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WAS PROMETHEUS UNBOUND BY CHANCE?
RISK, DIVERSIFICATION AND GROWTH
Daron Acemoglu
Fabrizio Zilibotti
95-12
Feb.
1995
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
WAS PROMETHEUS UNBOUND BY CHANCE?
RISK, DIVERSIFICATION AND GROWTH
Daron Acemoglu
Fabrizio Zilibotti
95-12
Feb.
1995
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
AUG 16
1995
libraries
:
5-1^
Was Prometheus Unbound By Chance?
and Growth
Risk, Diversification
Daron Acemoglu
Massachusetts Institute of Technology
and
Fabrizio Zilibotti
Universitat
Pompeu Fabra
February 1995
Keywords: Chance, Development,
Diversification,
Growth,
Risk Aversion, Underdevelopment.
Abstract
This paper offers a theory of development which links the degree of market incompleteness
to capital
accumulation and growth. At early stages of development, the presence of indivisible
projects limits the degree of risk-spreading (diversification) that the
to avoid highly risky investments slows
down
the idiosyncratic risks introduces high uncertainty in the
the
typical
economy can achieve. The
capital accumulation
development pattern of a society
will
and the
desire
inability to diversify all
growth process. As a result of these forces,
consist of a lengthy
accumulation" followed by take-off and financial deepening and
period of "primitive
growth. "Lucky"
finally, steady
countries will spend relatively less time in the primitive accumulation stage while with sufficiently
risk-averse
agents,
underdevelopment.
economies
We
that
receive
a series
of unlucky draws
may
get
locked into
also identify a pecuniary externality as a source of inefficiency in the
decentralized equilibrium that makes the average speed of development suboptimal. Further,
shown
that
our results generalize
to
an open economy and
that capital flows
may
it is
increase or reduce
the rate of industrialization.
We
are grateful to Dudley Baines, Abhijit Banerjee, Andrew Bernard, Peter Diamond, Jordi Gali,
Hugo Hopenhayn, Albert Marcet, Thomas Piketty, Julio Rotemberg, Xavier Sala-i-Martin and
Andrew Scon for useful discussion and to seminar participants in LSE, MIT, Columbia University,
Rochester,
CEPR
Pompeu Fabra
Tarragona, Virginia Polytechnic, Yale Development Conference and Universitat
for useful
comments. All errors are naturally our own. Financial Support from
(DGICYT PB93-0388) is gratefully acknowledged.
Ministry of Education of Spain
the
1)
Introduction
"The advance occurred very slowly over a long period and was broken by sharp recessions. The right
road was reached and thereafter never abandoned only during the eighteenth century and then only
by a few privileged countries. Thus, before 1750 or even 1800 the march of progress could
affected by unexpected events, even disasters.
"
Braudel (1962), p.
F.
be
still
xi.
This view of slow and uncertain progress between the tenth and early nineteenth century
shared by
many economic
historians, for instance
North and Thomas (1973) who describe the
fourteenth and fifteenth centuries as times of "contractions, crisis
who
Vries (1990)
and depression"
De
71) or
(p.
Age of Crisis". The same phenomenon
refers to this period as "The
is
observed
is
today; Lucas (1988) observes that whereas "within the advanced countries, growth rates tend to be
very stable over long periods of time. ..", for poorer countries
large changes in growth rates, both up
the process of
subject to so
development
is
and down.
painfully slow.
much randomness? The models
e.g. Azariadis
"
.
.
there are
many examples of sudden,
Furthermore, for
(p. 4).
Why
".
many of these
countries,
are the early stages of development slow and
of economic development based on threshold effects,
and Drazen (1990), may predict the slow progress
at the early stages
of development
but without additional assumptions have no implications about randomness of growth. This paper in
contrast argues that these patterns are predicted
by the neoclassical growth model augmented with
the natural assumptions of micro level indivisibilities
and micro
level uncertainty. In particular,
link the availability of diversification opportunities to the incompleteness of markets
depends on the process of capital accumulation.
highly
random and, by endogenously reducing
We
start
show
that will
most economies have access
projects, thus a significant part of the risks they face
of these projects are subject to significant
that
in turn
such links both make the early stages
the rate of return
from a number of observations
the next section. Firsdy,
We
which
we
on savings, slow down development.
be elaborated and empirically supported
number of imperfectly
to a large
in
correlated
can be diversified. Secondly, a large proportion
indivisibilities, especially in the
form of minimum
size
requirements or start-up costs. Thirdly, agents normally dislike
risk.
Fourthly, there often exist
investment opportunities that are less productive but relatively safe.
And
finally, societies at the early
stages of development have less capital to invest than developed countries. These features naturally
lead to a
number of important
implications,
(i)
at the early stages
of development, due to the scarcity
of capital, only a limited number of imperfectly correlated projects can be undertaken and agents will
seek insurance by investing in safe but less productive assets. As a result, poor countries will
endogenously have lower productivity and
this will contribute to their
a large part of the savings are invested in safe investments, the
undertaken will bear more of the diversifiable
highly random.
It
will also
risks.
This will
slow development,
more productive ventures
make
the earlier stages of
slow down economic progress even further, because
progress will be stopped by crises and failed take-offs,
1
(iii)
(ii)
many
since
that are
development
runs towards
chance will play a very important
role;
economies
enough
that are lucky
good draws
to receive
have more
at the early stages will
capital,
thus will achieve better risk-diversification and higher productivity. Therefore, although Prometheus
be unbound accidentally, chance will always play a key role in his unchaining and
will not
condemn him
permanent imprisonment.
to
how much
In our model, agents decide
safe asset with lower return.
However, not
projects.
minimum
may even
all
The
to save
rest of the funds are
and
used
how much
some of these
sectors.
money
to invest in a
to invest in imperfectly correlated risky
risky projects are available to agents at
size requirements that affect
of their
all
points in time because of the
The more
"sectors" (projects) are open,
the higher
is
the proportion of the savings that agents are willing to put in risky investments. In turn
the higher
is
the capital stock of the
economy,
sectors can be opened; thus development goes
diversification opportunities.
However,
this
the higher are the
hand
process
economy
in imperfectly correlated projects, the
is
in
incomes and savings, and the more
hand with the expansion of markets and
is full
better
of perils, because with limited investments
randomness and spends a
subject to considerable
long time fluctuating in the stage of low accumulated capital. At this stage economies that receive
grow while those
"lucky draws" will
will stagnate.
As
that are unfortunate
enough
of "bad news"
to receive a series
the lucky economies grow, the take-off stage will be eventually reached
larger stock of savings will be
made
and a
available to be invested in productive investments, or to use
Braudel's terms, to "open the sluice-gates
"
of the economy. Over the course of take-off, the number
of projects expands quickly, and the induced financial development provides better opportunities for
growth
diversification. Eventually, industrial
diversified
and
"the right
That take-off
baseline
economy
corresponds to
is
road thereafter
will eventually
is
reached with idiosyncratic risks being completely
will never
occur
is
due
be abandoned".
to the feature that the equilibrium path
unique and the stochastic process associated with
and "industrialization";
full diversification
stage and they will never abandon
it.
all
it
has a unique ergodic set that
economies
However, we show
will eventually reach this
that this feature only arises
preferences induce a sufficiently low rate of relative risk aversion. With
equilibrium stochastic process no longer has a unique ergodic
receive a series of unlucky draws
will
is
may
be trapped in underdevelopment.
quite different
technology
from others
Boldrin
(e.g.
indivisibilities
are
(1992),
it
is
more
risk-aversion, the
is
possible, thus they
has to be noted that the nature of this underdevelopment trap
literature that are
Matsuyama
the
attitudes
(1991)).
In
mainly driven by the form of the
our
case,
although
the
amount
in risky assets
and aggregate productivity
is
to invest
low.
Theoretically our model corresponds to a general equilibrium
commodity space because
sectoral
towards risk that condemn an economy to
underdevelopment; when only a few sectors can be opened, risk-averse agents are willing
a limited
when
This implies that economies that
a region from where no escape
development
in the
important,
It
fall to
set.
of our
the set of traded financial assets (or
economy with an endogenous
open
sectors)
is
determined in
We use the competitive equilibrium concept suggested by Makowski (1980)
equilibrium.
of economies. This equilibrium
therefore
We
agents are
all
still
is
Walrasian conditional upon the number of sectors that are open;
kept as price-takers and there are no unexploited gains in any activity.
and simple form of the
find that despite the innocuous
economy
competitive
is
inefficient
and
undertaken. The underlying problem
other open sectors as consumers
only do
we show
result that there is
It
may be
is
now
two
indivisibilities in this
economy,
the
form of too few projects being
this inefficiency will take the
that each sector creates a positive "pecuniary externality"
bear less risks
when
no decentralized market structure
that
conjectured that since our mechanism
we
can avoid
also establish the stronger
this inefficiency.
related to capital shortages,
is
on
they buy the shares of these sectors. Not
that the competitive equilibrium is inefficient but
We
be limited in the presence of international capital flows.
identify
for this type
its
validity will
turn to this problem in section 6 and
would make
factors that need to be taken into account. Decreasing return technologies
foreign capital flow towards poor economies, and yet, domestic capital would also flow out towards
richer countries in order to obtain outside insurance.
to a
two-country world. In
this
To
discuss these issues,
capital flows
more
economy becomes
into
richer, the direction of capital flows is reversed
and
Our model
interesting
is
form
matches
that
at the early stages,
one of the countries, thus capital flows create divergence; but as the world
that in
purpose of outside insurance, which
industrialization
an
development of Western Europe (see Neal (1990));
convergence increases. Yet, we also show
the
extend our model
case the problems that our paper emphasizes do not disappear and
the general pattern of development in this two-country world takes
the historical facts of the
we
is
many
quite
and as a
result, the rate of
cases, the outflow of domestic capital for
common
in
practice,
be harmful
will
to
to the welfare of future generations.
related to the
growing
literature
on credit and growth (among others Bencivenga
and Smith (1991), Greenwood and Jovanovic (1990), Greenwood and Smith (1993) and
(1994)). These papers
make valuable
contributions but derive
most of their dynamic
effects
Zilibotti
from
the
presence of fixed costs. Our model, instead, concentrates on the diversification role of the credit
markets (as emphasized by Gurley and Shaw (1955)) and argues that better diversification
opportunities enable the allocation of funds to the most productive investments.
One
result is that in
the presence of micro-level non-convexities (but without fixed costs of financial intermediation),
endogenous dynamics
most closely
that link
related to ours
is
economic growth
to financial
Saint-Paul (1992)
who
role of the stock market. Saint-Paul's
there will be
more productive
specialization.
ways; the degree of diversification
market incompleteness and growth
randomness
in
main argument
is
is
is
deepening are introduced. The paper
discusses the insurance (risk-diversification)
that
Our paper
when
differs
stock market insurance
from
his in a
is
possible,
number of important
endogenized, the dynamic interaction between the degree of
investigated and the implications for the role
and evolution of
development are derived. Also none of the above papers investigate the
links
between
and international
credit markets
capital flows.
Another
literature that relates to
our paper
pioneered by Allen and Gale (1988,1991) where the incompleteness of markets
is
is
the one
endogenized by
introducing costs of issuing financial securities (see also Bisin (1994) and Pesendorfer (1991)). In
our model too, the market structure
and the main focus
open
diversification
on
is
is
endogenized, but without explicit costs of issuing securities
the interaction
between the incompleteness of markets, the opportunities for
to agents
The plan of the paper
motivation to
and the process of development.
is
this investigation.
Section 4 shows
why
as follows.
The next
section discusses
some
historical
and empirical
Section 3 lays out the basic model and characterizes the equilibrium.
the decentralized equilibrium
is
not Pareto efficient and characterizes the Pareto
optimal allocation. Section 5 shows that in the presence of a rate of relative risk-aversion greater than
one, poverty traps are possible. Section 6 presents an open
economy extension of the model
in
which
international capital flows are allowed. Section 7 concludes.
2)
Motivation and Historical Evidence
Many economic
historians (e.g. Braudel (1962,1979),
North and Thomas (1973), De Vries
(1990)) emphasize that countries at the early stages of development experienced high degrees of
uncertainty.
McCloskey (1976)
Medieval England
the variability
was
0.347 and
at
certainly
due
calculates the coefficient of variation of output net of seed in
that "famines"
is
p. 3 12)
to
be
itself a
symptom of an
undiversified economy.
in the
visible to the
failed take-offs.
More
specifically,
"
he
West when there was an expansion of banking and credit so
naked eye (Florence 1300s, Genoa
late
1500s and Amsterdam
17 00s)... three substantial successes, which ended every time in failure or at any rate in
of withdrawal.
on
development of industry before 1750 as "subject to halts and
and points out the presence of
documents "three occasions
abnormal as
is
largely dependent
considerable evidence that non-agricultural activities were also subject to large
uncertainties. Braudel describes the
breakdowns" (1979,
was
to the fact that agricultural productivity
weather, but this heavy reliance on agriculture
Additionally, there
were occurring on average every 13 years. Part of
(1979, p. 392). Also the pattern of these failures
is
some kind
informative; these cities
grew
gradually by expanding the scope of industrial and commercial activities and the collapse took the
form of an abrupt end
ignited
whose resolution impacts
And
by a few bankruptcies. This
significantly
on
the rest of the
Among
suggestive of large undiversified risks
economy
1
.
also today, poor countries exhibit considerably higher variability of output (and of
consumption) than more developed economies. To support
1
is
the historical examples of sparks that triggered
this claim,
we
estimate an
more widespread
failures,
AR(2) process
we can
recall the
of 1557 which led to the decline of Genoa and Antwerp (Braudel (1984, p. 153)) , the
Clifford bankruptcy of December 1772, the starting episode of the crisis of Amsterdam (Braudel (1984, p.
Spanish
271), or
state default
more
recently the Austrian Bourse Crash of 1873 (Rudolph (1972, p.29)).
.
growth
for the
rate of
the period 1960-85
_
income per capita for each country, using annual data for 117 countries for
from
Summers and Heston (1991)
the
a
ran the regression equation AlbgGDP., = a(0/) +
innovation obtained from this equation,
for the corresponding country.
whole
the
set of
1
We
(1
we
Dataset. Formally, for each country,
Alog GDP
i
,_,
a(2i)AlogGDPil2
+
+
e
The
ir
then used to calculate the variability of the growth rate
e ijt , is
consider two different samples of countries. The former includes
17 countries, the latter only a subset of them, excluding those five countries
whose
estimated standard deviation of the innovations exceeds 0.10 (Iran, Iraq, Gabon, Somalia and
Uganda), since we suspect
political
unrest and wars to have been the
exceptionally high variability. In Figure
capita in 1960
on
the horizontal axis
on
the corresponding country
(tstat
=
goodness of
R =0.208
fit is
Analogous
for the first
results are obtained
consider pairs of observations
{GDPti
R
and
=
je,
+1
=0.232, for the second
J},
(i.e.
Indonesia,
-5.74) with the smaller sample.
that
of the one period-ahead growth innovation in country
the
is
GDP
as obtained
i
The
.
time
at
is -
2
by pooling cross-sectional and time
,
per
e iit for
very steady rate), the regression coefficient
at a
2
GDP
of the innovation
In particular, after controlling for an outlier
grown
their
solid curve traces the regression line of the (log)
-5.41) with the larger sample, and -0.25 (t-stat
2
plot the (log of)
of) the standard deviation
The
the vertical axis.
a very poor country in 1960 which has
0.26
we
using the smaller sample,
and the (log
GDP1960.
standard deviation on (log)
1,
main source of
series information.
t
We
and the absolute value
above (we have 117x23 such
We then fix some threshold income levels and divide the panel of innovations into four groups
where
according to increasing GDP ranges, i.e. {|e r+I jjGjGROf/i^} iff xk <GDPr r <xM
pairs).
..
(
jxj
k= 1,2,3,4
denote the thresholds. Finally,
we compute
absolute value innovations belonging to each group.
standard error of the
first
column shows
mean
in brackets
that the
(GDP
The
per capita in
average size of innovations
is
the sample
mean
for the (pooled)
results are reported in table 1, with the
US
countries).
The
decreasing with the income range,
i.e.
Dollars 1980 for
all
lower income ranges are associated with higher variability than higher income ranges.
GDP
Range
Avg. (innov)
No
< GDPj < 700
700 < GDPi ,< 1500
1500 < GD^ < 4500
4500 < GDPi
t
t
Table
2
To
control
1.
Variability
Nor
for the possible
Control
FE
0.046 (0.0025)
0.044 (0.0016)
0.007 (0.0048)
0.006 (0.0030)
0.039 (0.0014)
0.028 (0.0013)
0.000 (0.0026)
-0.010(0.0023)
°
Obs
404
687
894
706
convergence effects in
AR(2)
that
we
this
cross-sectional regression,
we added
a
ran for each country and this was found not to change
did the inclusion of the average growth rate as a regressor change the significant negative
coefficient of the initial
economies. Yet,
FE
N
of growth rate innovations (panel).
deterministic polynomial trend to the
the results.
control
Avg. (innov)
GDP.
A
source of concern
may be
this is unlikely to lead to as strong
interpret the evidence as very supportive of the
that
measurement error could be larger for poorer
1. Thus we
and significant a relation as shown in Figure
main implication of our theory.
FIGURE
HERE
1
we
In the second column (Control FE),
report the sample averages for
the corresponding
income
ranges after controlling for fixed country and time effects, by subtracting from each observation the
respective country and the time
deviations from averages,
correlation between
removed,
GDP
this finding
means (averaged across
some observations
levels
and growth
will
now be
t
and
respectively). Since
i,
negative.
As
we compute
the results show, the negative
variability remains. Since country averages
have been
suggests that our results are not only driven by cross-country variations, but
3
the time series variations are also consistent with our prediction
.
Recent empirical work by Quah (1993, 1994), and Benhabib and Gali (1994) provide further
On
evidence supportive of the general implications of our model.
the
very poor countries to achieve growth and improve their condition.
a country to experience "set-backs"
is
a decreasing function of
On
average (Table
1, p. 431).
in four
groups according to
their
GDP
income
Markov chain
difficult for
level.
Quah
transition matrix
per capita relative to the world
GDP
per capita relative to the world average):
Prob(=)
Prob(t)
x<l/4
1/4<x<1/2
1/2<x<1
-
0.76
0.24
0.52
0.31
0.17
0.29
0.46
0.26
Kx<2
0.24
0.53
0.24
2<x
0.05
0.95
-
Prob(l)
2.
seems
Table 2 below report a summary of his results for a 23-year transition
matrix 1962-1985 (where x denotes
Table
it
the other, the probability for
(relative)
its
(1993) studies the cross country dynamics of growth by estimating a
by classifying the countries
one hand
Estimated 23-year transition matrix (Quah 1993, p. 431).
we report the estimated probability that a country belonging to a
In the three different columns,
certain group falls to a relatively poorer group, remains in the
same group, or moves up
group, respectively. Observe that the probability for a country to get worse decreases as
to a richer
we
consider
higher income groups (also found by Benhabib and Gali (1994, p. 11)). These findings give further
support to the claim that the process of development
From
growth points
a different perspective,
at a
some
It
work on
at the earlier stages.
financial
decreasing relation between risk-premia and development.
borrowing and lending
lagged
of perils"
recent empirical
Sussman (1993), Sussman and Zeira (1993) and
3
is "full
rates applied
by banks
-
Zilibotti
development and
More
precisely,
(1993) show that the gap between
which can be taken as a proxy for risk-premium
-
should also be reported that regressions of the (log) of absolute value of innovations over the (log of)
GDP per capita also give
encouraging
0.20 (tstat=-8.91) when only a constant
is
results.
The point estimate of the
included, and -0.35 (tstat
=
-2.6)
coefficient of lagged
is
beyond
-
when we control for country and
GDP, in this case raises some
time effects. Yet, note that the non-stationarity of the regressor, the level of
inference issues the discussion of which
GDP is
the scope of this informal analysis.
are negatively correlated with the income level across countries, and across regions and time in the
United States. These findings give further support to the mechanism proposed in
A
possible explanation for the decreasing variability (and for the decreasing risk-premia)
technological. Countries at the early stages of development
are risky and
low productivity
(e.g. agriculture).
The adoption of new technologies
is
more
may
This however does not seem to be the whole story.
often subject to economic as well as scientific constraints.
that
were
and
attribute the failure to adopt these technologies earlier to the lack
Hobsbawm
(1968) goes even further and asserts that there was nothing
-
of the technologies
of monetary incentives.
new
in the technology of the
and most of the new productive methods could have been developed
British Industrial Revolution
at
150 years before.
On
the other hand, as an obstacle to expansion and growth, capital scarcities and limited
savings appear important in
and
that indivisibilities
poor
many
used in improving agricultural productivity were actually known in Medieval Europe
that
later
is
only have access to technologies that
North and Thomas (1973) and Rosenberg and Birdzell (1985) argue
least
this paper.
countries, there
many
instances.
capital shortages
is
Bagehot (1873,
were a
p. 4)
more than a century ago argued
significant hurdle to
overcome,
no spare money for new and great undertakings.
.
",
and finally
there
capital
is
is
"...in
scarce and diffused, the distrust of industrial activities
greater pressure for bigness...
".
The
"...in
even when they were
very promising. Gerschenkron (1962, p. 14) also echoed the same view in writing;
backward country
words,
in his
relatively
considerable
is
size of the required activity
a
was
certainly
a relevant factor in the minds of innovators; Scherer (1984, p. 13) quotes Boulton, James Watt's
partner, writing to
but would be so
certain
economic
if
Watt
the
was not profitable for just a few countries
that the production of the engine
whole world were the market. In
historians, for instance
line
with the view of capital constraints,
Wrigley (1988), argue capital shortages that led
to shortages
of energy to be the main reason for heavy reliance on agriculture and a key constraint on
industrialization, while a large part of the
development
literature, implicitly
or explicitly, defines
underdeveloped countries as those that have shortages of capital (see for instance Viner (1958),
Singer (1958), Lewis (1958))
4
.
The mechanism which we
4
offer in this paper
is
The important empirical question of course concerns
and the savings constraint
definitely
will apply.
For
crucially related to the attitudes towards risk.
at
what level the
be too large for a single village or even town. However,
if all
or even those of a small developing country today were pulled together,
have binding minimum
size requirements.
indivisibilities (non-convexities)
instance, the start-up investment for an industrial activity will
On
the resources of Medieval Europe
it is
unlikely that
many
sectors
would
the other hand, both actual transaction costs and those created
by incentive and enforcement problems imply that the relevant unit of accumulation will often not be a whole
continent nor even a whole country. Identification of the relevant unit of accumulation is at the heart of the
empirical relevance of this question and does not have an easy answer. The regressions reported above for
instance takes
relation
it
may be
as given that political borders are related to the size of the accumulation unit
quite imperfect.
-
though
this
Risk-aversion
institutions
is
recognized as an important factor in economic behavior, witness for instance the
developed
in
many
societies to deal with the
(Persson (1988), Townsend (1990)).
bear, even
when
this
It is
problems of insurance and risk-pooling
then natural that agents will try to reduce the risk they
has a cost in terms of expected return. In our model,
when
diversification
opportunities are limited, agents choose investments which are safe but less productive, for instance,
the storage technology
chosen because of
and the scattering of
their relative safety
5
fields
widely used in Medieval Europe, both of them
Braudel emphasizes that unproductive hoarding (storage)
.
More
frequently occurred in poor undiversified economies subject to capital shortages.
he writes (1979,
"Every society accumulates capital which
p. 3 86);
be saved and hoarded unproductivety, or
flow was not strong enough
its
true nature as
it
to
open
to replenish the channels
and 19th century Britain also
illustrates
how
then at
its
disposal, either to
of the active economy. .. If the
was almost
inevitably immobilized,
in the portfolios
between the eighteenth
all the sluice-gates, capital
were unrealized". The pattern of change
is
specifically
the use of relatively safe assets has decreased as the
array of available assets has expanded and the income level has increased, thus giving agents better
diversification opportunities through a wider variety of risky assets (see
Kennedy (1987),
Finally, this paper stresses that at early stages, lack of diversification gives
to "luck". In particular,
are successful, the
we emphasize
that if large
economy grows, more
an important role
and risky investments undertaken
risk-diversification
table 5. 1).
at early stages
becomes possible and new projects can
be undertaken. Here the historical role of the railways can be viewed as an example of the
importance of the success of large projects in opening the way to steady growth. For instance,
according to Alfred Chandler (1977),
development of
US
railways
it
was
also possible to finance other large scale projects
(although the main aspect emphasized by Chandler
On
a turning point in the historical
capitalism because after the efforts of the Wall Street to raise the necessary funds
required for the large railway investments,
deals).
constituted
is
the other hand, Spanish railroads attracted
that
Wall Street "learned-by-doing"
more than
15 times the
other manufacturing in Spain by the end of 1864; but, in contrast to the
in the
US
second half of the nineteenth century were very disappointing and
economy
the result
in all
experience, the returns
all
the sectors of the
Cameron
"Spain in the 1850's undertook a vigorous program of railway construction
promotion not unlike the
later
was a fiasco which
by at least a generation.
5
amount invested
suffered turmoil and capital scarcities as a result of these heavy losses in railways (Tortella
(1972, pp. 118-121)). Regarding this episode and a similar one in Italy,
writes,
to cut big
"
Russian program;
set
Italy
(1972, p. 14)
and financial
did the same in the 1860's. But in both cases
back the progress of industrialization and economic development
Another case of a large project's destiny getting interwoven with
common
that of
Medieval Europe and yet McCloskey and Nash (1976) argue
that storage was scarcely used but interpret the open field system and scattering of cultivation as an inefficient
technology adopted for insurance reasons.
Storage
is
widely believed to be
in
the
whole economy
that of the
is
Dutch versus the English East India companies (see Neal (1990)
and an earlier version of our paper for discussion).
The Basic Model and the Decentralized Equilibrium
3)
We
two periods. There
the
who
consider a simple overlapping generations model with non-altruistic agents
a continuum of agents normalized to
is
same generation are
The production
all identical.
1
in each living generation
side of the
economy
good sector and a continuum of intermediate sectors normalized
to
live for
and agents of
consists of a single final
1.
The
good sector
final
transforms the capital and labor of the economy into final output. The intermediate sectors transform
savings of time
t
into capital to
be used
In their youth, our agents
of this sector. At the end of
decision entails
how much
this
to save
at
work
time t+
1
in a final sector firm, receiving the competitive
in
some of
how much
and
of their savings to put in a safe asset. The safe
own
FIGURE
state j.
2
More
brought forward to the next period
by
is
is
in
the uncertainty
determined. The
final sector firms
summarizes the sequence of events
and old agents
our model.
let gj(Fj)
*m-
this,
we assume
to 1)
and project
form of the uncertainty. For
likely states (again with
formally,
t
HERE
crucial issue concerns the
continuum of equally
is
in their retirement period is rented
the rent they receive. Figure 2
The
therefore one unit of capital at time
r,
economy. After the investment decisions,
the risky projects of this
unravels and the amount of capital that
consume
rate
of capital at time t+1. Alternatively, they can invest part of their savings
r units
capital that agents
wage
period they take their consumption and saving decision. This
asset has a non-stochastic rate of return equal to
transformed into
without using labor.
measure normalized
denote the return from asset j
limdjMj —F.
if state
when an amount
that there is a
j
pays only in
Fj is invested.
Then:
j occurs and Fj>M(j)
aj
otherwise
where
R >
r.
Also
are equally likely,
if
we
we
will
Fj>M(j); then, investing
in
one
state
denote the probability density function over states by
have v(j)=l, for
in a sector
is
the safe asset (since
R>r); second,
that there is safety in
results
of the paper.
To
start with,
ignore the second requirement that
equivalent to buying a Basic
of nature. This formalization
features that will drive all our results;
all j.
is
first,
v(j), since all states
Arrow
Security that only pays
adopted for mathematical convenience and captures two
the risky investment has a higher expected return than
different risky investment projects are imperfectly correlated so
numbers. Other formulations with these two features will also yield
A
convenient implication of
this
formulation
is
all
the following; a
the
main
$1 equi-
proportional investment in a proportion p of projects (thus a total of $p invested) will yield
probability p and zero with probability 1-p. Formulated in this
economy
straightforward: invest
is
result will also
be true
in
all
way
the allocation
$R
with
problem of
the
the savings in each of the risky sectors simultaneously. This
more general models
as long as all these sectors exhibit decreasing or
constant returns to scale. However, in the presence of some projects with non-convexities, a trade-off
is
introduced between diversification and high productivity. In our case,
to the
presence of M(j), the
implies that
all
minimum
size requirements.
Thus, the technology of
intermediate sectors exhibit constant returns to scale but
certain size (M(j)) before being productive.
requirements, M(j),
is
We
This specification implies that
of the sectors, the
some
this
minimum
all
sectors
minimum
size
have no minimum size requirement and for the
rest
also
assume
that the distribution
of
j
{»•
<y
£™}
requirement increases linearly. Our results are not however
size
6
this linear specification
,
and the ranking of projects from lower
obviously in equilibrium, sectors with smaller
Also note
economy
sectors require a
to higher size
without loss of any generality and imposes no timing constraint; any project can be adopted
others.
due
given by (see Figure 3 for a diagrammatic representation):
M(j)=Max
dependent on
this trade-off will arise
that the
minimum
minimum
size requirements will
first,
is
but
be opened before the
sectoral size requirements introduced here
may be
thought as
operating at either the firm or the sectoral level.
Preferences over final goods
The preferences of consumers over
E, £/(c,,c
where
j
final
goods
is
defined as
M )=logc,+/3 J logc,Vy
(i)
represents the states of nature which are assumed, as noted above, to be equally likely. Each
agent discounts the future at the rate
/3
and has logarithmic preferences which
will give us a constant
saving rate. Note that, although the realization of the state of nature does not influence the
productivity of the final good sector, the final consumption of our representative agent depends on
this since
6
is
it
determines
Except the
result
how much
we
capital
he takes into the
final
will obtain in section 5 that the value of
1
good production
stage.
for the relative degree of risk-aversion
a critical level above which underdevelopment naps are possible. With a different functional form for M(n),
the critical value of the relative rate of risk aversion
would be
10
different.
Final good production technology
Output of a typical
final
good producer
is
given by
y,=AkX~
where lower case
letters
denote the levels hired by
a
(2)
this firm.
Bearing in mind that the labor and capital markets are competitive and that aggregate labor
supply
equal to
is
1,
the relative prices of these factors, respectively
w
t
and p„ are obtained as
w=(l-a)AK?
(3)
A V<X-\
P=aAK,
Organization of the Stock Market
We
will think of
Arrow
each intermediate sector as represented by a set of firms that
compete "a
la
Bertrand" by calling prices
(one unit of security entitles the holder to
R
units of
particular
how much
security and
t+
1
P
j
for each unit of security
capital in state j)
and each agent decides
of each traded security to buy. Naturally, for the sectors that have binding
requirements only one firm will be active but the potential of competition will force
below for
profits (see
the precise equilibrium concept).
firms function without using any additional resources.
For simplicity we assume
The intermediate
minimum
it
to
from
then lends
the
consumers (again by
size
make zero
that intermediate
sector trading can be thought
as a stock market, but for our present purposes this can also correspond to a situation in
financial intermediary collects the funds
sell the
which a
selling similar securities)
and
directly to firms.
it
we
In the analysis of this section,
will
assume
that inter-firm share trading
is
not allowed.
This assumption will rule out the situation where some firm or intermediary can buy-up the whole
"security market"
and
in essence,
will act as a restriction that financial intermediaries are only
it
allowed to offer Basic Arrow Securities. Although
no more than a simplifying assumption
concept of equilibrium. In section 4,
concept will
An
make
this
it
at this stage this
looks quite
that enables us to derive the
will
main
ad hoc,
it
is
actually
results using a simple
be shown that a natural refinement of our equilibrium
assumption redundant.
immediate implication of competition among financial intermediaries and of ruling out
inter-firm share trading
is
that investors will face a constant (linear) price for
marginal cost of supplying
it;
thus
P =1
J
for all securities that are traded.
each share equal to the
To
see this note that
Bertrand competition will lead to zero profits and in the absence of inter-firm share trading,
implies zero profit in each state of nature. If a firm
its
security, a competitor can enter
$R when
zero in
it is
and undercut
successful and since firm
all states
of nature other than
j is
demands a price higher than $1
this price.
The same
applies
if
for any unit of
a firm pays less than
not allowed to hold shares of other firms,
j.
11
this
it
has to pay
The Equilibrium Concept
A
competitive equilibrium for this
economy
defined as a proportion of open sectors and
is
a price function that assigns a price level to each open sector. All agents act as price-takers and the
number of open
sectors
is
determined from a zero-profit condition.
Definition 1: Consider the vector [n* ,{P*
P
sectors and
J
(i) erft
(ii)
*
t
prices of securities at time
(n,*,P*)=0 for
all j
<n,* (supply
no additional firm can enter
and make positive
Note
that
profits
we have
even
J
} 0£j£lli
n,*
formally;
denotes the proportion of open
This vector characterizes an equilibrium
t.
equal to
is
where
A
More
demand
in all the
open
iff
(n,*) sectors);
at
any time, offer a security conditional on a feasible production plan
if
consumers re-optimize.
we
defined the equilibrium as a dynamic concept. However,
will see that
because of the limited intertemporal interaction, the equilibrium prices and the measure of active
upon
sectors can be determined separately for each period, conditional
The
generation.
relation of this equilibrium concept to others
is
the earned
income of the old
discussed in section 4.
Individual Maximization Problem
We
asset
by
<£,
denote the saving of each individual
and
their investment in sector
j
by F
at
j
time
Since
.
t
t
by s„ the amount they devote
we have
also correspond to their respective aggregate amounts. Also
at
time
t
by
n,.
Each agent solves
we
a representative agent model, these
denote the proportion of sectors open
the following maximization
max
to the safe
problem
logc,Vy
logc,+/3f
Jo
(5)
subject to
+
4>t
\l
FidJ=s
(6)
t
cU=f/ (r<t>l+RF!)
t
F/=0 for
J
'
(7)
all J
j>n,
(8)
'
c+s<w
(9)
l
and taking
w
t
,
n,
and
also, in writing the
j
p, as
given, p j denoting the marginal product of capital in state
t
budget constraint of the individual consumer,
12
we have used
j
at
time
t.
the fact that
Note
P =l
j
t
for all
that
j
is
traded.
It is
particularly important to note that n,
among
taking individuals and will be determined by competition
aggregate resource constraint that
aggregate constraint
is
obtained by
is
summing
taken as given by these price
is
intermediate sector firms and the
Using
(6) across all individuals.
(8), this
expressed as
*FJdj=st -4>,
|
Finally, the marginal product of capital in equilibrium
is
(10)
determined as ^,=aA(r(j)+RF/) a
'
1
.
Individual Decision Rules
From
logarithmic preferences
we
directly obtain the following saving rule
—
s,=—
'
1+/3
rm
w,
(11)
'
In other words, the logarithmic specification implies that the exact risk-return trade-off will not
influence their saving rate. Before
decision,
Lemma
we
1:
state a
Let
J(t)
The main
each individual
Therefore,
Lemma
idea of this
is
be important throughout our analysis
lemma
is that,
at
t;
then
[
to
it
;
V jj' EJ{t).
,
t
because of Bertrand competition in the stock market,
purchase an equal amount of
sectors are open,
7
FJ =Fj
facing a constant price for each of these symmetric Basic
when j <n,
given by
that will
to the characterization of the individual's portfolio
be the set of securities traded in equilibrium
would want
therefore
utility is
simple
we move
will follow that
all
Arrow
securities
and
the securities that are being traded.
F =F Now
j
.
t
t
noting that the second period
log[f/,(RFi+r<l>)]dj=n}og[p](RFt +r4>)]+(l-n)log[p}(r<l>
l
)]
with p's treated as
given by the individual, the maximization problem can be written as
max n log(RF +r<t>) +( 1 -/2,)log(r<£,)
l
subject to s t =<£,+n,F, where again
(12)
l
n, is treated as
a parameter. Maximization yields the following
decision rules;
7
Since
we have
a maximization problem with respect to a continuum of choice variables,
should be read as "almost everywhere". However,
proofs of
Lemma
1,
this technical detail
Propositions 1-3, and Corollaries 1-3 are in Appendix A, the proofs of
Proposition 4-7 that are longer and
more cumbersome
authors.
13
all
does not affect any of our
are provided in the
Appendix B,
our claims
results.
Lemmas
available
The
2-6 and
from the
—
(l-n)R
=
(13)
s,
<t>,
R-rn.
'
R-r
'
Vj<n,
s,
(14)
R-rn
F/=
i
VJ>n,
Equation (14)
aggregate
demand
grows
as the
expressed diagrammatically in Figure 3 as a relationship between a pseudo-
for each risky asset (solved for the equilibrium price)
that are already open.
asset
is
It is
clear
from both
measure of sectors
diagram and equation (14)
the
open
that are
increases. This
is
sectors are open, the better are the risk diversification opportunities
consumers
to further
and the number of sectors
that the
due
demand
for each
more
to the fact that the
and the more willing are the
reduce their investments in the safe asset and increase their investments,
F
j
t ,
in the risky projects that are available to them.
We
make an assumption
also
to ensure that risky investments are sufficiently productive
relative to the safe asset. This will guarantee the uniqueness of the equilibrium path.
in section 5 that with a refinement of
arise
and assumption (A)
Assumption (A)
:
will
It
will
be shown
our equilibrium concept, multiplicity of equilibria will never
now
be unnecessary, but for
it
simplifies the exposition.
R>(2-y)r
Equilibrium
Proposition
1:
Suppose (A) holds and
let
K
*
characterized by a vector (^,*,{P ^) n</<n
.)
t
(R+ry)-
n* =min
t
be given. Then there exists a unique equilibrium
where P
*
t
{R+ryf-Ar
J
=l
for all jG[0,ni*]
(R-rXl-y)0(l- a )A
D
1+0
a
and
R
(15)
'
,1}
{.
2r
All security
demands are given by
FIGURE
This equilibrium
3
is
(14).
HERE
expressed as the intersection of the pseudo-aggregate demand of each
risky asset with the curve that traces ininimum size requirements in Figure 3.
when
s
value
1
t
=D
so that there
is
enough money
to
open
all
if
can be checked that
the intermediate sectors, n,* indeed takes the
and thus (15) defines a continuous function as drawn
roots of a second order polynomial;
It
in Figure 3.
Note
that n,*
is
one of the
assumption (A) were violated existence would be
14
still
guaranteed, but the two schedules F(n) and M(n) could cross
[0,1], so multiple equilibria could arise
Proposition
full stochastic
is
more than once
.
characterizes the equilibrium allocation and prices for given K,.
1
equilibrium process,
range
8
we need
to
To
obtain the
determine the equilibrium law of motion of
K
This
.
t
straightforwardly given as;
—-—-RTK" = aB {n*(K),r,R)TK
*,>
=
prob. \-n,*
R-rn,*
(16)
RYK? = oc{n*{K),r,R)YK?
where n *=n*(K )
t
t
is
whether the society
probability
time.
As
is
feature to note
when the
Moreover, the probability of
1-n,*).
economy develops,
the
can afford to open more sectors,
it
event, increases. Also
from
(16), the expected productivity of
development and diversification; as
proportion
of
sector
n,
event also changes over
i.e.
higher
and the
n,*,
the probability of a lucky
an economy depends on
its
level of
increases, the expected "total factor productivity" (conditional
open,
or
on
<f(n* ,r,R)=n*o G (n* ,r,R)+{\-n*)oB (n* ,r,R)=
the
economy. This
is
-
-,
(R-rn
stock
of
the
increases
economy),
well.
as
by
given
However,
*)
on good news, aG (n
,r,R), is independent of the
a feature of logarithmic preferences as the effect of
higher investment in risky assets as n increases
invested in projects that did not pay-off.
capital
—
interestingly, the productivity level conditional
diversification level of the
is
depend on
risky investments pay-off,
this
probability of transferring a large capital stock to the next period, that
the
n,'
that the level of capital stock next period will
is
lucky in the current period (which happens
or not (probability
n,*)
prob.
given by equation (15) and r=A(l-a)(3/(\+(3).
The important
on
in the admissible
We will
exactly offset by the fact that
is
return to this issue
no money was
when we consider more
general
preferences in section 5.
To
(i)
QSSB:
formalize the dynamics of development,
"quasi steady state" of an
sectors invested
(ii)
QSSG:
The
by
this
we
define the following concepts;
economy which always has unlucky draws
economy never
in the sense that the
pay-off.
"quasi steady state" corresponding to an
capital stocks corresponding to these
economy which always
two quasi steady
*( KQSSB))
receives good
states will respectively
r (l- n
T=5
Prl>;
K QSSB = J r(l-/2*(fg™))
RY
QSSB
f
{
Yet, for
st
<D,
uniqueness
is
R-rn*(K
guaranteed even
if
15
(A)
)
is
news
.
be
(17)
1
J
violated (see section 5 for
more
discussion).
1
K QSSC = fffp\-FZ
In particular,
state;
if
a point,
uncertainty can be completely
economy
reached, from which the
if
condition for this steady state to exist
is
removed
(18)
(i.e.
will
n(K QSSG ) = l),
From
never depart.
that the saving level that
sufficient to ensure investment in all the intermediate sectors.
there will exist a steady
equation (15), the
corresponds to
K QSSG
should be
Thus
a
1
D<T T^R T^
We
will refer to this particular case of (18) as
When
this steady state exists, the pattern
At very low
in Figure 4.
capital levels (region
positive growth. Then, there
draws
-
although
some
when
i.e.
some of
this level
time;
when
is
the concavity of the production function guarantees
II) in
not a steady-state,
when
bad news
the
is
increase (but bad
even when
it
economy has
all
it is
As
grow and
finally the
K ss
grow towards
it.
On
the other hand,
bad shocks, and since
in case they receive
since
more
is
KQSSB
is
.
Yet as good news are
draw next period
will also
invested in risky projects
enters region EH,
all
risks,
and
9
Note
).
will typically
idiosyncratic risks will be
invested in
all
sectors)
and we
will
.
established that an
specification of the
form of
is
true
economy
model has
economic
that the
more
that receives a series of lucky
draws
will
be
the unrealistic prediction that set-backs in the pre-take-off
disasters, taking the
sectors are open,
economy back
the less will
willing to put in the safe asset for insurance purposes. Therefore, a
bad
to a very primitive stage
be the amount that agents are
realization,
though
now much more
unlikely, will have devastating effects. In contrast, the set-backs observed in the historical development
process that were mentioned in the introduction do not have this feature. This feature of the model can be
avoided by altering the specification for the structure of returns of the risky assets. An example would be to
assume that in every period, a fixed measure z > of all possible projects have positive return, with each asset
having the same probability of success. In
this case the probability
gives a positive returns would be decreasing in the
z.
number of
of a catastrophe in which no risky asset
active sectors and, tend to zero as
But, bad realizations and crises in which a small proportion of the risky assets
with positive probability until
;
HERE
we have
of development. The reason
will
open and an equal amount
4
The current
KQSSB
are separated by
II
exposed to large undiversified
economy
FIGURE
stage can only take the
QSSB
the probability of a lucky
is still
observe a deterministic convergence to
far,
and
a level of capital stock just above
grows, the economy
(since all sectors are
So
K
I
a point around which the economy will spend
economies
news now becomes more damaging
experience some set-backs.
removed
which growth only occurs conditional on good
increasing in the level of capital stock, the probability of a set-back
received, the capital stock will
9
of development will typically look as represented
the risky investments paying off. Regions
of capital
19)
.
they are above this level, they will go back to
highest
that
I),
a range (region
is
they are below this level,
the probability of
is
Kss
<
full diversification is
reached.
16
is
successful, can
n tends
still
to
occur
unchained from
and
its
low productivity "primitive accumulation" stage and
As Figure 4
industrialization.
full
may be
-
which
Suppose (19)
1:
What
will
measure
happen
will take longer
it
and
plim ^ a> (K )=K ss
then
is satisfied,
l
is
l
randomness of the growth process? To answer
to the
to look at
and
probabilities n*
(1-n*).
from
clear
deterministic
=
logT
determine
how
considered;
(i)
+
+
(a-Ologtig
a.
on
(20)
after
volatility,
the capital stock of the
measure evolves as a function of n* (and K).
economy develops, more money
is
as;
log (cKn'flQ)
removing
Vn
economy).
Two
invested in risky assets,
the
be entirely
will
Define the conditional variance of a as
equivalent to conditioning
this volatility
as the
with respective
B (n*,,)
<7
induced by the neoclassical technology,
determined by the stochastic component
is
G (n\.) and
Then, taking logs, we can rewrite the dynamic equations (16)
"convergence effects"
conditioning on n*
<r
equation that capital (and output) growth
this
this question, the
the conditional variance of the "total factor productivity" defined above.
Alog(K[+1 )
is
in this case is just a point. Therefore,
and take-off will occur almost surely, though
Let <j(n\.) be a random variable that takes the values
It
the
painfully slow for countries that are unfortunate.
Corollary
natural
exists
economy
suggests, given the specification of this
equilibrium stochastic process has a unique ergodic set
no underdevelopment trap
will reach full diversification
(observe that
We
want
to
forces have to be
(ii)
as
more
sectors
are opened idiosyncratic risks are better diversified. Corollary 2 establishes that at later stages the
variance
is
always decreasing in the capital stock. At early stages the result
some parameter configurations
may be
the first effect dominates
is
ambiguous; under
and early stages of the development process
associated with an increase in the variability of economic activity, whereas under
configurations, the variance
Corollary
2:
(a)
Vn =
(b)-(i) If
(b)-(ii) If
may be
Var{a(n
*
some
other
decreasing throughout.
,.)
\
n
*) =
n
*
(1
-n
*
)R 2
R(R-r_) ~i
2
R-rn
—
t<——
7>_R
dV
_^<0,
then
,
,then
3
v£,
.
K s.t. n*{K)=JL.<\
—.<0,
n
dK.
,and
2R-r
2R-r
VK,>K
'
;
whereas
—->0,
n
VK,<K.
8K.
Therefore, our model predicts that the conditional variance of the growth rate
uniformly decreasing with the size of the accumulated capital (case
shaped relation with respect to
productivity of risky projects
it
is
(case (b)).
much
More
precisely, if either
(a))
7
is
either
or exhibits an inverse U-
is
sufficiendy large, or the
higher than that of the safe asset, then the relation will be
17
monotonic.
Endogenous Growth
It is
this
straightforward to extend our analysis for endogenous growth and the
would be
growth
industrial
To do
growth.
we drew
that the distinction
will
this
now
more
fit
the final
y=AkXwhere K,
It
is
between primitive accumulation, take-off and
earlier
economy
nicely since at the last stage the
we can simply modify
main return from
good production function
a
K?-
will exhibit steady
(3) to
a
the aggregate stock of physical capital in this
(2D
economy.
suffices to note here that all our results carry through to this case
and the law of motion
of the capital stock takes the form;
"^—LIrYK,
Kr
prob.
<
RYK
can be seen that as
a constant and
4)
n,
reaches one, the growth rate increases until the
non-random growth
22)
prob. n*
t
It
IV
R-rn,
economy eventually reaches
path.
Pareto Optimal Allocations and Sources of Inefficiency
The
Socially Optimal Portfolio Decision
In this section
we
will explain
why
the equilibrium of section 3
when
characterize the optimal allocation and discuss
first, it
has to be noted that
we
this
type of inefficiency
is
not Pareto optimal,
may be
are only dealing with the issue of static efficiency.
important. Yet,
The
additional
source of inefficiency arising from the overlapping generations aspect will not be discussed in this
section,
though
we
will return to
it
later.
For the purpose of the analysis of static efficiency, a social
planner chooses the allocation of resources in order to maximize the welfare of the current
generation.
will also
The planner would
also solve the
program given by
(5),
with the only difference that
n,
be a choice variable and (10) will become an additional constraint. The constant saving rule
of the individual maximization will again apply due to the simple form of the preferences. Yet, since
it
no longer follows
that the
same amount
will
be invested in
18
all
open
sectors, (12)
changes
to
n,
max
\og{RF{ +r4>)dj+(\ -«,)log(r0,)
[
(23)
We now
characterize the solution to this problem.
Proposition
Let
2:
n FB {s)=n *(s) V s
>D. The
given
be
n'CsJ
by
n F\s)>n*{s) Vst
then
(15),
allocation of funds in the first best
is
<D
and
,
as follows
t
3j;<n FB(s )s.t.Fl=F
t
l
FIGURE
5
ifj<j;;F!=M(j)
n FB(s ,)>./,>//
if
ifj>n FB (s,).
and F/=0
;
HERE
Figure 5 gives the diagrammatic form of the first-best allocation (represented by the shaded
Note
area).
equilibrium.
qualitative properties
that the
The economy
steady growth but progress
is still
is
as a
faster
externality
become more
more
due
make
buy
(i.e.
exist.
assigns a price to each
demand
this
for security
externality
is
more of
j
A
full
commodity
at
time
t,
the risk
to
reach the
additional sector opens, all the
and as a
diversified)
However,
it is
important to note that
economy because
non-convex and as a result a
is
Arrow-Debreu
defined as a price mapping P* that
equal to zero, and for
all
the project level
full
P * > 0,
(project) in each time period such that for all
concept of equilibrium assigns a price level to
result, risk-
endogenously determined in equilibrium.
this is
Arrow-Debreu equilibrium
is
best can be seen
first
because the amount of undiversified
is
not internalized in our
edj(P*),
more
state.
As an
the existing securities.
the aggregate production set
Equilibrium does not
each
economy
to missing markets.
markets are not assumed to be missing, but
indivisibilities
dynamic
the transition equation looks considerably
attractive relative to the safe asset
willing to
The pecuniary
best are very similar to our
total return is different in
them are reduced
risks associated with
averse agents are
on average. Also
for the failure of the decentralized
form of pecuniary
existing projects
first
characterized by three stages; primitive accumulation, take-off and
complicated than (16), because the
The reason
of the
j
all P,
j
t
the excess
*=0, ed (P*)<0. Note
j
that
t
commodities, irrespective of whether they are
being traded or not. The non-existence problem can be explained using Figure 6. In the top part,
have the supply and demand of a security for a sector
supply
is
that has
no minimum
size requirement.
panel has the supply and
the
is
demand schedules
for a sector with positive
discontinuous because at no price
minimum
size. If as
and the market for
demand.
The
horizontal at $1 because competition forces prices to be equal to marginal cost (as in
standard competitive analysis) and the marginal cost of investing $1 in this sector
supply
we
drawn
in Figure 6,
supply x units of this security
demand
is
sufficiently
is
below
1,
excess supply. This
19
low there
supply
is
is
$1.
The bottom
size requirement.
we can
this security is missing. If price falls
If the price is larger than 1, there
minimum
is
is
when x
below
no equilibrium price
zero and there
the reason
is
The
why
is
excess
a decentralized
equilibrium exists only conditional on the number of sectors that are open and the auctioneer has to
look for a fixed point of a mapping that has
be seen
that,
because due
to the correlation
n, in it
is
however not a surprising
(It
can also
of the sectors, micro level indivisibilities are translated
into aggregate non-convexities, lotteries will not
This
as well as the prices of securities.
result, full
be useful and
in this equilibrium will not
Arrow-Debreu equilibrium
is
be used).
too strong a concept for
an economy with endogenously determined commodity space (proportion of open projects) and
is
not
normally used in these circumstances (see for instance Hart (1979), Makowski (1980), Allen and
Gale (1991)).
FIGURE
The
HERE
6
alternative equilibrium concept that
of a competitive situation; in particular,
that
keeps
all
offered instead captures
agents as price-takers and
the salient features
all
all
the gains
from trade
can be exploited via a decentralized trading procedure are exploited. The main distinction
the requirement that for
number of open
essentially the
Makowski
non-open sectors edj<0
sectors
same
as the
at P,j
=0
is
that
replaced with the condition that the
determined by a zero-profit condition
is
is
10
.
This equilibrium concept
is
one proposed by Makowski (1980) (also used by Allen and Gale (1991)).
defines a Walrasian Equilibrium as a feasible competitive allocation sustained by the set
of traded commodities.
the
it
we
added condition
A
Full Walrasian Equilibrium
that "no
firm sees that
it
(FWE)
can increase
assuming that the set of marketed commodities other than
can further be noted that the
spirit
(1976) and in the next subsection,
its
we
then a Walrasian Equilibrium with
profits by altering
own
its
of the equilibrium concept
is
is
will
its
trade decisions
remain the same"
(p. 228). It
also related to Rothschild and Stiglitz
will use this idea to refine the equilibrium concept to deal with
the issue of inter-firm share trading.
Inefficiencies
It
be related
that
Under Alternative Market
Structures and the "Reactive" Equilibrium
could suspected that the failure of the decentralized mechanism to achieve efficiency can
to
our assumption of no inter-firm share trading. In
under very weak and plausible assumptions, there
efficient allocation as
is
an equilibrium. And secondly, we
this subsection,
no market
will
show
we
structure that can support the
that with a simple refinement of
our equilibrium concept, even in the presence of inter-firm share trading,
characterized in Proposition
1
is
will first establish
the
equilibrium
the unique equilibrium.
10
The important point of this equilibrium concept we offered as compared to Arrow-Debreu is not that
there is no price for the non-traded securities but that this price is not the same as the price that an entrant
contemplates to receive if it enters. In our context, there are well-defined prices conditional on entry and an
entrant would enter in case it would make positive profits at those prices. There are also prices that would
induce the consumers to choose zero demands. However, because of the non-convexity, the marginal entrant
is never a "small" player and these two set of prices are not equal to each other.
20
note
First,
"superintermediary"
that
we allow
if
may emerge and buy up
absent, the superintermediary
it
firms
would be able
all
to
buy
of
shares
the
other
firms,
a
giant
of the industrial sector. If further competition were
consumers. In particular,
to offer non-linear prices to
could issue a bond which replicates the structure of returns required by the optimal allocation, and
this
would destroy our previous equilibrium
bond
to the previous equilibrium portfolio.
since agents prefer a portfolio consisting entirely of this
However,
no decentralized equilibrium can achieve the
not sufficient to lead to efficiency and
this is
Thus,
first-best (without intervention).
Corollary 3: Suppose intermediary firms can costlessly trade shares and there
intermediate sector.
Then
is
free-entry into the
the Pareto optimal allocation cannot be sustained as a decentralized
equilibrium.
The
shown
can be given as follows. In the Pareto optimal allocation,
intuition for this corollary
individual faces equal prices, he
would
like to
we know
But
in Figure 5, each individual needs to hold a "unbalanced" portfolio.
have a balanced portfolio. Thus,
it is
entrant to enter the no-minimum-size sectors, charge a price sufficiently close to
representative individual, hence
making positive
profits.
1
that if
an
possible for an
and
sell to the
Therefore with free-entry, the Pareto optimal
allocation cannot be sustained. Hence, despite the fact that our
model
is
quite similar to a
convex
model, there are serious inefficiencies associated with the interaction of the endogenous commodity
space and the
indivisibilities;
and these
We can now establish the
firm share trading
is
stronger result that even
be avoided by any market structure.
when
the "ad hoc" assumption of no inter-
relaxed, the equilibrium characterized in Proposition
equilibrium, provided that
reactive equilibrium,
inefficiencies cannot
we
1
remains the unique
refine the equilibrium, concept along the lines of Riley's idea of
suggested as a refinement of Rothschild and Stiglitz (1976)'s
originally
equilibrium concept. According to Riley (1979) to destroy a proposed equilibrium,
possible to introduce a contract that
more
contracts are added
concept in our model
buy up
which
can enjoy a
profitable and
by a new entrant a
clear:
on
the whole,
whole or a large part of the
the
it
is
(i) is
(ii)
posteriori.
it
intuitive appeal
seems implausible
profit. In fact, this equilibrium
must be
does not become unprofitable when yet
The
industrial sector, if
it
it
of
this
equilibrium
that a firm or intermediary will
knows
that there is
no equilibrium
at
concept can be more formally justified with
small costs of making contract offers. Also note that using the same argument of the proof of
corollary 3,
it
can be shown
that with
our original equilibrium concept and interfirm share-trading,
there exists no equilibrium and this suggests that as in the Rothschild-Stiglitz insurance market, the
strategy spaces of agents are too large and a tighter equilibrium concept
To
contracts
state
our idea more formally,
C when
consumers have access
let
7r(C|CUC)
denote the
to these contracts
21
is
required.
total profit
and an additional
made from
set
C
' .
Then;
a set of
Definition 2 (Reactive Equilibrium): Consider the vector
of open sectors
at
time
and
t
intermediaries. This vector
(i)
«/ (n,\C *)=0
all
J
t
t
open
the
is
t
{C,}) ta0
where
the set of contracts being offered to
an equilibrium
denotes the number
consumers by the
financial
iff
<n,* (given the set of contracts, supply of security
all j
n,*
j is
equal to
demand
in
(n,*) sectors);
C
3
(ii)
for
is
C
(n,*,
on
based
VC'MC"
a
|C*UC'U c// )>0,
feasible in the sense of
ed
j
t
production
feasible
tt(C'\C*\]C'{}C")>0
(n'„{C'UC'UC"})<0
such
plan
,
7r(C'|C*UC')>0
that
and
and that conditional on C",C remains
for all securities
j
offered
by
the set of contracts
CUCUC".
Proposition 3: Suppose intermediary firms can cosdessly trade in shares. Then, the equilibrium
characterized in Proposition
Since
that
1
is
the unique Reactive Equilibrium.
we have demonstrated
that there
a potentially important source of market failure and
is
no decentralized market mechanism can solve
policy
subsidies or regulation that
would support high
type of policy
is
in fact not too different
some
stages of development in
a large
required.
from
the pattern of industrial policy
heavy industries
Germany, with
at the
we
observe
at early
state intervention, there
was
expense of light industries, Gerschenkron
(1972)).
Towards Risk and Underdevelopment Traps
5) Attitudes
Our
Cameron
in
may be
indivisibility projects. It is interesting to note that this
countries (for instance in
amount of capital invested
(1962, p. 15), see also
problem, government policy
easy to characterize and can be seen to take the form of direct
The necessary government
is
this
results so far
have been derived using logarithmic preferences. This has proved
a very convenient functional form as
it
to
be
induces a constant savings rate. However, these preferences
are very special as they imply that intra-temporal preferences are also logarithmic and thus the rate
of relative risk-aversion
analysis and
it is
is
that
is
when
Most of our
more general
results will hold with
key role
preferences.
more general
agents have a rate of relative risk-aversion greater than
An economy
is
that receives a series of
low
that only a limited
averse consumers of
assets,
yet, risk-aversion plays a
We
this
non-ergodic. That
unlucky draws
number of the
economy decide
and the productivity of the
is,
may
1,
in
our
turn to
preferences, the only
Corollary 2 which proved the ergodicity the equilibrium stochastic process.
equilibrium (stochastic) process that
sufficiently
And
instructive to investigate the implications of
this issue in this section.
exception
constant and equal to one.
We
we may end-up
show
with an
underdevelopment traps are possible.
reach a level of capital stock that
is
risky sectors can be opened. In response, the risk-
to invest only a small proportion
capital stock
22
is
of their wealth in risky
endogenously low, and therefore growth
is
prevented.
To
illustrate these features,
we
consider the following preferences
logc.+piog^c^y-odj)™
where 0>O.
Note
(24)
von-Neumann-Morgenstern but of
that these preferences are not
the Kreps-
Porteus variety (see Kreps and Porteus (1978)). They are adopted so as to enable us to separate the
impact of attitudes towards risk from intertemporal substitution. The form
we have adopted
rate of intertemporal substitution equal to one, thus a constant savings rate
risk-aversion need no longer equal
It
among
can also be noted
that
1,
but
Lemma
1
is
11
still
gives
but the rate of relative
given by 6 which can take any positive value.
still
applies as
it
was derived only relying on competition
intermediaries and the form of the technology. After
some simple
algebra, the optimal
portfolio can be characterized as
(25)
max
subject to n
F +(f>=s
1
l
t
where
n, is
again taken as given by the representative consumer. Then;
i
m
*>=
(26)
r
(R-rn^+rnttl-ny)
1
i
i
[(r(l-n,))~M*-™,)" 7 ]",
Fr =
i=i
r
27>
:
{R-rnp' *rn {<i\-n)r)
<
?
l
Notice that
it
is
when
B-*\,
the solution
is
identical to that obtained in section 3. Yet, in general,
not possible to find a closed form expression analogous to (15) for the proportion of open
sectors. Also, in this case static multiple equilibria
concept, that
is
rate of risk aversion, 6, is greater than
we
when we use our
original equilibrium
all
sectors
4 shows
that this
can only occur when the relative
one and the economy has a
and more importantly,
numbers of open
sufficiently high
amount of
that the multiplicity of equilibria disappears
when
allow inter-firm share trading and use the reactive equilibrium.
11
Generalizing the preferences to allow for other forms of intertemporal substitution would also be
interesting, although
as
arise
there exist alternative equilibrium configurations with different
sectors for a given state of the system. Proposition
savings to open
may
more
somewhat cumbersome.
becomes possible.
If there are
diversification
23
precautionary savings, for instance, these will
fall
Proposition 4:
Suppose no inter-firm share trading
(a)
equilibrium of Definition
(i)
An
if
0>1,
D> j,>5
if
st
(iii)
If
6<
3
,
>s
and
,
>D
1
0>O.
all
such
that,
<D
then
,
is
n* ,n*
Suppose interfirm share-trading
-
denote the
static equilibria as
with part
start
convex when 6 >
open
sufficient to
(a)
1
n= 1.
But
sectors. Proposition
this is
cases, F(n) intersects
M(n)
intersect
economic
because
at
is
is
unique for
s
all
allowed, then for
all
>D
=1
t
;
.
values of 6,
of the sectors open where n,*=max
st
<D.
3.
Since
there
and
{n,*(i)}
F(n=l)=s and M(n=l)=D,
1
t
This, together with the fact that F(n)
when
the geometric intuition
no longer true for larger
levels of risk aversion,
when
M(n) from below
st
n=l. Since F(0)>0,
at
level of n,
then
is stricfly
the stock of savings
is
not
established that, under assumption (A), in the logarithmic
=D;
some lower
<n
n*
that
* *
=1.
rules out the possibility of multiple crossings
,
all
n,*
and go back to Figure
case the equilibrium was also unique
F(n) at
n*
open such
unique;
is
characterized in (a) above.
always vertically below M(n) when
is
sectors
and
exists
t
Let us
F(n)
*
equilibrium
satisfied, the
a unique Reactive equilibrium with a proportion
n,*(i)'s
equilibrium
the
then there exists a unique equilibrium with
and assumption (A)
1
st
if
then there exist two equilibria with
(b)
is
then;
,
equilibrium exists for
If
(ii)
1
allowed and use the concept of
is
is
that
M(n)
and we prove
static
steeper than
such
that in
F(n) must also
this implies that
and therefore, we have
is
The
multiple equilibria.
intuition is that if all agents invest a lot in the risky assets, all sectors
can be opened and,
risks are diversified, the representative agent wants to invest a high proportion of his
all
savings (in our case,
less in risky assets,
all
and
of
it)
in risky assets.
no
still
if
also
an equilibrium
at
which agents invest
can
assumption (A) did not hold, a similar multiplicity would also
apply to the logarithmic case of section
(b)
is
profit opportunities are left to individual entrants (unless they
coordinate their decisions). Note that
However, part
But there
3.
of Proposition 4 shows
of equilibria only has limited relevance because
that,
it
although interesting per se, this multiplicity
can be shown
that if
we
allow interfirm share
trading and apply the concept of Reactive Equilibrium as in Proposition 3, the multiplicity will
disappear and the economy will always choose the Pareto preferred equilibrium which
the highest
enter,
number of open
buy a
sectors.
Whenever we
sufficient proportion of sectors
our baseline analysis
is
the one with
are in a Pareto inferior equilibrium, a firm can
and offer a larger
that the function n*(Kt) will
is
portfolio.
have a discontinuity
The only
difference
from
at s t =D, so that take-off will
be more abrupt than before.
Despite the fact that the
of limited
interest,
we
will
underdevelopment traps, which
static multiplicity
now show
is
more
that
associated with a high degree of risk-aversion
there
interesting
is
is
an issue of "dynamic multiplicity" or
and robust. The stochastic process associated with
24
equilibrium will be unique but will posses
the law of
motion for the
more than one ergodic
capital stock. After
some
set.
To
analyze this issue, consider
algebra, this can be written as;
TK? = oB {n*(K),r,R,e)TK?
with
prob 1-n,
(27)
K,rl
TK" = aG(n *(K ),r,R,6)TK"
with
t
prob
n,
(R-rn)((l-n )ry+rn (R-rn)
t
where T
is
the
t
same constant which appeared
productivity of the capital stock conditional
logarithmic preferences,
Lemma 2:
it
The term
on good news and
it
aG{n *(K),R,r,6)
dn
>«)0
measures the
can be recalled that in the case of
turned out to be a constant. In this case
0>(<)1
Lemma
in (16).
we can
state;
.
2 implies that with high risk-aversion as the number of open sectors increases, the
On
productivity of the investment rises.
the other hand,
decreasing marginal product of capital, thus
parameter configurations, the net effect
increasing in the level of capital.
It is
is
we have two
the neoclassical technology
Under some
counteracting effects.
such that the marginal product of capital
then possible to find a region from where the
implies
is
locally
economy
not be able to escape; in other words, an underdevelopment trap: once the capital stock
is
will
low
enough, even in the presence of only good shocks, the economy will not grow beyond a certain
point.
Also note
that in
models where the aggregate technology
is
endogenous growth case discussed above, the marginal product of
increasing, thus such traps will be
Proposition 5: Suppose (19)
(a,A,P,y,R,r,D);
Expressed differently,
trivial set
if
capital will
be everywhere
more common.
is satisfied,
3[K\K~],
linear in capital, as in the
and
9>
K+<K"<K
SS
,
1.
Then, for a generic subset of parameter values
K E(K\K")
s.t.
t
the rate of relative risk aversion
is
=>
K ..e(K\K ++
t
)
,
V/>0
.
greater than 1, there exists a non-
of economies characterized by a non-ergodic equilibrium stochastic process. The technical
intuition of this proposition
conditional
can be obtained by noting
on receiving good news
locally increasing with K,,
stock for which
APK=
1
is
given as
that the
growth
rate of this
APK(K) = oG {n*{K),R,r,d)YK?~\
which can be when 6 >
1
,
we can have
(multiple quasi-steady-states conditional
25
If
economy
APK(K)
is
multiple values of the capital
on good news). The corresponding
dynamics are described by Figure
K
increasing with K. For
when
it
similar
receives only
t
in the right
good shocks,
argument using aB
,
At
7.
K ++
the average productivity of capital
neighborhood of
negative.
is
a result, [K ,K
possibility
++
will
]
growth
rate
K+
of the economy, even
different
is
words, an underdevelopment
from a
static
K ++ A
.
bad shocks,
below which the economy cannot
set or in other
is
fall.
trap.
As
This
and captures the
multiplicity,
development which were pointed out by David (1985)
in the context of
in different contexts.
FIGURE
It
be an ergodic
dynamic lock-in
and Arthur (1989)
the
the productivity of the capital stock in the presence of
of underdevelopment trap
possibility of
,
one and
is
Thus the economy cannot grow beyond
establishes the presence of a certain level of capital,
+
K
++
7
HERE
should also be emphasized that the nature of
this
underdevelopment trap
others in the development literature or elsewhere also for the underlying
Azariadis and Drazen (1990), Boldrin (1992),
Matsuyama
(1991)).
is
economic
different
from
intuition (e.g.
Underdevelopment traps are
usually driven by increasing returns. Yet, in our model, at earlier stages of development agents have
access to technologies which are as productive as those available to advanced countries. However,
the scarcity of capital leads to a lack of diversification opportunities,
making
the adoption of high
productivity technologies risky, and in response to this, agents choose an aggregate technology that
is
relatively less productive,
model micro
when
hence making the
state
of underdevelopment permanent. Thus, in our
level non-convexities only play a role in conjunction with risk aversion.
risk aversion
is
For
not high enough underdevelopment traps can be ruled out. This
following proposition (note that corollary
Proposition 6: Suppose (19)
is
satisfied,
1
above
and
is
0<1
is
this reason,
stated in the
a special case);
,
then pliml^ aa (K)=K ss
.
This proposition establishes that with low levels of risk-aversion, underdevelopment traps are
not possible, thus
6) International
We
makes
it
clear the role that risk-aversion plays in the existence of these traps.
Capital Flows and Development
have so
far dealt with a closed
economy and
abstracted
from the
possibility of capital
flows. This can be an important omission. For instance, in the process of industrialization of Western
Europe, capital flows were important. The areas that emerged as industrial centers were often
attached to (or were themselves) financial centers that attracted capital
and also subsequently provided
capital
to
from other regions and towns
them (Braudel (1979,1984), Neal (1990)). These
observations point out to the importance of analyzing the role of diversification and capital scarcity
in the context
of a model where inter-country capital flows are possible. Such an analysis will also
give us an indication of the importance of the mechanisms
26
we have developed
in this paper for the
development problems today
A
further consideration
Europe are not
is
more important
in the presence of relatively
dynamics of capital flows
that the
easily explained
by
international capital flows.
development process of Western
in the
Models based on a decreasing returns
the standard theories.
technology predict strong flows to poor countries whereas models with increasing returns imply the
opposite. In practice, the picture
more complicated. While
is
London
strong net flows from continental Europe towards
nineteenth century,
Encouragingly,
Before
this is the
we move
open economy and
problems, then
from
direction
the
all
same
(see
center),
were
in the
Neal (1990) chapter
11).
as the basic prediction of our analysis in the next subsection.
can flow
we can note
that if the country
we
are dealing with
is
a small
and out without any transaction costs or enforceability
in
our results disappear. This
capital shortages
more developed
(the
was reversed
of net flows
to the analysis,
capital
in the eighteenth century there
of course natural; our main mechanism
is
and the small open economy with perfect
capital mobility
seem
shortages away. This characterization of events does not however
experience of developing countries today.
Many
derived
is
assumes
capital
to accord well with the
of those try to attract foreign capital but often are
not very successful and also they and even frequently try to limit the extent to which domestic capital
is
allowed to go abroad. Further, as
we mentioned
in the introduction, capital shortages are often
mentioned as an important problem in the context of development. The findings of very imperfect
consumption insurance across countries
is
another indication that the presence of frictions to
international capital mobility. In the rest of this section,
free capital flows
a)
A
we
will analyze a two-country
world with
and then a small open economy with enforcement problems.
two-country model. Neoclassical technology
Suppose
That
is,
that there are
two countries with
identical technologies as described in section 3.
both countries have the same production function and access to the same set of investment
opportunities. Therefore if the state of nature
countries and also the
by equation
(l)
12
.
We
minimum
size requirement
assume
further
project
is j,
of
compete a
pays a rate of return equal to
la
R
in both
M(j) in both countries as given
this project is
that capital flows
as in the previous sections, intermediaries
j
between the two countries are
free. Also,
Bertrand in the financial market but the assets
sold by these intermediaries can be bought by consumers of both countries. Further, since the
do not care whether
investors naturally
the assets they are investing are domestic or foreign.
Let us denote the amount of risky investment in sector
of risky investment in sector
asset of country
12
It is
1
and
<f>2
j
of country 2 by
which case there
will
j
.
minimum
be additional gains from
from these
of country
Similarly, let
the investment level in the safe
of course possible to allow the
prefer to abstract
F2
j
<$>
x
capital flows than the
by F^ and
amount
These variables should
vary between the two countries in
ones emphasized. But in
in order to maintain the analysis as tractable as possible.
27
the
be the investment in the safe
asset of country 2.
size requirements to
1
this section
we
have time-subscripts but these are dropped
of
this section will deal
economy country
Lemma 3:
is
two countries
is
that with perfect capital mobility
indeterminate.
To
we had
A final
lives for capital,
will call the larger
same
economy
in the closed
price.
The
case.
prices of traded
determined by the technology because of the constant returns aspect. This implies that
characterize the individual consumer's decision in a simple form.
V (^
max f J'logUW, +RF{)
2
+
'
by choosing
{F, j },
{F2
j
}, <£,,
(f>
2
2
i
+/?F/)]# +
f
^"logUO^,) +p2 (RFi+r<t> 2 )]dj
logry, (/•<£,) +f/2 (r(t> 1 )]dj
f
and subject
to the constraint
^'F(dj + ^'^Fidj + ^ +
where pi
is
the marginal product of capital in country
open
n, sectors
in country
and
1
+
n!
is
taking the
countries as given and as in the closed economy, he
different states
as
parameters.
i
n2 sectors open
equi-probable. Note that the consumer
the
we
avoid these problems,
All risky assets that are traded in equilibrium have the
securities are
and one period
most
2.
This result mirrors the one
we can
problem of fund allocation between the two countries.
static
we move on
point to note before
the identity of the
with the
in this section not to complicate the notation since
2
=S
in state
j.
(29)
We
also adopt the convention that
in country
2 and as before
all
number of
sectors that are
open
the states are
in the
two
treating the marginal products of capital in
is
although
Further,
<t>
it
not important for the consumer's
is
maximization problem which sectors will be open, as in the closed economy, sectors with smaller
minimum
size requirements will
We
Lemma
can
now
4:
(0
(ii)
The
V
V
(jj')
1
(jj
s.t.
) s.t.
j<j'
<n
=»
same
F( =F{'
=»
x
is
,
F{ =
this
problem.
problem more
closely;
open
in both
Fi = F^
consumers
will
problem as
subject to
28
j
and
j'
are
never be happy to purchase different
two sectors are only open
4 we can then write
this into the
Fi'
straightforward. If two sectors
price, then
assets. Similarly, if
Lemma
and we have already incorporated
n <j<j'
intuition of these results
amounts of these
first
characterize the solution to the individual maximization
countries and they trade at the
hold. Using
open
in
one country, the same
will again
max« log[Pl (l)(r</. 1+ /fF
Fv F2 ,G,<t> v 2
1
1
)
+ p (l)(r^ 2+ /?F )] + « log[p (2)(r^
2
2
2
1
I
)
+ p (2)(r<A +i?G)]
2
2
(30)
<t>
+ (l-/i -« )log[p (3)((^
2
1
1
n,(F,
where p
{
denotes
n,
< j <n, +n2
+nF2 ) +n 2 G+(j>
marginal product in country
(q) denotes the
q=3
and
denotes
p.(l)= aA(r4> i +RF)''-
1
j
> n, +n2
p (2)=p
Lemma
5:
(i)
The
capital
capital,
KF +
1
first
= RF2 +
r<£,
part of the
would be higher
could be increased.
in
An
we can
lemma
.
)
+ p (3)(r^» ))]
2
2
=s
+<t> 2
l
(3D
q where
in state
i
q=l
(3)=aA(rcj>
i
this
r
p 2 (2)=c^(r^ 2+ i?G)-
i
(32)
1
maximization problem and substituting for the
establish;
r<£ 2
(ii)
.
F2 >
F[
and
follows directly as
>
<£,
if it
<j>
2
.
(iii)
G > F^
were not true the marginal product of
F2
one country and by changing, F, and
informal intuition for second part of the
,
output in the
total
lemma
is
all states j
that since
G
country
1
is
due
than in the states
j
to the fact that in the states
<n, and
The general form of equilibrium
FIGURE
Lemma
6:
(i)
If n,
8
+n2 -*
j
for insurance reasons,
that is implied
by
this
£
(n l7 n,+n2 ), there
it is
Lemma
is
also
shown
this
lemma
is
that
>
<£ 2 .
G
above
F,.
in Figure 8.
HERE
1
,
then n[ -*
1
.
(ii)
If n,
<
1
,
then n2 > 0.
The
intuition
full diversification, the effect that
leads to
In words, one of the countries cannot reach full-diversification before the other.
of
<£,
lower return in
beneficial to increase
is
< n,
received
is
in country 2, the insurance role of safe investments in this country is less important, thus
Finally, the last part
q=2
denotes j<n,,
Therefore
l
l
Next, using the first-order conditions of
marginal product of
1
when one of
the countries
is
near
concentration becomes small relative to the difference between the marginal products of capital in
the
two countries and
the additional value of a dollar
sectors open. Therefore, consumers will always
is
want
relatively
to invest
more
more
a result, the two countries will reach diversification at exactly the
Now
combining
all
these
lemmas we can summarize
29
the
in the country that has fewer
in the smaller country
same
main
and as
time.
findings of this section as;
Proposition 7: The equilibrium of the two-country model always takes the following form;
<
If s
2D;
(i)
(ii)
One of the
No
(iii)
If s
>
countries always attracts
country opens
As
s
all
projects; n,
more of the
+n2 <
all their
projects
and open more sectors; n2 > 0.
1.
approaches 2D, both countries reach
2D, both countries open
capital
simultaneously.
full diversification
and each project receives the same amount of
capital.
Our
taking the
analysis so far has characterized the equilibrium of the static
amount of savings,
s
t
To
as given.
,
problem of fund
obtain the full dynamic equilibrium,
allocation,
we need
to
determine the amount of savings and also provide the transition equations as in the closed economy
case.
Because of the logarithmic preferences, the aggregate amount of savings
will
be a fixed
still
proportion of wage income and the dynamic transition equations of our economy can simply be
obtained by determining wage
income.
However,
overlapping generation structure leads to some
from a point of
capital scarcity
Proposition 7 (or
Lemma 6)
extreme simplicity of the two-period
this
artificial results.
Consider the world economy starting
and equal wealth in the two countries. Then, the
implies that
more
capital will
go
to
one of the countries and thus we can
claim that capital flows create forces that work in the direction of divergence. However,
j
<n, occurs, because
and the workers
left in
will
r^, -l-RF, =r<£ 2 +RF2> the
two countries
will
Then
in the subsequent period, again
of the countries, thus the capital stocks will not be equalized, but
all states j
as those in the poorer one.
realistic direction will
allowed here,
this
avoid
problem
<n
{
the
,
young
have exactiy the same output
this
problem. In particular,
will not arise.
is
more funds
it is still
if
the
model
there are
will
be invested
level
one
an unattractive feature of
that take
it
more intertemporal
motives for our agents.
level
towards a more
linkages than
Here we sketch a simple way of allowing for
to introduce bequest
in
have the same income
in the richer country will
However, many extensions of
complicating our analysis, which
utility
a state
if
have exactly the same income, hence there will not be any income inequality
the following period.
our model that for
part of
first
More
without
this
precisely, the
function (2) can be changed to
£/(c,,c,
where b t+1
is
+1
M
,b) =log(c,) +/3£[log(c
the bequest that generation
t
)
M
+/dog(fc
)]
(33)
agents leave to their off-springs and
fi is
a parameter that
captures the strength of this impure altruism. In this case, the saving decision changes to
(34)
i+/3(W)
but since
it is still
analysis applies.
constant and independent of the rate of return expected between
And,
if
countries have different income levels at time
30
t,
even a
t
and
state j
t-t- 1
<n
lt
,
all
our
does not
young of
lead to equalization of incomes, since the
the richer country will have received
more
bequests than those of their poorer neighbor.
With
formulation, the general dynamics of the world
this
obtain from Proposition 7. First of
enough
there does not exist
all, if
countries to reach full diversification, neither country will have
news
all
will affect both countries, but, in general, not symmetrically.
us consider a situation where both countries have
initial
economy
capital in the
of
Thus bad
another observation
that are small
close to each other. In this case, the possibility of foreign capital flows
world for both
sectors open.
its
To make
endowments
are straightforward to
and also quite
leads to divergence
first
across countries; one of the countries (in our case, country 2), attracts capital from country
However, as
n2 becomes
the
world economy receives a
arbitrarily small,
historical evidence
at the early stages
while enhancing
it
from
good shocks, n,+n2 and
series of
and the two countries
the industrialization of
start
And
in the other.
n,
both approach
1.
1;
converging. Therefore, in line with the
Western Europe, our two-country model predicts
of development, capital flows slow
let
down development
at later stages, the relatively
that
one of the countries
in
backward country receives
capital
imports from the advanced economy, consequendy capital flows contribute to fast (but of course, not
immediate) convergence.
(b)
A
two-country model. Constant returns to capital.
Romer
In the presence of endogenous growth a la
at the
end of section
3), the rate
of return on capital
the productivity of the remaining capital.
of the world will go to country
effects that arise in this type of
the
poor country flowing
2.
This
is
(i.e.
this is straightforward: all the capital
obviously unrealistic but
is
1
and the young of
this
also instructive of a set of
to capital, all the capital of
beneficial for the current old (rentier) generation of
both countries and the young of country 2, but as a result of
country
is
economy. In the case of constant returns
to the rich country
model we analyzed
constant and capital flows will not influence
The implication of
is
the version of the
this, there is
no
capital invested in
country are impoverished. Therefore, although capital flows are
welfare improving from the viewpoint of the current generation, they lead to important intertemporal
inefficiencies
and as a
who
result, future generations
very high price. In our analysis so
efficiency, nevertheless this case
far,
shows
we have
are
assumed
not concentrated
that these considerations
to
remain
on
in this country
pay a
the intertemporal aspects of
become very important
in the
presence of open economies and from the viewpoint of intertemporal efficiency, capital flows can
be quite harmful.
(c)
Small Open Economy with Enforcement Problems.
Now
briefly consider the case of a small
open economy. In the absence of transaction costs
associated with foreign capital flows and enforceability problems,
31
it is
obvious that
all
the sectors
immediately open. However,
will
it
was argued above
that this is not
an interesting case and does
not accord well with the existing observations concerning the experience of developing countries.
Instead, suppose that there are enforceability problems. In particular,
considering
To
sovereign,
is
if
has the right to repudiate on the debt and often
it
the
it
country that
will
we
are
choose to do so 13
.
give the main ideas of this case, simply assume that the foreign investment that comes into a
country will not be paid back. Naturally, in
small open
put their
economy
money
of interest because of the capital outflows. Domestic investors can always
is still
and get a fixed
in foreign assets
interesting implication
is
be no capital inflows. However, the
this case, there will
rate of return (since the country
is
small).
of investments abroad increases 0, and
that as the productivity
(normally) dimmish, therefore, the rate of industrialization will
fall.
This
is
F
t
An
will
the static effect of capital
flows out of the country which improves the welfare of the current generation. However, the
dynamic
fall
effects are
and there
is
constraints are
more harmful
even
still
to development.
less capital to
limiting the
As
capital flows out of the country, the
wages
be invested in the following period. Assuming that enforcement
amount of foreign
capital that
can come
in, this
process will quickly
take the wages in the country to a very low level and future generations will be impoverished and
industrialization will be prevented.
How
enforcement constraints
that
even
are.
As
effects
when
strict the
in the other case of harmful capital flows, this extension points out
in the presence of capital scarcities, the possibility of international capital
have adverse
7)
on how
serious the consequences of this process will be depends obviously
movements may
other market failures are also present.
Conclusion
Indivisibilities
of development.
We
and non-convexities are often recognized as important factors
in the process
argue that the interactions between diversification and indivisible projects will
crucially influence the development experience of a country. In the presence of non-convexities, an
economy with
limited resources will not be able to invest simultaneously in
extent that these sectors have imperfectly correlated returns,
it
will not
many
and
to the
to benefit
from
sectors,
be able
diversification. This lack of diversification will often bias the portfolio towards safe but
projects and will therefore slow
it
will
will
the process of accumulation.
However,
as the
economy grows
have more resources, because more projects are undertaken, the composition of investment
change towards
riskier
and more productive projects. This
to a take-off stage. Take-off
13
down
low return
will eventually bring the
and the associated financial deepening
economy
will finally result in steady
Because of the two-period overlapping generations aspect, the old generation will always want to
repudiate completely on the debt.
It is
possible to incorporate these type of enforcement constraints in models
with more forward looking behavior and obtain
strictly positive
developing countries (e.g. Bulow and Rogoff (1989)).
32
but limited foreign direct investment into
industrial growth. In this process
random events
are very important. Since at the early stages of
development, the economy only invests in a few sectors,
We
long in the primitive accumulation stage.
risk-averse,
sufficiently
"underdevelopment";
at
a
show
draws
bad
of
series
also
can easily be unsuccessful and spend too
it
that
condemn
can
Arrow-Debreu equilibrium
fails to exist
form of too few
and
Due
required government policy
to
permanent
is
show
and price-taking, there
as a result the
that this source
economy on average
of inefficiency
is
robust across
it.
The form of
in the
development
necessary to deal with
is
subsidize large projects as
to
to sectoral non-convexities, the
we observe
many economies.
Although we argue
will
We
open and
market structures and government intervention
experience of
society
in all equilibria with competition
sectors being
spends too long in primitive accumulation.
different
a
that the investigation
of the interactions between risk and
indivisibilities
be important in understanding the process of development, a number of issues require further
thought and research. First, our model allows perfect risk-pooling and yet, indivisibilities
limit the access of individual agents to financial instruments
agents;
it
will
be interesting
income distribution
at the
to investigate
how
may
also
and may prevent risk-pooling across
with the evolution of diversification and
this interacts
aggregate level. Second, our model does not grant a role to financial
intermediation, and implicidy assumes that the process of intermediation
all
are
from primitive accumulation.
In our model, an interesting inefficiency also arises.
inefficiency in the
economy
agents in this
a certain level of the capital stock, the economy will never invest enough
in risky projects to take-off
is
when
costless
is
and
efficient at
stages of development. In the light of the findings of the recent literature, one could try to
consider explicidy the role of credit and stock markets (a step that
Third, though
we show
mechanism, one would
suffer
from
in section
we
are taking in on-going work).
6 that international capital flows do not necessarily undo our
like to assess empirically to
capital shortages induced
by undiversified
what extent underdeveloped countries today
A
risks.
related issue
is
to investigate
part of the high variability of growth rates of poor countries can be reduced
by
whether
diversification.
Finally, the
most pressing and probably
explicitly in
such a model and attempt to determine the size of an accumulation unit together with
fruitful
extension
is
the degree of indivisibilities that will influence such a unit.
reach a
much
to incorporate
Such an extension would enable us
better assessment of the empirical importance of this
current development problems.
33
enforcement problems
mechanism
to
in the context of
APPENDIX A
Proof of
P{>
F
All
would lead
=Fi'
,
VjJ'EJit)
Proof of Proposition
PJ-P^'-l
P{<
to further entry; if
of nature j are equally likely and
states
J
t
1: Free-entry implies that in all equilibria
the positive profits
1
a loss).
thus
Lemma
all
(if
then the firm in question makes
1
securities
have the same price,
QED
.
Consider the allocation characterized by (n=n*
1:
v jj' EJ(t)
,
,
{^=1} 0S
-
Sni
where
-)
n,*
is given by (15), and asset holdings are given by (13) and (14). Conditional on n,*, both consumers
and incumbent firms are optimizing. Thus we only need to check whether all opportunities for
profitable entry are exploited. First, observe that the entry cannot be profitable unless
where v indexes the
Next note
potential entrant.
F'(0=
r(R ~ r)
>0
assumption
F'(n,')<DI{\ -7),
(A),
V/i,*
is
have
r(R ~ r)
F"(Q=-2
,
when an
in with
(Al)
(R-rn,')
F'(l)= r/(R-r)
<
which
D/(l-y)
is
implies
always less than the slope
additional sector opens, the increase in the investment in the
not sufficient to cover the
no additional firm can come
<0
3
Geometrically, the slope of F(.) in Figure 3
.
of the M(.) function. Thus
risky asset
we
>1,
that
(R-rn;)2
Under
V
P,
minimum size requirement of an additional sector. Therefore,
Pv (.)> 1. This proves that the characterized allocation is an
equilibrium.
For uniqueness note
that all equilibria
have
to
be characterized by the intersection of F(n) and
M(n) or by
(R+ri)±
(R+ry)2 -4r
(*-r)(l-7)
(A2)
V7/g
2r
when s >D, n,* = l and s, is given by equations (4) and (11) in the text.
Denote the solutions to (A2) by n, and n2 with n,>n2 When assumption (A) is satisfied,
n, > 1 for all values of s < D. But in this case n= 1 cannot be an equilibrium since all sectors cannot
be opened. Thus when s<D, there is a unique equilibrium given by n,* as in (15) given by the
as long as n,*G[0,l]. Also
t
.
when assumption (A)
s>D. Thus when s>D, there
smaller root of (A2). Next note that
is satisfied,
than or equal to
is
1,
for
all
This establishes uniqueness.
Proof of Corollary
1:
the smaller root
n2
is
greater
only a unique equilibrium with n*=l.
QED
The law of motion (16)
implies that for
34
< K < K ss
t
=>
K >K
tfl
t
conditional
T
on favorable
Also, (16) implies that for any Ko there exists a sequence of good
realizations such that K, reaches K* in finite time, where K* is such that n*(K*) = 1 Once this capital
realizations.
.
Furthermore,
K
economy converges
(16) and (20) ensure that the
level is reached,
ss
is
deterministically
K ss
to
.
the only absorbing set of the system. Since any finite sequence of "good"
realizations occur with positive probability, then convergence to
Kss
is
guaranteed almost surely.
QED
Proof of Corollary
of a(n(K),r,R)
The
2:
and from
Sign{
calculation of
(16).
Vn
follows
straightforwardly
the
definition
Then:
—dV
n
*
-}=Sign{(l-n *){R-rn *)-n*{R-rn •)+2r(l-/i
*
dn
from
*)/i
(A
y
^
'
=Sign{R-2n*R+rn*}
Thus
if
p
n*>
,
the variance
is
decreasing in n*(K).
We
also
know
that
n*>y;
therefore,
if
lR—t
p
y>
,
then the variance
always decreasing in
is
n*.
Otherwise,
it
will
be non-monotonic
zR—
(inverse U-shaped, with a
maximum
monotonic function of the
capital stock
Proof of Proposition
2:
By
at
n
*
p
=lR—t
K, the
).
Since n* as determined by equation (15)
rest of the
proof follows.
substituting constraint (9) into the utility of the individual, let us define
(A4)
J
Evaluating V(.) at the decentralized equilibrium
</>,
from
a
QED.
V{nt ,{Fi}A)=[\og{RFt ^)dj^\-n)\oz(r<l>)
for
is
we
the budget constraint,
- i.e.
at
n,=n*, F,j =F * for
t
all j
<n, and
substituting
obtain
'^••^ Q
(AS)
*,-£„,•
Fi
and
dV(n,',Ft ')
dn
,
=log(/fF,* + r(5/ -/i/ F,*))-log(r(5,-n r *F,*))
(A6)
F *(rfr,-/i,*F,>fiF,'(l-ii,')) ^ n
A
t
(RF; + rist -n;F;))(srn;F;)
Since
we can
reduce
F
j
t
decentralized equilibrium
for
is
some
intra-marginal sector and increase
not Pare to Optimal.
35
t^,
V(.) can
go up, thus
the
To
characterize the first-best, construct the Lagrangean
B
L(n ,{Fi})=
t
f
log(^/ +
J°
and differentiate with respect
F
to
we
j
t
,
rW (l-",)logK)
+
obtain the following
R
first
(A7)
order conditions
-\V,=0
(A8)
RFt+r<f> t
Now
do not have a binding minimum size requirement, i.e. F >M(j), the
corresponding Lagrangean multiplier is zero, thus F, =F, for all such j. It is straightforward to see
that all sectors with M(j)=0 fall in this category since, by its concavity, V cannot be maximized
when no investment is put in these sectors. But for the rest of the sectors (i.e. those with a positive
multiplier), F =M(j) must hold. Thus the form in Proposition 2 follows. We only need to show that
j* is strictly smaller than n,™. Were this not the case, an equal amount would be invested in all
sectors and we would get the decentralized equilibrium which is suboptimal as shown above. QED
for all sectors
j
that
j
t
j
j
t
Proof of Corollary
Suppose there
3:
exists a
Pareto optimal allocation. Take a sector
Proposition 2, F"
among
<
P (where P
that
market structure with free-entry that implements
does not have a
minimum
size requirement.
the optimal holding of the asset with the largest
is
those in which the social planner would invest). This implies that the
representative agent
facing for the marginal unit of
is
n and
facing for the marginal unit of
MP >
j
j,
MP
j
,
is strictly
the seller of these securities has to
-the
Then, by
minimum
shadow price
size
that the
greater than the price he
make non-negative
is
profits,
But then because there is no minimum size requirement for j, an additional firm can come
but still above 1. If this price is sufficiently near 1, the representative
in, set a price below
consumer will want to buy further units of this security, thus the firm will make positive profits.
j
1
.
MP
QED
We need to show (i) the equilibrium of Proposition 1 is a reactive
no other allocation that can be supported as a reactive equilibrium. For this
the maximum price that a consumer would be willing to pay for a
us also denote by
Proof of Proposition
equilibrium;
proof, let
(ii)
there
3:
is
MP
marginal unit of security
(i)
Let
need
C
to
ir(C
|
be the
show
set of contracts that maintains the allocation of Proposition
U C" ) < 0.
by a balanced portfolio
some
prices
that disturbs
But by
that has
C,
3
C"
such
1,
denote the set of these
Now we
1,
\
1
C
as an equilibrium.
C
\J
definition, the equilibrium of proposition
MP=MP Vj.
must be greater than
ir(C"
that,
1
U C" )>0
We
and
cannot be disturbed
Then consider an unbalanced portfolio to disturb C,
thus some of the prices must be
be attracted to
to make positive profit,
securities by the set J. Then for
C
|CUC')>0, some consumers must
but for ir(C
lower than
C
that for every
C
CU
j.
thus Vj' such that
MP' >
1,
C
denote the set of these securities by
C
can be bought
two possibilities; (a) different securities in
(other hybrid cases can be handled
separately; (b) there is only one combined security in
similarly). Also denote by nc the number of sectors open with the set of contracts C (i.e. the number
of open sectors in the equilibrium of Proposition 1) and nc- the set of sectors open to support the set
J'.
have
to allow for
C
36
C.
of contracts
C
For
to disturb
no balanced
greater than r^ (since
C
and make positive profit we know that i^. must be strictly
C and if i^. <nc, the consumers will hold
portfolio can disturb
a balance portfolio).
C"
such that in
all
j'£J\
implies that, the firm offering the set
C
will not sell
(a)
some of
Consider
profitability
to
is
sold at the price 1+e. This
any of the securities in J' and since at least
below 1, it will make a loss.
constitutes
have a price P and by definition of the
the other securities in J are sold at a price
(b)
want
a unit of security
Let the combined security that
of C
have
' ,
P>
P'>P
1
.
Then, consider
for all
C
such that P' =
C"
1
+e < P for
all j
'
6 J'
.
The consumer
will
C offers more of security
purchase of the combined security
C a level no
j6 J and j'GJ',
whereas by construction
J
Thus consumers will reduce their
in
to
greater than F(ncO. But by definition of the equilibrium of Proposition 1, we know that F(nc) is not
sufficient to cover the minimum size requirements of sectors n^-no therefore the supply of these
securities is equal to zero whereas the demand is strictly positive, thus
violates feasibility and
than
J'.
C
is
C
not a valid disturbance to C.
(ii)
Now
consider an allocation supported by a set of contracts that
is
different
from
the equilibrium
By definition, MP is not constant (otherwise we end up with the allocation of
which in this case is just a
Then
at least for some j, MP> 1. Then, there exists
Proposition 1).
Basic Arrow Security j that will sell at 1 +e and for small enough e, it would make positive profits.
remains feasible and does not make negative
Now we show that for all profitable reactions C",
consists of Basic Arrow Securities, if C" undercuts C, then no units of
profits. First, since
are sold and hence, it makes zero profits. Next, since C" is undercutting C\ more units of Basic
Arrow Security j mst be demanded, therefore feasibility will not be violated. Thus, for all other
of Proposition
1
1
.
C
C
C
allocations other than that of the Proposition
C
1,
a valid disturbance
37
C can be found. QED
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39
Figure
1
7
8
-2.0
-2.5-
-3.0
3
-3.5-1
I
-4.0
-4.56
9
10
Log(GDP 1960)
YOUNG
^^ofufwete^
Rent capital
to final sector firms
Capital
Income
(pt+1 tt+1)
Figure 2
Ct+1
—
F(n,s)
M(j)
n*
1
j,n
Figure 3
n=l
Bad draws
>
Gooddraws
>
>
k
!>
—>
> >
3
0388
Figure 4
—»—^-^s
<
< <
n*
1
Figure 5
p
Sector with no
minimum
D
S
Figure 6
size
—
—
MIT LIBRARIES
3 9080 00927 9594
n=l
Bad draws
—<—<
> > > >
^-£
<
<
m
— — —^ —
>
>
>
+
k
Good draws
>
>
rv
—•»—
<
++ < <
k
>
> >
;
—<-<
<
_ss
£
Figure 7
G
F2
LARGER COUNTRY (2)
n1
1
n1+n2,
SMALLER COUNTRY
(1)
f
7966 00
\
Date Due
!
T
ih-9fi-K7