# A note on pyramid building

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<ul><li><p>A NOTE ON PYRAMID BUILDING*t </p><p>Zivia S. Wur te le </p><p>It is of interest for planning purposes to determine the effect on the rate of balanced </p><p>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. </p><p>Let u s assume that we have a closed Leontief-type economy which is divided into n </p><p>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 </p><p>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 </p><p>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: </p><p>1J t t . </p><p>0 . </p><p>A 2 </p><p>where Ae (e=l, . . . , k) are square, indecomposable matr ices of order ne, with k </p><p>n = c ne7 e=l </p><p>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 </p></li><li><p>378 Z. S. W U R T E L E </p><p>0 (2 1 (I/d - (I-A)-' (A+B))x = 0. </p><p>By means of some fairly straightforward algebraic manipulation, it can be shown that Eq. (2) is of the form: </p><p>I * (I/d-CJ -1 0 where the matrices Ce = (I-Ae) </p><p>vector of the outputs xz + n + </p><p>block. </p><p>(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 </p><p>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- </p><p>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: </p><p>m </p><p>1 e Suppose d = d </p><p>e j ' I 1 0 0 0 </p><p>0 0 </p><p>J J </p><p>(4) Ye 0 = ( I/dj - Ce)-' ze , 0 where ze is a nonnegative vector. A necessary and sufficient condition for the solution ye to </p><p>be positive is that de >d . </p><p>balanced growth rate d. a r e added the sectors of the ( j - l ) s t group. There a r e two cases to </p><p>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) </p><p>would be physically impossible. In evaluating a plan for enlarging the economy by the addi- </p><p>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. </p><p>j th Suppose that to an economy consisting of the sectors of the jth through k groups, with </p><p>1 ' </p><p>J </p><p>J through kth groups st </p><p>lThe argument of this paragraph r e s t s heavily upon the resul ts in [ 11 and [ 41 </p></li><li><p>A NOTE ON P Y R A M I D BUILDING 379 </p><p>BIBLIOGRAPHY </p><p>[ 1 1 G. Debreu, and I. H. Herstein, "Nonnegative Square Matrices, '' Econonietrica, Vol. 21, No. 4, October, 1953, pp. 597-607. </p><p>[ 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, </p><p>561 pp. </p><p>[ 4 1 M. Woodbury, 'Characterist ic Roots of Input-Output Matrices, I' Econoniic Actlvlty Analysis, 0. Morgenstern, ed., 1954, pp. 365-382. </p><p>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. </p><p>* * * </p></li></ul>