optimum growth with scale economies, review of economic letters

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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.



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



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.



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

Economic

Studies

(June 1962).

[2] Chilton,

C. H. (ed.).

Cost

Engineering

n the

Process

Industries

New York, McGraw-

Hill,

1960).

[3]

Gale, D.

and Sutherland,

W.

R. "Analysis

of a One

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

?.



17/17

570 REVIEW OF ECONOMIC

STUDIES

[4] Haldi, J. and Whitcomb, D.

"

Economies

of

Scale

in Industrial Plants

",

The

Journal of Political Economy(August 1967), 373-385.

[5] Hirschman, A. 0. The Strategy

of

Economic

Development (New

Haven, Yale

University Press, 1958).

[6] Kelley, J. M. General Topology (New York,

Van

Nostrand,

1955).

[7] Koopmans, T. C.

"

On the Concept of Optimal Economic

Growth ",

Pontificae

Academiae Scientiarum Scripta

Varia (1965), 225-300.

[8] McFadden, D.

"

The

Evaluation of Development Programmes ",

The Review

of

Economic

Studies,

January

1967,

25-50.

[9] Manne,

A. S.

Investments

or

Capacity Expansion (Cambridge,

M.I.T.

Press, 1967).

[10] Moore, F. T.

"

Economies

of

Scale: Some

Statistical Evidence

",

Quarterly

Journal

of Economics (May 1959),

232-245.

[11] Ramsey, F.

P.

"

A

Mathematical Theory

of

Savings ", Economic

Journal(December

1928), 219-226.

[12] Rosenstein-Rodan, P. N.

"Notes on the Theory of the 'Big Push'

",

Ch.

3

in

H. S. Ellis (ed.) Economic Development for Latin America (I.E.A.

Conference)

(London, Macmillan, 1961).

[13] Scitovsky, T.

"

Growth-Balanced or Unbalanced?

"

in Abramovitz (ed.) The

Allocation of Economic Resources (Stanford, Stanford University

Press, 1959), 207-

217.

[14]

von

Weizsacker, C. C.

"Existence of Optimal Programs of Accumulation for an

Infinite Time Horizon ", Review of Economic Studies (April 1965),

85-104.

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