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7/26/2019 Optimum GRowth With Scale Economies, Review of Economic Letters
1/17
Review of Economic Studies, Ltd. and Oxford University Press are collaborating with JSTOR to digitize, preserve andextend access to The Review of Economic Studies.
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Optimal Growth with Scale Economies in the Creation of Overhead CapitalAuthor(s): M. L. WeitzmanSource: The Review of Economic Studies, Vol. 37, No. 4 (Oct., 1970), pp. 555-570Published by: Oxford University PressStable URL: http://www.jstor.org/stable/2296485Accessed: 29-12-2015 07:34 UTC
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2/17
Opt imal
Grow th
w i t
c a l e
conomies
i n t h
Crea t i on o
Ove r h e ad
Capital
1. SUMMARY
Following closely the approach
to optimal economic growth taken
in the work of Frank
Ramsey [11], a highly simplified two-sector model is presented in which the " overhead
capital
"
sector
exhibits increasing returns to scale. Basic
properties of
the optimal growth
path are discussed.
From an economic standpoint, the
model might be relevant
in
bearing
on some basic issues of development
programming. Mathematically, this
kind of a model
has an interesting
structurebecause it is a combination
of convex and concave
sub-problems.
2. INTRODUCTION
In the
context
of development economics it is useful
to distinguish two types
of
capital
according
to how round-about a
role each plays in producing output.
One type,
the
quantity of which is denoted K, is the ordinary directly productive quick-yielding capital
which, when it
is combined with labour, creates output
according to classical laws
of
pro-
duction. A second kind of capital,
Kl, is the indirectly
productive infrastructure
which
lays
down
the basic
framework within
which directly productive economic
activities
can func-
tion. Capital
of this variety has come in for increased
scrutiny by development economists.
At
least
in
part
this is due to the growingsuspicion
that capital, comprising
those essential
services
without which ordinary production cannot operate,
plays an especially important
role
in
the early
stages of economic
growth.
For
the purposes
of this paper the total capital stock
of
the
economy
is
thought
of
as
being partitioned
between two
sectors-K.
belonging to
the
ac
ector and
Kl
to the sector.
This being
the case, it becomes
a
fair question
to
ask
for operational
criteria
which can be
used
to
distinguish
ac rom capital. Unfortunately
it is difficult to
be
precise
about this
issue. For one thing it depends upon
how aggregative
a
view one
is
prepared
to
take.
Considering
an entire economy
on
the
most
general level, ,Bmight
consist
of all social
overhead capital including public
service facilities for education,
scientific
research,
sanita-
tion
engineering,
public health,
and
law enforcement,
agricultural
overhead
such as
drainage
and irrigation systems, and
hard
public
utilities like
transportation, communications, power
and
water
supply
installations.
A
somewhat
more satisfactory interpretation might
limit
/
to the hard public utilities. There
is even an interesting way
of looking
at
this
model
which
restricts the economic scenario
to
manufacturing
and
treats as structures,
cx
as
producers'
durable equipment.
For the purposes of this paper probably the most useful formulation is the middle one
which treats
as overhead capital for producers' services.
In any case,
the basic features
are taken to be
the
following.
1
For
their helpful comments
I
would like
to thank D.
Cass,
T. C.
Koopmans,
and
A. S. Manne.
The researchdescribed
n this paper was carried out under grants
from the National Science
Foundation
and from
the
Ford
Foundation.
555
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3/17
556
REVIEW OF ECONOMIC STUDIES
(i) Capital
of
thef
type is strongly complementary
with
cx.
Investment
in cx
apital will
be productive
only
if
it
has
been preceded
by
sufficient
investment
in
,Bcapital.
(ii) The ,Bsector is
highly capital intensive
and
usually
consists
primarily
of structures
and
installations.
It
is
typically
characterized
by
a
significantly higher capital-labour
ratio
than the cx ector.
(iii)
There are substantial economies of scale
in
creating capacity.
The main reason
is that due to
indivisibilities there
is obvious cost
lumpiness
involved
in
creating
a trans-
portation, communications, or
power
and water
supply
system
as
a whole.
Geometric-
engineering
considerations
are
also
important
in
the
case
of
many
structures
because
the
cost of
an
item is
frequently
related
to
its surface area while
the
capacity
increases
according
to its
volume.1
(iv)
Both ,B
and cx
apitals are specific to the
role
for which
they
have
been
created and
cannot be
shifted.
3. THE BASIC MODEL
The
highly stylized
economy
under consideration is centralized and closed.
A
single
homogeneous
output, denoted Y, is produced which is
perfectly general
before it has been
committed,
and can
be
used for
any purpose.
The
planners
seek
to maximize
welfare
by
appropriately manipulating
the
available
instruments-in this case the destination
of
final
output.
For
simplification
the
following
are
assumed:
stationary
labour
force
and
popu-
lation,
tastes
independent
of
time,
constant
technology,
no
capital
deterioration.
As
an
abstraction of
proposition (ii),
it
is postulated that
,Bcapital
has
only negligible
manpower
requirements.
This
makes the
labour
allocation
problem
trivial
because
all
available workers
will
be
assigned
to work with
cx
apital.
If capital were abundant at time
t,
Y(t) would depend only on the stock of Ka(t)and
the labour force.
Since the latter is treated
as constant, the production
function
in
this
case
could
be written as
Y(t)
=
F(Ka(t)).
Decreasing
returns
to a
single
factor
implies
that
F(K.)
is concave.
Purely
for
convenience,
we assume
that a
first
derivative exists and that
F'(K,)
>0 for all
K.
>
0
With
K,(t)
plentiful,
the
production
function would
simply
be
Y(t)
=
Kp(t).
Note
the
implied asymmetry
in
capital measurement;
K.
is
gauged by
the usual criterion of
real
production
cost,
whereas it
will
prove
useful to
quantifyK,
in
capacity
units. Of course
strict identification of K, with " capacity " would be possible only if there were a negligible
elasticity
of
substitution
between
K,
and
F(K.),
a condition which we
readily
assume
following (i).
In the general case,
Y(t)
= min
{F(K,(t)),
Kp(t)}.
With
cx
apital
all
investment
goes
into
capital
formation
in
the usual direct
form
Ka
=
ial
where
I.
denotes investment in cx
apital.2
However, with capital
there is a
meaningful distinction between
capital accumulation
and investment. Because of the presumed increasing returns to scale described in (iii) it
will
typically
be
better not to invest directly
in capital. Rather it will
pay
to
first
accumu-
late what
could be thought of as either a
generalized
inventory of materials or as
projects
1
In addition
the usual
internaleconomiesof specializationand
informationhandlingmay be
present.
Note
that the increasingreturns
relates only to
the design stage when the amount of
installedcapacityis
treated as
variable. Ex post the
size of an installation s considered o
be fixed and
unalterable.
2
A
dot
over
a
variable
denotesdifferentiation
with respect
to
time. Variables
may
not
be
explicitly
specified
as
functions
of time
if
this interpretation
s otherwiseclear.
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4/17
OPTIMAL GROWTH WITH SCALE ECONOMIES 557
in progress, denoted X. Only after a while should some generalized inventory be trans-
formed into new K; available for operation.'
Let AX represent a portion of generalized inventory earmarked for conversion into
operating ,Bcapital. Naturally 0 _ AX
0. It is assumed that lim G(AX)
=
so, G(O)
=
0,
Ax
Xoo
and
lim
G(AX)
=
0.
...
(1)
AX-O+
AX
Something
like the latter condition is
necessary
to
ensure
that
economies of
scale are taken
advantage of and that in fact generalized nventories must be accumulated for this purpose.2
We
will also
find it useful to work with the
function
H, defined as the inverse of G.
H
can be interpreted
as an
investment cost function relating the cost in cumulated output
units of
a
given capital
increase
according
to the schedule
AX
=
H(AKp)
=
G-1(AKp).
Displaying decreasing
unit
costs,
the
continuous, monotonically increasing,
concave cost
function
H is defined for all
AKp
>
0
and
possesses the properties lim
H(AK.)
=
co,
AK-co
H(AK
~ ~ ~ ~
K
H(O)
=
0,
and
lim
H(AK)
=
AKP-O+
AKp
Before
turning
to the
main
problem,
we
digress
in
the next
two
sections
to
consider
a
pair
of
related
problems
whose solution will
prove
useful in
characterizing
an
optimal path
for the
general
case.
4. OPTIMAL GROWTH
IN
A
MACROECONOMIC MODEL
The
social
utility
of
consuming
amount
C(t)
at
time
t
is
taken to be
U(C(t)).
The
instantaneous utility
function U is monotonic
increasing,
concave and
differentiable.
For
simplification
the
condition
lim U'(C)= oo
is
imposed, guaranteeing
non-zero
consumption
for all
time.
Finally,
it
is
necessary
to
make
a
boundedness
qualification
of the form
sup
U(F(K,))
=
B
0.
It
is
easy
to
see that a
superior
policy
is firstto
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12/17
OPTIMAL
GROWTH
WITH SCALE ECONOMIES
565
invest only
in X and then to coalesce
H(AKp(T))
nits of
X into
AK,(T)
as soon
as
X(T)
-
X(t)
units of
X have
been
accumulated,
say
at
time
'
K,,(t),
if
X(t)
=
0, and
if
z
is
the first
time after
t
when
AX> 0,
then
AX(T)
=
X(T).
This condition
requires
that all
X built
up
from zero
for the
purpose of increasing
K,
must
be
coalesced
into AK,,
all
at once.
Suppose
to the contrary that
AX(T)
=
y
0,
D
>
0
and
7
=-
C
>
1
are
given
constants.
U,
With
the
specific
parameterization
(46),
(47)
the
minimum
cost
capacity
schedule
has
a
particularly
simple
characterization.
When
capacity
must be
increased
(because
no more
slack
exists),
it
is
always
incrementedby
the same
constant
percentage
of existing
capacity.'
We prove this interesting result by considering a schedule {ti, AK(tj)}which is a candi-
date for
minimizing
present
discounted cost
f.
Without
loss of
generality it can
be
pre-
sumed
that
K(ti)
=
Y(ti),
so
that no
extra
capacity is
installed while
excess
capacity
is
already
in
place.
*=
Z
q(tj)H(AK(tj))
i = 1
00
i-1
-
Zi l[K(O)+
E
AK(tj)]-
s
A
[AK(ti)]a
i=l
j=
=1AK(O)'a-;
1+
AKt)~FA(~1 ...(49)
j
i
=
i
K(O)
L
K(O)
]
Because
0
can be
written in
the
form
(49), it is
apparent
that the cost
minimizing
values
of
AK(tj)/K(0)
are
independent of
K(0).
Now
consider
the problem
of
finding
a
least cost
capacity
schedule
which
begins
at time
tn
with
capacity
K(tn)
instead of at
time
0 with
capacity K(0). This is a
sub-problem of
the
original. The
cost function
for the new
problem
can be
written
in a
form
identical to
(49) except
for
obvious index
renumbering
and
the
interchanged roles
of
K(tj)
and K(0).
But the
optimal
incremental
capacity sequence
expressed
in
units of
initial
capacity is
independent
of
the initial
capacity
level.
Hence,
for an
optimal
path
{1j,
A
K(?j)},
AkRt) _
Ak(Q1)
1, 2.
KQi)
K(0)
1
Note
that if g(t)
(and
hencer(t)
also) is
constant,
an
optimal
policy
would call for
scheduling
extra
capacity
increments at
equally
spaced time
intervals
of length
l
In
I +
I.
This constant
cycle
time
g
pK(Oe
result was
provedby
Srinivasin
n
Manne
[9]
for
the
special
case mentioned
above.
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16/17
OPTIMAL
GROWTH WITH SCALE ECONOMIES
569
Let
_AQ) i_=
1,2,
...,
be the constant fraction by which capacity is always incremented. It is not difficult to
demonstrate
that
8y48a0.
9. CONCLUDING REMARKS
The time spent in a
typical big push period will be
H(AKfl)/i.
A Ramsey growth
phase
will last approximately
AK/Ytime units.'
The fraction
of time spent in
big push
stages
will
be
approximately
H(AKp)
K
AKFK
H(AK
O)
1
H(AIK)
+
~Kpi
H(AKO)
A
KP_
F(
H(AKP)+1
I
Y 1~I
F'Kzk
AK#
The
earlier the stage of development,
the higher the anticipated
values
of
F'(KD)
and
H(AKp)/AK
2 Thus, it
is to be expected that the percentage of time
spent
in
big push
stages should decline
over time.
This quantifies the generally accepted
notion that infrastructure
is
somehow
a
much
more important ingredient in the growth of
an underdeveloped
than
of a
mature
economy.
The
increased significance of big push stages
during the early years of development
means
more time spent in
no-growth stagnant
consumption phases awaiting
the
completion
of
overhead facilities. Of course the present model over-emphasizescertain structuralrigidi-
ties,
but the conclusions accord well with the
customary feeling that the
creation
of
social
overhead capital is a
more formidable
barrier to growth in a less developed economy.
Cowles Foundation M.
L.
WEITZMAN
Yale
University
First version
received May 1969;
final
version
received
December
1969
REFERENCES
[1]
Arrow, K.
J. "The
Economic Implications
of Learning
by Doing
", Review of
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[2] Chilton,
C. H. (ed.).
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n the
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New York, McGraw-
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[3]
Gale, D.
and Sutherland,
W.
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Good
Model of Economic
Development
",
in Dantzig
and
Veinott
(eds.) Mathematics of
the Decision
Sciences,
Part 2 (Providence,
American
Mathematical
Society, 1968).
1
This
is just a
first approximation
whichwill
be increasingly
accurate
as
Y
is close
to
being constant
over
the
Ramsey growth
phase.
We are comparing
a big
push period(corresponding
o
a
single
Ramsey
time point) with the
growthphase
which
directlyprecedes
or follows
it. Both I and Y
are evaluated
at
this
singleRamsey
time point.
2 It has alreadybeen noted
that
d-
F'(K)
= q0
would be sufficient
o demonstrate
hat
AKa
increasesover
time. Note
that
in Section 8
H(AK0)/1AK,l
declines
over time
irrespective
f
the
sign
of
?.
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17/17
570 REVIEW OF ECONOMIC
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