# A note on pyramid building

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A NOTE ON PYRAMID BUILDING*t

Zivia S. Wur te le

It is of interest for planning purposes to determine the effect on the rate of balanced

growth of adding to an economy industries whose outputs are used neither directly nor indi- rectly for the production of consumption goods. Examples of such industries may be found among defense industries; and a classic example is Keynes' pyramid building.

Let u s assume that we have a closed Leontief-type economy which is divided into n

sec to r s and which is largged so that the outputs of one period are the inputs of the next. Let A and B, respectively, be n-order matr ices of technical coefficients a.. and capital coefficients

b.. . Let x be a colunin vector of the outputs xi (i=l, . . . , n) of the sec to r s during the period 1.l

t. W e assume that (1-19) is a Leontief-type decomposable matrix and that the industries of the economy have been aggregated into sec to r s so as to achieve a matrix of technical coefficients of the form:

1J t t .

0 .

A 2

where Ae (e=l, . . . , k) are square, indecomposable matr ices of order ne, with k

n = c ne7 e=l

where all entr ies above and to the right of Ae are zero, and where the las t sector of A is labor. Thus the sec to r s in the eth block Ae, for e

378 Z. S. W U R T E L E

0 (2 1 (I/d - (I-A)-' (A+B))x = 0.

By means of some fairly straightforward algebraic manipulation, it can be shown that Eq. (2) is of the form:

I * (I/d-CJ -1 0 where the matrices Ce = (I-Ae)

vector of the outputs xz + n +

block.

(Ae + Be) a re indecomposable and ye (e=1, . . . , k) is a + I ' * * * 7 xo of the sectors of the eth n1+n2+ . . . + 1 2 . . * e- 1

Let l/de be the maximal characteristic root of Ce ; and assume that i f m + e, d 4 de. We now make the additional assumption that the matrix of coefficients in Eq. (3) is not separa-

ble. (The separable case may be studied by first dividing the matrix into nonseparable sub- matrices. ) The elements of C are positive. If Eq. (3) holds for nonnegative yo, then d must equal d for some e such that 1 l e O . Consider now the eth set of equations of Eq. (3) where e > j. These can be written in the form:

m

1 e Suppose d = d

e j ' I 1 0 0 0

0 0

J J

(4) Ye 0 = ( I/dj - Ce)-' ze , 0 where ze is a nonnegative vector. A necessary and sufficient condition for the solution ye to

be positive is that de >d .

balanced growth rate d. a r e added the sectors of the ( j - l ) s t group. There a r e two cases to

distinguish. If dj-l I d . , uniform growth is possible with positive outputs for all sectors of the enlarged economy, and the new growth rate is djml. On the other hand, if dj-1 >d. , uni- form expansion with positive outputs for all the sectors in the (j-1)

would be physically impossible. In evaluating a plan for enlarging the economy by the addi-

tion of industries whose outputs are not usedl directly or indirectly for consumption, i t is thus appropriate to determine whether it is compatible with balanced growth and positive outputs for all the sectors of the economy.

j th Suppose that to an economy consisting of the sectors of the jth through k groups, with

1 '

J

J through kth groups st

lThe argument of this paragraph r e s t s heavily upon the resul ts in [ 11 and [ 41

A NOTE ON P Y R A M I D BUILDING 379

BIBLIOGRAPHY

[ 1 1 G. Debreu, and I. H. Herstein, "Nonnegative Square Matrices, '' Econonietrica, Vol. 21, No. 4, October, 1953, pp. 597-607.

[ 2 1 J .M. Keynes, The General Theory ~~ of Eniploynient, Interrs t , and Money, 1936. [ 3 1 W.W. Leontief, and others, Studies in the Structure of - the - - Anierican Economy, 1953,

561 pp.

[ 4 1 M. Woodbury, 'Characterist ic Roots of Input-Output Matrices, I' Econoniic Actlvlty Analysis, 0. Morgenstern, ed., 1954, pp. 365-382.

Z. S. Wurtele, 'Equilibrium in a Uniformly Expandang Closed Leontief Type System, I ' I 5 1 Review of Economic Studies, Vol. 28, No. 1, October 1960.

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