A Leontief Model of Inter Regional Economic Growth

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A LEONTIEF MODEL OF INTERREGIONAL ECONOMIC GROWTH

Michael Carlberg*

Abstract

Savings are invested in the region offering the best return,thus increasing its stock of capital. Simultaneously labour grows,moving to the region which pays the highest wages. On which pathdoes this system of regions develop? Theoretical analysis showsthat the time path depends on Leontief technology, propensity tosave and on the growth of labour. In the case of isolated regions adynamic equilibrium only exists by chance; in the case ofintegrated regions, however, a stable equilibrium is likely toexist. There is no need for regional policy in this setting, since themarket allocates efficiently and distributes equitably.

I. Introduction

Interregional economic growth is based on free trade, capital movements

and labour migration. Products are shipped to the region paying the best price.Savings come to be invested in the region yielding the greatest return, thusincreasing its stock of capital. At the same time labour grows, moving to theregion which offers the highest wages. This continuous process raises a numberof questions: How are capital and labour allocated to regions? Is there a trade-off between interregional equity and aggregate efficiency? At what speeds dothe regions develop, and what are the underlying factors? Does a dynamicequilibrium exist, and if so, will stabil ity prevail?

A good deal of analysis has been performed on regional planning models(Rahman (5), Mera (2), Sakashita (7)), on market models treating the region as anannex of the nation (Botts and Stein (1)), and on foreign trade models withoutcapital movements and labour migration (Oniki and Uzawa (4)). Of course, realeconomic growth is subject to market forces, and the nation is nothing but asystem of regions closely linked by product and factor mobility. Hence thepurpose of this paper is to discuss a market model of interregional economicgrowth, focusing on free trade, capital movements and labour migration. Underneoclassical technology, however, there would be the risk of seeing all activitiesmove to the most productive region. A simple solution to this problem is givenby introducing Leontief technology. This assumption can be made without muchloss, since the problem at hand is interregional and not intersectoral substitution.

*Sozialoekonomisches Seminar der Universitaet Hamburg, Federal Re-

public of Germany.

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II. Static Analysis of Allocation

For the sake of convenience consider two regions (i = 1,2) producing ahomogeneous output (Yi) by means of two inputs (capital Ki, labour Li). Though

spatial diffusion may spread technical knowledge, we assume a distinct Leontieftechnology in each region; in doing so, we take into account that regions differwith respect to natural resources (including land, environment and climate). Sooutput depends on inputs and on fixed technology coefficients (capital-outputratio vi, labour-output ratio ui):

K LiYi= rain ( vi , ui ) 9

Let the nation's endowment with capital K and labour L be given. How does themarket allocate capital and labour to regions? The answer is given by theEdgeworth box in figure 1. Region 1 operates on ray i, region 2 on ray 2; theirintersection yields the equilibrium (A). This allocation is feasible, and there willbe full employment of capital and labour. From figure 1 follows the conditionfor production to take place in both regions (interior solution):

ui L ujv K vj ( 1 )

0 1

L

2

I

v IK

0 2

F i g u r e ] : A l l o c a t i o n o f C a p i t a l a n d L a b o u r

If capital and labour endowment is inappropriate, however, thenproduction is restricted to a single region (corner solution); this case is tr ivialand uninteresting as far as interregional economic growth is concerned. Now

will this static equilibrium be reached by the market? Turning back to figure 1suppose the initial state is B, where capital is abundant. If the addition toincome in region 1 exceeds the reduction in region 2, then firms in region 1 canoffer higher input prices and thereby bid away resources. Since capital is

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abundant, the change of income is governed by migration ALl:

ALl AL 1AYI = Ul ' A Y2=- u2

If u L< u2, then the increase in region 1 is greater than the decline inregion 2. Provided however capital is scarce and labour is abundant, then thecondition for stability is v 2 u 2 , v 1< v 2 (2)

But if both capital and labour are more productive in region i than in region j,then, of course, all activities will tend to region i. In static equilibrium, regionaloutput is determined by national factor endowment and by tlechnology. Whennational input grows, there may be a change in regional shares: ~

uj K-vjL Yi uj K-v.Lj

Y (uj-ui)K- (vj-vi) Li uJ vi-uivj , ai: = - - =

Here the question arises, whether market allocation is efficient or not. Thisproblem can be stated as a linear program with the objective functionmaximising national output under the input constraint:

Y: YI +Y2-* maxYI' Y2

VlY l+v2Y 2


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Y2

Kv 2

L~2

F i g u r e

Y2

\ K

K L

v 1 u I2 : P r o d u c t i o n i n B o th R e g io n s

\

F i g u r e 3 : P r o d u c t i o n i n a S i n g l e R e g i o n

>

Y1

U . U .

i L ]

v --? < t ~ < ~ a n d

u I


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Hence the capital constraint shifts parallel to the right, as can be seen fromfigure 4; so the optimum moves from A to B. Output n region 1 grows, whileoutput in region 2 declines with the relocation of capital and labour to region 1.

At the same time labour grows, moving to the region which offers the highestwages. Let the natural rate of labour growth in region i be invariant; weightingthese regional rates of labour growth n by the spatial distribution of labour thenprovides the national rate of labour growth n. Labour is allocated to regionsaccording to the labour-output ratios ui:

al u I n I + a 2 u 2 n 2n= a l u I + a 2u 2 ' Lo ent=ulY1 + u2 Y2 " (4)

Y2~

nstable

~ ~ J / equilibrium

>Y1

Figure 4: Growth Path of the Economy

Due to migration the labour constraint shifts parallel to the right as well,thus the optimum is situated in D. As a consequence, output and income expand,part of which is saved; and the savings again come to be invested in the regionyielding the highest return. Simultaneously abour continues to grow, moving tothe region which offers the highest wages. This continuous process raises thequestion, whether a dynamic equilibrium does exist, and if so, whether stability

will prevail. Turning back to figure 4, is there a time path with both regionsexpanding at the same speed? And supposing he economy is off this time path:Will there be a tendency back to equilibrium? If, for example, labour grows toofast, then region 1 runs out of capital and labour, production being ultimatelyrestricted to region 2. These questions will now be looked into more closely.

Steady-state growth is defined by capital and output expanding at auniform and invariant rate. On the other hand, labour grows at the naturalrate. Under Leontief technology, full employment will only prevail if th~expansion of capital and output is concordant with the natural growth of labour:

3:~ stands for the growth rate of X.

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activities tend to the most productive region, or will the economy return toequilibrium? The answer is given by the time path of the model, which ischaracterized by the differential equation (3),(4):

ntsl Y1 + s2 Y2 = Vl D Y1 + v2 D Y2' Loe = UlY1 + u2 Y2 "

Starting from an arbitrary initial state, the stock of c~pital expands at thenatural rate in the long run, provided the inequality is met:

u I s 2 - u 2 s 1

u I v 2 - u 2 v I

Under this condition the market finds the dynamic equilibrium on its own, so theconcept of equilibrium gains in importance. If, however, propensities to savediffer too much as compared with capital-output ratios, then all activities willtend to the most productive region. How does this error adjustment mechanismwork? Suppose the growth of labour accelerates, then capital becomes scarce.Consequently capital relocates to the more efficient region, until the additionallabour is fully employed. Hence he overall capital-output ratio declines so as torestore equilibrium. Yet in case the shift of activities has gone to the high-consuming region, the overall propensity to save declines as well. Provided thisdecline exceeds the fall in the capital-output ratio, then the threat of Myrdalinstability becomes reality.

On this foundation some conclusions can be drawn concerning trade,capital movements and migration. Savings Si, consumption C and investment I

obviously follow from the relationships:

a i v iS i = s i Y i , C i = ( 1 - s i ) Y i , I i - a l v l + a 2 v 2 I .

Provided the demand for consumer and investment goods exceeds output, thenthere will be an import (Y~;) of goods. And if savings fall short of investment,j~then there will be a capital import (Sji):

Y j i = C i + I i - Y i ' S j i = [ i - S i / Y j i = S j i "

The analysis reveals, that in equilibrium commodity imports are financedby capital imports. If, however, a zero balance of trade (capital balance) shouldbe required, then allocative efficiency would suffer a cut-back. In dynamicequilibrium, the rate of commodity import (capital import) will be invariant:

u _ S i iYi - S i = n v i - s i = c o n s t .

4See appendix 2.

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Interregional migration adjusts discrepancies between the supply of andthe demand for labour. The addition to labour supply D L.. results from thenatural growth_ of labour n L:,. and from immigration Lji; the'addit ion to labourdemand D L i conforms to natlonal growth:

s dD L = n i Li + Lji , D L i = nL i

If in region i the natural growth of labour falls short of the expansion ofoutput, then there will be an influx of labour. In dynamic equilibrium, the rateof immigration is invariant, too:

L..

j1 = n - n. = eonst.Li i

Alternatively the parameters n and n can be viewed as the (Harrod-neutral) rate of technical progress, thus relaxing the assumption of Leontieftechnology.

IV. Interregional Distribution of Income

Within this set-up, income distribution is based on interest rates ri, wagerates wi, and output prices pi o Let firms set prices in such a way that revenuecovers cost:

YiPi = Ki ri + Li wi ~ (6)

Products are shipped to the region paying the best price, consequently outputprices coincide (excluding transportation costs):

pl =P 2 = 1 .Output prices are set to unity in order to simplify notation. In addition,

savings come to be invested in the region yielding the greatest return, therebyequalising interest ra tes . Wage rates converge too, as labour moves to the regionoffering the highest wages. In this sense, market mobility provides forinterregional equity:

r l = r 2 = r , W l = W 2 = W 9

Wit h t h i s , e q u a t i o n ( 6) c a n b e r e s t a t e d i n t e r m s o f L e o n t i e f t e c h n o l o g y :

v i r + u i w = 1 .

T hi s y i el d s t h e r a t e o f i n t e r e s t a n d t h e r a t e o f w a g e s , w h i c h o b vi o u s l y a r ei n d e p e n d e n t o f c a p i t a l a n d l a b o u r e n d o w m e n t :

u 2 - u I Vl - v 21"- ~ W =

u 2 v 1 - u 1 v 2 u 2 v 1 - u 1 v 2

I t is i n t e r e s t i n g t h a t i n t e r r e g i o n a l i n c o m e d i s t r i b u t i o n i s d u a l t o t h e l i n e arp r o g r a m i n s e c t i o n 2 . T h e i n t e r e s t r a t e a n d t h e w a g e r a t e a r e d e t e r m i n e d s o a st o m i n i m i s e c o s t , b u t f a c t o r i n c o m e m u s t n o t f al l b e l o w o u t p u t v a l u e :

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Z = K r + L w - . r a i nr , w

v i r + u i w > 1

r > 0

w > 0 o

Figure 5 plots the dual: The goal function (dashed line, at an angle of-K/L) should reach a minimum under the constraint, the feasible set is shaded,hence the solution is A. Factor prices do not vanish (interior solution) as long as:

u i u .

< L v 2 o r u I > u 2 , v l < v 2

Once more this is the condition for production to take place in bothregions, cf. (I) and (2). In this case the interest rate and the wage rate do notdepend on capital and labour endowment, but on Leontief technology.

W

I

I

u 2

\

/

W

1 1

v 1 v 2

>

r

F i g u r e 5 : R a t e o f I n t e r e s t a n d W a g e R a t e ( S h a d o w P r i c e s )

V. C o n c l u s i o n

Interregional economic growth depends on free trade, capital movementsand labour migration. The goods are shipped to the region paying the best price,savings come to be invested in the region yielding the greatest return, and labour

moves to the region which offers the highest wages. This behaviour is good bothfor the individuals and for the economy as a whole, since it implies equal wagerates (and interest rates) in all regions, simultaneously maximising nationaloutput.

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In the case of isolated regions a dynamic equilibrium only exists bychance, and if so, there will be the threat of knife-edge instability. In the caseof integrated regions, however, a steady-state is likely to exist; and if adisturbance occurs, there will be a tendency back to equilibrium. Output andi n c o m e g r o w a t t h e s a m e r a t e i n a l l t h e r e g i o n s , a l t h o u g h t h e r e g i o n s d i f f e r i n

L e o n t i e f t e c h n o l o g y , p r o p e n s i t y t o s a v e a n d n a t u r a l g r o w t h o f l a b o u r ; t h e s e

f a c t o r s a l s o d e t e r m i n e t h e s p e e d o f e x p a n s i o n . Y e t i f t h e n a t u r a l g r o w t h o f

l a b o u r i s t o o f a s t o r t o o s l o w , t h e n a l l a c t i v i t i e s t e n d t o t h e m o s t p r o d u c t i v e

r e g i o n . I n t h i s c a s e o f M y r d a l i n s t a b i l i t y , t h e r e w i l l b e a n e e d f o r r e g i o n a l

p o l i c y . T o f a c i l i t a t e e x p o s i t i o n , t h e a n a l y s i s h a s b e e n c o n f i n e d t o t w o r e g i o n s ,

b u t t h e f i n d i n g s a p p l y i n t h e c a s e o f m a n y r e g i o n s a n d m a n y i n p u t s , t o o .

APPENDIX I

The amount of capital and labour required to produce Yi depends ontechnology:

K =v I Y1 +v2Y2 ' L =u IYI +u2 Y2

Conversely, how much can be produced, given the nation's endowment withcapital and labour? Solve the above equation for Yi !

APPENDIXII

Substitution gives:d Y2

(UlV2-U2Vl) d t (Ul s2- u2 Sl)Y2= ( s l - nv l ) L ~ent

Integration by varying the constant yields:

c I L o eC2Y 2 = n - c 2 e n t + c 3 w i t h c o n s t a n t s

s I - n v I u I s 2 - u 2 s 1

c I - , c 2 = .

UlV2-U2V 1 u I v2 - u 2 v I

1.

2.

3.

4.

REFERENCES

Botts, G. H., J. L. Stein. Economic Growth in a Free Market. NewYork: Columbia, 1964.Mera, K. A Multiregion Multisector Model of Equilibrium Growth, inPapers of the Regional Science Association, Vol. 21, 1967, p. 53.Mera, K. Income Distribution and Regional Development. Tokyo:

University, 1975.Oniki, H., and H. Uzawa. Patterns of Trade and Investment in a DynamicModel of International Trade, in Review of Economic Studies 32, 1965, p.15.

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5o

6 .

7.

R a h m a n , M . A . R e g i o n a l A l l o c a ti o n o f I n v e s t m e n t , in Q u a r t e r l y J o u rn a lof Economics~ Vol. 77, 1963, p. 26.Richa rdson, H. W. R egion al Gro wth Th eory. London: MacM il lan 1973.Sakash i t a , N . Reg iona l Al loca t ion o f Pub l ic Inve s tm en t , i n PaP ers andProceedings~ Regional Science Associat ion, Vol. 19, 1967, p. 161.

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A Leontief Model of Inter Regional Economic Growth

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