~ is neither too large nor too small. This example means that a poverty trap
will exist if the fraction of subsistence consumption in GNP is sufficiently sensitive to past
income whenever that income is within a critical range. The trap prevents sustained growth
because, as income expands inside that range, the young adjust sharply upward their oldage subsistence consumption and saving increases only a little. In fact, both saving and
investment remain at the modest levels consistent with the low steady state at kl.
Another example, pictured in Figure 4(c), illustrates how habit formation may set up a
poverty trap. We start with a continuous, piecewise differentiable utility function
1)(Cl, C2)
=
log c l / f l ( c O + log c2 - A ( c l , c2),
(14a)
which we require to satisfy the following restrictions:
ill > 0
and A ( c l , c 2 ) = O
ifcl < ~
fl(cO =
(14b)
f12 > ~61 and A ( c l , c 2 )
= [(f12 -/~O/fllfl2]logc2
ifcl >
POVERTYTRAPS
461
r
C1
(a)
k2
kl
45 ~ _
.............................~. C
y
xt=O
I~
kt
(bl
~
1+132
..-- f /
" ~_....
9"~176
~ , "
45"
. . . . . . . "~
~ . - ~ 1 7 6 1 7 6 1 7 6 1131
7 6.Wt
-
1+131
:'w,e
kt
(c)
Figure 4. Impatience traps. (a) Subsistence consumption. (b) and (c) Dynamic effects of habit formation.
462
AZARIADIS
The indifference map of this function exhibits increased patience when cl surpasses the
critical value ~; the relevant indifference curves are continuous but not differentiable at
c~ = ~. Consumers in this economy turn out to save
st =
slwt
if
wt<(l+/~l)c
wt - ~ if wt c [(1 + ill)c, (1 +/32)~]
s2wt if
wt > (1 +/32)6
(14c)
where
sa =/31/(1 +/31) < sz =/32/(1 + 132)
(14d)
Dynamic equilibria are solutions to the piecewise differentiable difference equation
kt+l =
SlW(kt) if
kt <_ fr
w ( k t ) - ~ if kt E [kl, k2]
s2to(kt) if
kt > k2
(14e)
whose phase diagram appears in Fig. 4(c), and shows how a poverty trap is related to
saving at the low rate sl. The critical values (kl, k2) in (14e) are uniquely defined from the
equations
w(lci) = (1 +/3i)~
3.2.
i = 1, 2
(14f)
Technology Traps
Suppose now that in equation 6 we set y = 0, and that subsistence consumption X t =
(x(, x~) is uniformly zero; hence fertility is endogenous and constant at n = 1, and labor
income again equals the wage rate. Equation 1 1 now reduces to
kt+l = s w ( k t ) .
(15)
In addition, suppose that the aggregate production function is Leontieff with intensive
form
f ( k ) = A m i n { k , y}, A > 1/s, 2/ > 0,
(16a)
where A and }, are constants. The Euler's law of distribution says that the rental rate ut and
the wage rate wt must satisfy the zero-profit condition
kut + wt = f ( k t )
(16b)
at any competitive equilibrium.
The difference equation 15 is easily seen to possess the usual two steady states: kt = 0,
Vt, supported by factor prices (ut, wt) = (A, 0); and kt = y > O, Vt, supported by the
price vector (ut, wt) = (A - 1/s, y / s ) . What is rather unexpected is a third steady state
POVERTY TRAPS
463
45 ~
say
kt
(a)
45 ~
kt+l
kc
kt
(b)
k~l
45 ~
k
kt
(c)
Figure 5. (a) Technology traps. (b) Demography traps. (c) The role of increasing returns.
464
AZARIADIS
> y, Vt, which corresponds to excessive use of capital and to factor prices
(0, yA).
Figure 5(a) explains the origin of this steady state. If the economy starts out at k0 > y,
the capital excess drives the rental rate to zero and national income goes entirely to labor,
which saves a fixed fraction s of it; 9 hence kt = s A y for all t >_ 1. On the other hand,
k0 < y means an excess supply of labor, which cannot be absorbed by a wage reduction;
then wages are driven to zero and so is saving. This implies kt ~- 0 for all t > 1. From
the shape of the phase diagram in Figure 5(a), it appears that the two extreme steady states
k = (0, s A y ) are asymptotically stable while the middle one k = y is not stable.
Whatever we conclude from this example is robust to small changes in the technology:
for any CES production function with elasticity of substitution r < 1, small increases in
the capital-labor ratio will accompany large changes in the wage-rental ratio, i.e., in saving.
That will come about through simultaneous large increases in wages and large drops in the
rental rate. Dynamical equilibria for low elasticities of substitution are well described by a
smoother version of Figure 5(a), that is, by something akin to Figure 2(c).
In either one of those diagrams, the attractor at k = 0 should be broadly interpreted as a
low-level trap rather than literally as a state of zero output. To convert one into the other
we simply add a constant to the production function (16a), permitting production to occur
without physical capital at very low levels of economic activity.
kt = s A y
3.3. OpenEconomy Traps: An Example
Poverty traps that stem from limited substitutability between capital and labor services
at the national level may be overcome at the world level if countries can trade laborintensive intermediate goods for capital-intensive ones at a given price ratio. Following an
unpublished paper of Trejos (1992), we regard final output as the outcome of combining
two intermediate inputs, which are in turn produced from two primary inputs---capital and
labor. Primary inputs are assumed to be not traded, whereas intermediate ones may or may
not be. By maximizing out intermediate inputs in each country, we derive a reduced-form
production function linking final output to primary inputs at the national level, and ask the
following question: How does the elasticity of substitution between capital and labor in the
reduced-form national aggregate production function depend on the international tradability
of intermediate inputs?
The reduced-form production function is easy to define. Denote by X b X2 the intermediate inputs, by F1 (KI, L l), F2(K2, L2) the constant returns technologies by which they are
produced from capital and labor, and by J (X l, X2) the final output technology. If there is no
trade in intermediate inputs, we may define a reduced-form aggregate production function
for final output in the home country from:
Y = H ( K , L) =
s.t.
max J [ F 1 ( K 1 , L l ) , F 2 ( K - K l , L - L1)]
(KbLl)
(17)
K ~ e [ 0 , K], L L e [ 0 , L].
where (K, L) are aggregate input stocks in the home country. On the other hand, if intermediate inputs are internationally traded at a price ratio p > 0 for intermediate good 2 in
POVERTY TRAPS
465
terms of intermediate good 1, then the aggregate production function becomes
Y = G(K,L,p)
=
max
(Kt.LI.MI)
J [ F I ( K I , L1) + M1, F2(K - K1, L - L l ) -- pMl],
(18)
where M1 denotes imports of intermediate input 1 and p M l is exports of intermediate
input 2.
It is easy to see why the open-economy production function in equation 18 will generally
have a higher elasticity of substitution than the closed-economy one in equation 17: small
open economies exchange, at fixed terms of trade, relatively capital-intensive intermediate
goods against relatively labor-intensive ones and, therefore, are less prone to fall into
development traps. The empirical implication we draw from this insight of Trejos is that
we can use measures of foreign trade openness (i.e., total trade in intermediate products
as a fraction of GNP) to help explain the clustering of middle-income economies into
high-growth and low-growth groups.
3.4.
Demographic Traps
Economic models of fertility have drawn the attention of growth economists ever since
Nelson (1956) observed that endogenous fertility is a plausible proximate cause of persistent
underdevelopment. We have by now many models of fertility, like the one distilled in
equations 6 through 8, in which offspring are a normal consumption good. Costs of being
a parent include the material resources and time we invest in our children, that is, the direct
consumption parents forgo as well as the opportunity costs of reduced leisure or less active
participation in the labor market. As capital accumulates and wages rise, the opportunity
costs of parenting also rise; significant increases in marginal costs may cause the large drops
in fertility, which we have come to know as demographic transitions.I~
Demographic transitions are associated with increased labor market participation by
childbearing-age women; they result in rapid deceleration of population growth which,
in turn, improves output per worker in the steady state. This result can be derived from
the model of endogenous fertility set out in equations 6, 7, and 8. We put (x~, x~) = 0 to
eliminate the impact of subsistence consumption, and simplify eqs. (9a), (9c) and (11) to
L(nt+2)nt+lkt+l = sy(wt, nt+l)
(19a)
--nt+lYn(Wt, n t + l ) / y ( w t , nt+l) = y/(1 +/3)
(19b)
These are two equations in the state variables (kt, nt). The first one clears the asset market;
the second one asserts that the partial elasticity of labor income with respect to fertility
is constant. To study dynamical equilibria, we solve (19b) for n/+l as a function of wt,
substitute the outcome in (19a), and obtain once more a first-order difference equation in
capital per worker.
Fertility in this economy with logarithmic utilities depends crucially on the earnings
schedule y ( w , n). Suppose that the time and resource costs of parenting are linear in the
466
AZARIADIS
number of children, e.g., that
y(wt,
nt+l)
=
w t -- p t n t + l -- Atwtnt+l
(20)
where (p,, A t ) are, respectively, the unit resource and unit time cost of parenting in period
t. Each household takes these costs as given parameters.
From the first-order conditions (19a, 19b), we obtain now
[y(wt+l, n t + 2 ) / W t + l ] n t + l k t + l = s w t - s(pt + A t w t ) n t + l
(21a)
y / ( 1 +/3 + y)
(21b)
=
(Pt q- A t t o t ) n t + l / t O t ,
which implies
y(w,, nt+l)lwt = (1 +/3)/(1 +/3 + y)
(21c)
Solving out nt+l and nt+ 2 from equations 21a and 21b leads to the first-order equation
kt+l = [Pt + Atw(kt)]s(1 +/3 + y ) / y
(21d)
The dynamics of endogenous fertility in this linear/loglinear economy is quite straightforward. If parenting takes no time away from the labor market and the marginal resource
cost of a child is constant (e.g., At = 0 and Pt = P, V t ) , then the capital labor ratio is also
constant, that is
kt = ps(l + ~ + y ) l y
Vt
(22a)
At the other extreme, suppose that fertility involves a pure time cost, e.g., At = A and
Pt = O, Vt. Then (21d) yields
kt+l = [As(1 +/3 + y ) / y ] w ( k t )
(22b)
In neither case is there scope for a demographic poverty trap if w(k) is a well-behaved function. In general, though, equation 22b allows a technological trap of the type encountered
in Section 3.2, whenever the elasticity of substitution between capital and labor is less than
unity. Equation 21c shows that, if this were to happen, fertility would be high and wages
would be low in this technological poverty sink.
Constant parameter values (p, A) lead to a phase diagram of the sort shown in Figure 5(b),
with two steady states, 0 and/r
A), corresponding to each parameter vector (p, A) of
marginal parenting costs. Note that the positive, asymptotically stable state k is an increasing
function of these costs; higher costs of rearing humans naturally lead to the adoption of
more capital-intensive technologies.
Poverty traps occur in this economy if the marginal cost vector (p, A) is sensitive to the
lifecycle income or the wealth of the childbearing generation. Suppose, in particular, that
(Pt, At) are weakly increasing functions of wt and, hence, of kt. Figure 5(b) diagrams one
POVERTY TRAPS
467
easy, but fairly robust, example:
At
=
Pt =
A
Vt
Pl if kt < kc, = p2 if kt > kc,
(23a)
(23b)
where kc is an intermediate critical value.
Endogenous shifts in fertility costs now support two asymptotically stable states: one
at/~1 is associated with high fertility and low per capita income; the other at fr has these
properties in reverse. A similar outcome would be caused by shifts in the marginal time
cost A, except that the phase diagram in Figure 5(b) would suffer a change in slope instead
of in intercept.
Fertility declines in this setting accompany upshifts in marginal parenting costs, implying
that these costs are not convex at the aggregate level. Is the poverty trap in Figure 5(b)
due to a nonconvexity in the childrearing technology? Strictly speaking, a nonconvexity
in private parenting costs is not needed to set up this trap; as we know from Becker et al.
(1990), we can obtain a similar result if we replace the technological nonconvexity with one
that affects the accumulation of human capital. This change to equation 6 would express
preferences over both the number and the quality of children, that is, the stock of their
human capital. Better children are more desirable to parents, cost more to raise, and supply
more efficiency labor units than do low-quality offspring.
Poverty sinks in this case follow naturally from the assumption that quality is a normal
good: in an indigent economy parents choose to endow their offspring with low-quality,
low-productivity education. In a similar vein, Eckstein et al. (1989) show how a trap may
also arise in an economy without preferences over the quality of children--e.g., in one
with parental preferences like equation 6---if a demographic transition takes place at a
certain interval of the wage rate. The argument here is less convincing than the quality-ofchildren story: it requires the substitution effect from a wage rate increase, which raises
the time opportunity cost of parenting, to overwhelm the corresponding income effect,
which would boost fertility, as wages move up to the range we have observed in industrial
democracies over the last half of this century. Both fertility and the labor participation rate
of fertile women are very wage-elastic inside this range. Outside it, the argument concludes,
the income and substitution effects will be comparable in size, making fertility and labor
participation nearly independent of wages.
Substitution effects from wage rates seems to me to be less credible than other stories
as causes of demographic transitions, even though they are consistent with postwar labor
market experience in the most developed countries. A pronounced lessening of fertility
and increased female participation in the labor market have occurred, at considerably lower
wage rates, in a number of developing nations in Latin America and East Asia.
A more believable explanation starts from the lessening of traditional family and clan
bonds that has occurred in the last 50 years, and the devolution to the state of many traditional
family functions like schooling and old age care. Following Dasgupta (1993, ch. 16), we
may amend the basic fertility model in equations 19 and 20 to allow the community at large
(i.e., the "economy" in our terms) to share with the parents some of the costs of rearing
children. This used to be common practice in traditional societies which shared the fertility
costs with, but left childbearing decisions up to, the parents. Childbearing in this story
468
AZARIADIS
involves a social subsidy, which relieves the parents' private costs; the typical subsidy per
child becomes bigger as the size of the family grows.
This cost externality injects a strategic complementarity into fertility decisions, of the
sort Cooper and John (1988) lucidly identified: additional births lower private childbearing
costs for everyone. One possible outcome of rewriting equations 21 and 22 to capture
this complementarity, already conjectured by Dasgupta, is the existence of several Paretoranked steady states. The one with the lowest capital-per-worker ratio and highest fertility
rate is a traditional poverty trap.
4.
Nonconvex Economies with Complete Markets
This section examines states of persistent poverty arising from increasing returns to scale,
external effects and complementarities operating at the industry, sectoral, and national level.
We start out with increasing returns in the production of consumption goods or intermediate
inputs; continue with external effects and complementarities in the creation of knowledge,
the accumulation of skills, and the funding of public education; and finish with two-sector
models of industrialization under increasing returns to manufacturing.
4.1.
External Increasing Returns
Nonconvexities in the technology typically result in discontinuities, kinks, or nonconvexities
in the equilibrium dynamics of economic growth. For example, Romer (1986) and Klenow
(1993) study competitive equilibria in the representative, infinite horizon growth model with
the external increasing returns to scale described by the following production function:
g(k j) = A ( k ) f ( k j)
j = 1. . . . . J
(24a)
Here j is an index of firms and A(k) is a scale factor that depends on the economy-wide
capital stock 11
J
k ---- ( l / J ) Z
kj
(24b)
j=l
To simplify matters, suppose that A is the following step function
A(k) =
AI>0
A2 > AI
ifk<k,
ifk >
(25)
for some intermediate critical value/~.
When producers are identical, competitive equilibrium in this economy is described by
two policy functions, Pl: R+ --+ R+, P2: R+ ~ ~+, such that p2(k) > pl(k), Yk > 0,
much like those appearing in Figure 5, e.g.,
kt+l =
I pl(kt) ifkt < k,
[p2(kt) ifk, >/~
The fixed point of the smaller policy function is a poverty trap.
POVERTY TRAPS
469
When returns to scale are strongly increasing, poverty sinks are almost inevitable. Boldrin
(1992a) demonstrates this by grafting a technology like equation 24a onto in a standard overlapping generations model like the one described by equations 9-10, from which fertility
and subsistence consumption are both removed. Assuming a technology
y~ =(k~)(k[)'~
O < o t < 1,
ot+y > 1
(26a)
for each producer i, and a constant saving rate s ~ [0, 1], the asset market clearing condition
becomes
kt+l
:
s(1 - ~)k~ +•
(26b)
The saving rate s may be thought of as the parameter/3/(1 + 13), representing the fraction
of wage income saved by an overlapping generations economy with income vector (w, 0)
and utility function log cl + / 3 log cz.
As Figure 5(c) points out, this equation has a stable steady state at k = 0 and an unstable
one at k = ~:--if we take seriously the assumption that ot + y > 1, i.e., that social returns
to scale are increasing and large. Any equilibrium capital stock sequence starting above fr
becomes unbounded, leading to perpetual growth. On the other hand, sequences starting
below/r converge to the origin which again acts as a poverty sink.
Boldrin (1992a) also shows how multiple stable steady states may occur if the scale
function A ( k ) in equation 24c is a complicated increasing function. Suppose for instance,
that the representative household in an overlapping generation economy has utility function
U(CI, C2) = CI + /3C2,
(27a)
the depreciation rate equals 1, and the technology is
g ( k i) = A ( k ) ( k i ) ~
(27b)
Then the equilibrium interest rate equals the consumer's rate of time preference, 1//3 - 1.
All equilibria in this economy are stationary ones, with the steady-state capital stock solving
the equation
k = [ot/3A(k)] 1~(l-a)
(27c)
Every solution to this equation is a fixed point of its right-hand side, that is of an arbitrary
increasing function; it is easy to imagine many such fixed points 12 for a suitably chosen
function A.
4.2.
Public Debt and Crowding Out
Economies with high returns to scale have within them the seed of perpetual growth. As
capital accumulates, increasing returns maintain the momentum of the expansion process by
enlarging the income of savers and by preventing the rate of return on capital from failing
470
AZARIADIS
to values that would encourage current consumption to rise too rapidly at the expense
of investment spending. The previous section pointed out one pitfall in this mechanism:
returns to scale may not be high enough, at low stocks of wealth, to keep an expansion
going.
Recent work by Grossman and Yanagawa (1993), Michel (1992), and Azariadis and
Reichlin (1996) explains how a rapidly rising stock of public debt may also stymie perpetual
growth. This conflict--known as crowding out in macroeconomics--does not have to come
from large public sector deficits as it typically does when returns to scale are constant.
Growth under increasing returns to scale has the pathological side-effect of raising interest
payments on the existing stock of public debt and displacing claims on private capital from
household wealth portfolios. Even a small initial amount of national debt may in principle
balloon to the point where it reduces the growth rate permanently, or even stops it cold,
unless the government quickly withdraws from circulation all public debt in private hands.
A quick way to see this is by adding a small amount of public debt in the economy
described by equation 26b, and reexamining its asymptotic equilibria. Suppose, for the
sake of concreteness, that the population is constant and bt is the stock of debt issued in
period t - 1 and maturing at t. Public debt is thus rolled over each period; government
purchases and taxes are both zero; the depreciation rate is 8 6 [0, 1].
The basic equations for this economy are
kt+l -'1- bt+l = s(1 -- ot)k~+u
(28a)
kt+l = b t R t
(28b)
Rt = 1 - ~ + etktu
(28c)
where we assume
/z~ot+y-
1 >0,
~ <s(1-ot)
(28d)
The first one of these equations equates total household wealth with claims on capital plus
claims on the govemment; the second one is a public sector budget identity equating new
debt with principal and interest on maturing debt. Equation 28c expresses the interest rate
as the private marginal product of capital net of depreciation. Substituting the last equation
into the first two and solving, we obtain
kt+l + bt+l = s(1 - ot)k] +u
(29a)
bt+l = [c~k~ + 1 - ~]bt
(29b)
Once more set 6 = 1 and check that the dynamical system consisting of equations 29a
and 29b has three steady states
(k, b) = {(0, 0), (~:, 0), (k*, b*)}
(30a)
POVERTYTRAPS
471
th
bl+l'bi
b,
,L
k~
kt
Figure 6. National debt with increasingreturns to scale.
with
(k)~ = [ s ( 1 - ~ ) ] - I
(k*) ~ =
ot - I
b* = k*[(k*/~)
(30b)
> (k)U
u -
l] > o
(3oc)
Note how the inequalities in (30b) and (30c) are implied by the ones in (28d).
Figure 6, the phase diagram for this economy, hints that (0, 0) is a sink, while the other
two steady states are both saddles. In fact, it is easy to show that the origin has the real
eigenvalues (0, 1 - 8) = (0, 0); the state (/r O) has eigenvalues (1 + / z , 1 - c) where
c = 811 -- (k/k*) u] = 1 -- (it~k*) ~ ~ (0, I)
(30d)
Associated with the state (k, 0) is a stable manifold that slopes up in the state space and
is marked SM in Figure 6. For each initial capital stock k0 > /c there is now a value b0
of public debt such that equilibrium capital sequences (kt) fail to grow without bound,
converging instead to [r Even a small amount of debt may now defeat strongly increasing
returns to scale, converting the perpetually growing economies of Figure 5(c) to ones that
remain forever in the neighborhood of k.
4.3.
Invention and Innovation
Economies of scale in research and developmcnt are another potential source of persistent
poverty because these economies link the profitability of research and development expen-
472
AZAPdADIS
diture to the overall size of the market. Benthal and Peled (1992), for example, argue that,
if firms spending a fixed amount of resources were able to shift their production possibility
frontier by a given proportional distance, then technical progress would be more rapid in
large economies than in small ones. Large economies may amortize with relative ease the
fixed costs of research and development over the numerous beneficiaries who share the
benefits of R&D.
To see this reasoning in its simplest guise, we begin with an economy consisting of a
constant number of producers, each of whom may engage in R&D at a fixed resource cost,
z > 0. Let the multiplicative scale factor Ot > 0 describe the technology inherited at t
from the previous period, and y > 0 be the potential rate of technical progress. Production
of a perishable consumption good requires one indivisible unit of entrepreneurship and a
variable amount of labor, denoted L. Output is
if no R&D takes place
| ( 1 + y ) O t L t i f z is spent on R&D
[Ot L~
Yt
=
(3 la)
where ~ 6 [0, 1]. Labor is hired competitively at a wage rate wt, and its total supply is
fixed at L ~
Equating the marginal product of labor with the wage rate provides each producer with
profit
(3 lb)
7~t --~ A ( O t / w t ) 1/(1-et)
if no R&D takes place, and with
Jrt = A[(1 + y)Ot/w~] 1/(1+'~) -- z
(31c)
when z is spent on research. Here we have defined
A = ot~/(l-~)(1 - or)
(31d)
Research and development will take place in equilibrium if, for each t, the cost of R&D
is smaller than its contribution to profit, e.g., if rrt > Jr/'. This implies
z _< A[(1 - / ) l / ( l - ~ )
_ 1](Ot/wt)l/Cl-a)
(31e)
If we combine this equation with the labor market clearing relation
wt ---=(1
+
(310
Y ) O t L t -1
and normalize labor supply at L ~ = [or(1 + y)]l/(l-~), we obtain two equalities that must
apply to all equilibria with positive R&D:
(3 lg)
wt = Ot
0 t >_ 0 --- a / A [ ( l + ~ ) l / ( l - a )
_
1]
(31h)
POVERTY TRAPS
473
Therefore, if the initial state of technology is 00 > 0, then wages and output per worker
grow at the rate y > 0, while 0o < 0 means that the economy remains in a poverty trap of
zero growth. Initial size is of critical importance, both in this story and in the next one.
Galor and Tsiddon (1991) show how the international diffusion of a nonconvex technology
may set up a trap for technology followers. This situation arises when innovations in the
"lead" country afford the follower country a menu of new and old technologies all of which
the follower country keeps using. Switching to the new technology leads to a growth
takeoff in the lead country while the follower, never quite ridding itself of traditional modes
of production, remains behind in perpetuity.
What is interesting here is that the poverty trap is entirely due to the innovation and would
not occur if the technology did not advance. To see how this works out, we go back to
the standard overlapping generations model with identical, two-period-lived households,
constant population, and Cobb-Douglas utility function. There are two countries, indexed by
i = A (advanced) or i = B (backward) producing one perishable good from two immobile
inputs, capital and labor. Technology in the advanced country is the discontinuous function
in Figure 7a, i.e.,
gA(k) =
A o f ( k ) if k <
A , f ( k ) if k > / c
(32a)
with A1 > A0 > 0, and/r a critical point at which knowledge suddenly advances.
The follower country may imitate at zero cost whatever technology is available to the
lead country; hence the follower's production function is a convexification of the leader's.
Given the lower critical value fr defined in Figure 7b, we obtain
gS(kB) =
Aof(kB) if either k A < k" or (k A > k andk 8 e [0,/r
A l f ( k B ) ifk A >/c and k B >/r
OAof(k) + (1 - O)Alf(Ic) ifk a > [c andk B ~ [}r
(32b)
where 0 ----(k - kS)/(k - k).
The poverty trap becomes an attractor whenever k 8 is in the intermediate interval (/},/c) for
it is in that interval that the marginal product of labor and the marginal product of capital
(and of technological innovation) are constant. While the backward nation absorbs the new
technology, capital-embodied technological innovation remains the mainspring of growth,
and the reward to capital absorbs all additional output. This leaves the wage bill stagnant,
keeps saving from expanding, and traps the economy in the neighborhood of a low-level
attractor.
Technological leadership and sufficiently high total factor productivity guide the leading
country to the high attractor ka shown in Figure 6b. Formally, the equilibrium sequence
(kta) satisfies
k a+l = { s A o w ( k t a) if ka <
s A l w ( k A) if kta > ~:
(33a)
474
AZARIADIS
Y
All
fjlf~
/Jff~
A
k
~<
K
(a)
kt§
45*
k
ks
k
kA
kt
(b)
Figure 7. Technology diffusion. (a) Production possibilities. (b) Dynamical system.
where w ( k ) =- f ( k ) - k f ' ( k ) , and s 6 (0, 1) is a saving rate. Given some technical
assumptions, 13 we have
lim k A = ka for any k a > 0
(33b)
t --+OO
i.e., technical innovation will take place in country A and all output will be produced using
the new technology, as shown in Figure 7b.
POVERTY TRAPS
475
The same diagram explains why the follower country cannot advance beyond k8 from
any initial position k~ ~ (0, [r Formally, we have
ktBl : Sll)tB
(33c)
with the follower's wage rate being
w~ =
{
Aow(k~)
if either k a < [c or (k a _> [c and kt8 < k)
Atw(kt B)
i f k a > k" andkt 8 >/~
[(it - kS)/([c - k)]A0f(/~) + [(k n - 1r
- k)]Alf(k)
otherwise
(33d)
It is easy to see that the wage rate wfl is independent of the capital-labor ratio in the interval
(k, k) over which both the old and the new technology are in competing use.
A constant wage bill means stagnant savings and no growth for economies placed by
history in the interval (k, it).
^
4. 4.
Human Capital Externalities
One basic presumption in the theory of human capital is that an individual worker's stock
of skills, or flow of labor services per unit time, depends not only on education and formal
training but also on things learned as a by-product of social interactions. In the home as well
as the workplace, we continuously learn from family members, colleagues, and co-workers
so that our index of skill depends, among other things, on the quality of our environment.
External effects from social interactions, between generations at the aggregate level and
between households at the neighborhood level, are one of the mechanisms of economic
growth that have figured primarily in work on nonergodic growth theory by Azariadis and
Drazen (1990), Durlauf (1993a) and others. In all this work, nonconvexities in the accumulation of human capital make the private rate of return depend on some broader human
capital aggregate. A crude description of what is at work goes as follows: private rates of
return are quite low in environments short on human capital, quite high in environments
where skills are abundant. Nonergodicity is directly caused by this sensitive dependence of
private yields on aggregates; groups, neighborhoods, or nations in deep poverty have great
difficulty overcoming their initial circumstances because the state of poverty contains in
itself the individual incentives that perpetuate it.
A case in point is the following simple example drawn from Azariadis and Drazen (1990)
in which the opportunity cost of human investment is forgone labor supply. Consider a
small open economy with a constant population of identical, two-period-lived households
whose time endowment is (1, e) units, and e > 0. The openness assumption means that
factor prices are exogenous to this economy; we assume, in particular, that the gross yield
on loans is a constant R > 0, that the wage rate per efficiency labor unit is also constant at
w > 0, and that the capital labor ratio is yet another constant k > 0. The vector of human
476
AZARIADIS
capital x t (i), that is, of efficiency labor services per unit time for a member i of generation
t is
xt(i) = (X~, xt+l)
i
(34a)
The corresponding income vector is
yt(i) = (w(1 - zt/)x~, wex~+l)
(34b)
where z] ~ [0, 1] is the fraction of time endowment young individual i = 1, 2 . . . . . J
spends in school or training, and 1 - r t is the fraction spent in gainful employment.
Human capital acquisition at time t obeys the technology
atx~h(r])
X[+ 1 =
At = A(xt)
1
J
(34c)
(34d)
-
xt = ~ i~l x;
(34e)
where xt is the economywide average human capital, At is the scale factor that is increasing
in xt and h: [0, 1] --+ R+ is an increasing, concave production function. For the sake of
concreteness, we assume that the scale factor exhibits a threshold externality of the form
A(x)= / A0>0
/
if x < 2
A1 > A0 if x >
Rational individuals choose the level of human investment r[ to maximize discounted
lifecycle income
y, = w(1 - r/)x~ + (we/R)x]+ 1
(35a)
subject to (34c), and taking as given both the aggregate state xt and their own starting stock
x~. Inverting the first-order condition
R/(eAt) = h ' ( r t )
(35b)
for this problem, we obtain
rt/ = z(At)
(35c)
where r: N+ --+ [0, 1] is the inverse marginal-product-of-human-capital schedule.
Open-economy equilibrium sequences for an economy of identical households consist
of eqs. (34c), (35b) and x~ = xt u Depending on the beginning-of-time stock of human
POVERTY TRAPS
477
capital x0, all of these sequences are balanced growth paths, e.g.,
X t + l / X t ~-
Aoh(r0) if x 0 < s
Alh(rl) if xo > s
(36)
Here r0 > 0 and rl > ro satisfy the first-order conditions from above, e.g.,
h'(ro) = R / ( e A o ) , h'(zt) = R / ( e A 1 )
(35b')
Clearly, human capital favors faster growth which accelerates once the economy reaches
the threshold stock 2.
A poverty trap occurs in this example if
Aoh(ro) < 1,
(37)
that is, whenever the initial human capital is insufficient to sustain a positive growth rate in
per capita income.
4.5.
Public Education
Similar mechanisms are present in economies where the opportunity cost of human investment is forgone consumption rather than labor supply. A case in point, studied by Perotti
(1993), concerns public education, as an input in the production of human capital, which
must be paid for by taxing current income. Income tax rates are set, in much of this work,
according to the wishes of a median voter in a particular community--a local school district
or the economy as a whole.
Median voters find that generous funding of public education is in their interest in a
rich community but not in a poor one because, at an average tax rate, the tax base in the
wealthy community is already large enough to sustain the education expense needed for
rapid growth. This is most obvious in a simplified version of Boldrin's (1992b) aggregative
model of public education within a small open economy with a constant population of two
period-lived agents. The time endowment vector is now (1, 1), factor prices (R, w) are
constant, and the basic structural equations are
Yt = xt
(38a)
xt+l/xt = zt + 1 - ~
(38b)
zt = Otxt
(38c)
The first of these is a constant-returns-to-scale aggregate output, which ignores the services of physical capital; the second one is the human capital acquisition technology, which
equates the rate of growth in human capital with per capita public expenditure on education;
and equation 38c is a government budget constraint expressing the equality of public spending zt with the product of a tax rate, 0t times income, Yt. The parameter ~ in equation 38b
is a depreciation rate.
478
AZARIADIS
A poverty trap would occur here even if the tax rate were an exogenous constant, i.e.,
Ot=OE[O,
1]
Vt
(39)
In that event, the dynamics of human capital is governed by the first-order equation
xt+l = Ox2t + (1 - 8)x,
(40)
The smaller of the two steady states {0, 8/0} is an asymptotically stable trap, attracting all
trajectories starting in the interval [0, 8/0). Trajectories outside that interval are unbounded.
Majority voting on the tax rate leads to qualitatively similar outcomes. To simplify matters,
we assume that the median voter is a member of the young generation who recognizes the
effect of his decision on aggregate outcomes but takes as given the choices of subsequent
median voters, t4 Specifically, the median voter in period t chooses Ot ~ [0, 1] to maximize
after-tax lifecycle income
o)t = w x t ( 1 - 0t) + wxt+t (1 - 0t+l)/R
(41)
subject to the constraints (38b) and (38c), and taking 0t+l as given.
Two possible voting outcomes are
Ot = [ 0
if (1 - O t + l ) x t / R < 1
indeterminate if ( 1 - O t + l ) x t / R = 1
(42)
The political-economic equilibrium corresponding to Ot = 0 is again a poverty trap if the
starting value of human capital is low, i.e., if
X t =
(1 -- 8)txo
Ot = 0 Vt,
xo < R
(43)
A sufficiently high initial stock x0 implies positive taxes and positive investment in human
capital. Dynamical equilibria are now solutions of the system
xt+l = (1 - 8)xt + Otx 2
Or+l= 1 -
R/xt,
(44)
Xo > R
This system has a steady state
0 = 1/(1 + R / 8 ) , x = 8 + R ,
(45)
which turns out to be determinate 15 under the mild assumption that R < 2 + 8. The
stationary tax rate depends on the depreciation rate and on the world interest rate, with
predictable signs: a high interest rate reduces the present value of future capital income
whereas a high rate of obsolescence for human capital calls for greater investment flows.
Public education is to some extent a substitute for private education. For families that
cannot borrow enough to finance educational expenses, voting on public education has important consequences for their future earning ability, and therefore for both the distribution
of income and its average growth rate. We take a more detailed look at this problem in Part
Two.
POVERTY TRAPS
4.6.
479
Industrialization Under Increasing Returns
Early pioneers of economic development like Rosenstein-Rodan (1943) stressed the importance of a sufficiently large manufacturing sector as a prerequisite of successful industrialization attempts in Europe. In modern terminology, development becomes self-sustaining
once the ratio of value added in manufacturing relative to agriculture surpasses a critical
value. Subsequent writers like Murphy et al. (1989) and Matsuyama (1991) took this message to heart, building formal models in which agriculture produces at constant returns to
scale while manufacturing benefits from increasing returns.
The intuition behind this argument is quite simple: at low levels of manufacturing output, labor productivity in manufacturing is modest, which encourages workers to stay in
agriculture. Any favorable external shock or policy measure that manages to enlarge the
manufacturing sector also improves labor productivity in that sector, drawing from agriculture a stream of laborers looking for better pay. Increasing returns will keep industrialization
going until it exhausts labor productivity gains in manufacturing.
Following Matsuyama (1991), we focus on a small open economy with a fixed and constant
interest rate, r > 0, perfectly mobile physical capital, and labor that can migrate at no cost
from agriculture to manufacturing but cannot move from one country to another. Production
takes place without capital in this economy, with constant returns in sector A (agriculture)
and external increasing returns to scale in sector M (manufacturing). Technologies are
Y,~ = L~ for producer i = 1. . . . . I
(46a)
Y~ = h(L)L~
(46b)
and
for producer j = 1. . . . . J
(L~, L j ) are labor inputs of individual producers;
L = ~ L~
j=l
(47a)
is total manufacturing employment; and h is an increasing concave function such that
h(O) = 0
(47b)
Populating this economy are overlapping generations of two period-lived agents, each
with a time endowment of labor (1, y) that is supplied inelastically to the labor market;
assume that y < 1. Each generation has size 1 and contains a continuum of agents indexed
by ~ ~ [0, 1], which measures comparative work advantage in agriculture. Specifically, type
r supplies an endowment vector of (r, 2/r) efficiency labor units to sector A or alternatively,
a vector of (1 - r, y(1 - r)) efficiency units to sector M.
At the beginning of the first period of life, every worker has to make an irreversible and
indivisible career decision whether to seek a career in sector A or sector M. Assuming a
well-functioning credit market and taking wage rates to be given at (wA, wff), agent r will
480
AZARIADIS
choose agriculture if and only if the present value of lifecycle income is greatest in sector
A, that is, iff
z t ( w tA + aWA+I) > (1
-
"Ct)(IOM "1-a w ~ l )
(48a)
where
a = y/(1 + r)
(48b)
Since the wage rate in each sector equals the corresponding marginal products of labor, we
have from (46a) and (46b) that
w a = 1,
(49)
w M = h(Lt)
This allows us to rewrite (48a) in the form
rt >_ zt =
h ( L t ) -I- a h ( L t + l )
1 + a + h ( L t ) -t- a h ( L t + l )
(50)
Suppose that the weakly increasing function F describes the cumulative distribution of
types on the interval [0, 1], so that
F(r) = Oforr~[O,f]
(0, 1) for r ~ [f, 1)
= lforz>
(51)
1
In other words, there is no population mass in the interval [0, f]. Then labor supply to
manufacturing at time t consists of F ( z t ) units by the young generation plus y F ( z t - l )
units by the older generation e.g.,
(52)
Lt ~- F ( z t ) -t- y F ( Z t - l )
Labor market equilibrium requires the simultaneous satisfaction of equations (50) and (52),
viz.,
h ( L t + O = [(1 -t- a ) z t
-- (1 --
z t ) h ( L t ) ] / a ( 1 - zt)
Lt+l : F ( z t + l ) + y F ( z t )
(53a)
(53b)
Figure 8(a) shows that this dynamical system is likely to have three steady states
(z, L) = {(0, 0), (z~, L~), (z~, L~)}
(54)
of which the first is a preindustrial state describing a purely agricultural society; the second
one is a low level manufacturing state; and the third one is a high-level manufacturing
state. All three steady states represent intersections between the labor supply schedule
POVERTY TRAPS
481
Z
L = (1~) F ( Z ) ~ ~
=
L) )
Z2*
71"
Lr
L
L2*
(a)
Zt
Lt=Lt.1
I-t
(b)
Figure 8. Industrialization with increasing returns to scale. (a) Steady states. (b) Dynamics.
L = (1 + •)F(z), and the increasing, concave arbitrage schedule
z = h(L)/[1
-t-
h(L)]
(55)
describing how aggregate manufacturing employment affects the relative wage parameter
wMII ( ff3a "]- if)M)'
482
aZARIADIS
The dynamics of industrialization, even in this simple example, are quite complicated;
Figure 8(b) hints whyJ 6 For some values of the parameters (y, r, F), it is possible that the
extreme states are saddles with stable manifolds as drawn in Figure 8(b), and the middle
state is a source. This means that there are initial positions of this economy from which it
may converge to either the preindustrial state (0, 0) or the high state (z~, L~) depending on
whether public expectations are pessimistic or optimistic. The absence of industrialization
in this class of economies is a form of expectational coordination failure; public beliefs in
preindustrial societies simply cannot conceive that industrialization is feasible, and these
beliefs turn out to be self-confirming.
Both history and beliefs are important in the evolution of this economy. Growth paths
are influenced by both initial capital and long-term expectations.
5.
Summary and Extensions
To refresh the reader's memory, here is a list--arranged by broad cause--of the types of
poverty traps surveyed in this essay:
1. consumer impatience bred from habit-formation, or subsistence consumption
2. low elasticity of substitution between physical capital and labor
3. distortions in international trading of intermediate inputs
4.
a strong wealth effect on unit costs of child rearing
5.
locally strong technological external returns to scale in producing consumption goods
or in accumulating human capital
6.
slow displacement of old technologies by new ones
7.
a strong income effect on public funding of education at the national or neighborhood
levels
It was one generation ago that such development economists as Kuznets, Gurley and
Shaw, Goldsmith, and McKinnon noted in historical data the relationship between material
progress and the state of financial markets, both in cross-sectional and time-series data.
Since then many theoretical models have stressed the role of credit market imperfections
as an obstacle to rapid growth, and to convergence among nations in the standard of living
or its growth rate. The message from this entire literature is that financial deepening, in the
form of smoothly functioning credit markets, is a prerequisite for economic development.
Azariadis (1996), reviews what modern growth theory has to say about the impact of
financial depth on persistent poverty, surveys the role of monopolistic competition, evaluates
some of the empirical evidence, and discusses the implications of poverty traps for economic
development policy.
POVERTY TRAPS
483
Acknowledgments
An earlier version of Section 3 circulated under the title "Development Traps in Convex
Economies." I acknowledge, with pleasure but without implication, discussions with Steve
Durlauf, Oded Galor, Mike Magill, Chris Pissarides, Mikko Puhakka, David Romer, Manrique Saenz, Manuel Santos, Bruce Smith, Bob Solow, Danny Tsiddon, and two anonymous
referees. Also useful were comments from audiences at RAND, USC, University of Texas
at Austin, Brown, UCLA, Hebrew University, the First IMOP Conference in Athens, ITAM,
UC-Riverside, Claremont Graduate School, and the National Central University in Taiwan.
Financial support from the Human Capital and Mobility Program of the European Union is
also acknowledged.
Notes
1. Unconditional convergence fails a number of formal tests conducted by Barro (1991), Durlauf and Johnson
(1995), Quah (l 996), and others.
2. Azariadis (1996) surveys convex economies with incomplete factor markets, monopolistically competitive
economies, and concludes by reviewing the empirical and policy implications of poverty traps.
3. See, for example, Cooper and John (1988), Galor and Zeira (1993), Boyd and Smith (1994), and Azariadis
and Smith (1994).
4. The two growth rates differ due to secular changes in the employment ratio.
5. For analyses of capital taxation and growth, see Chamley (1981, 1986), Judd (1985), and Lucas (1990).
6. Recent work along this line inchides Londregan and Poole (1990), Grossman (1991), Benhabib and Rustichini
(1996), and Alesina et ai. (1996).
7. Multiple steady states are known to exist in neoclassical growth models with variable rates of time preference
or nonconvex preferences. Early work in this area includes Koopmans (1960), Koopmans, Diamond, and
Williamson (1964), Beals and Koopmans (1969), Kurz (1968), Liviatan and Samuelson (1969), and Mantel
(1993). Growth with habit formation is treated explicitly by Ryder and Heal (1973) and Boyer (1978) under
the assumption that current utility flow is jointly concave in current and past consumption. Orphanides and
Zervos (1994) show how dropping thejoint concavity assumption in these models may lead to multiple steady
states.
8. The literature on endogenous fertility in static and dynamic environments is quite large. Relevant theoretical
references include Becker and Barro (1988), Becker, Murphy, and Tamura (1990), and Galor and Weil (1996).
9. We ignore for the time being the paradox of why workers save at a zero gross rate of interest. The gross yield is
positive whenever capital lasts more than one period; and if the elasticity of substitution is small and positive
(rather than zero), the rental rate remains bounded away from zero no matter how much capital we start with.
10. This increase has a substitution effect that limits labor supply and an income effect that favors labor supply;
the standard assumption is that the substitution effect dominates. For an economic analysis of demographic
transitions, see Sundstrom and David (1988) and Azariadis and Drazen (1993).
11. We may think of the function A(k) as describing learning by doing or the state of knowledge embodied in
physical capital.
12. A sufficient condition for a unique positive fixed point is that the function [A(k)] l/(l-c') be increasing convex
or increasing concave.
13. We also assume that the function sAow(k) has no positive fixed point in [0,/c] while the function sAlw(k)
has exactly one positive fixed point in ]~+.
14. This equilibrium concept is called "open loop" or "myopic Nash" in game theory, for it ignores the strategic
links between current and future plays, which are embodied in more sophisticated concepts like subgame
perfection. See Fudenberg and Tirole (1991), pp. 130-134 for discussion of this issue.
484
AZARIADIS
15. More precisely, the steady state in (45) is a saddle with one stable eigenvalue in the interval ( - 1 , 0) if R < 6,
which means damped oscillations en route to the steady state from any x0 > R and an appropriate 00 e (0, 1).
16. Matsuyama provides an excellent treatment of how this class of models behaves asymptotically.
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