OCR Text |
Show ( 15) 5 kojAii = ^^ V1 jAjj + ( part ignored), where rl H £ L = hoi - = AU' ^ d Wj = Vii dXj J' dDi 3Xj J Forming the ratios we get: ( 16) Sh°, jAji = A + ( part ignored), ( i = 1, . . ., n), 2VlJAji where the left- hand term is the ( joint) marginal value of water in terms of income * A similar term for the employment criterion is shown in the third section of this paper * As a final consideration in this section, we want to summarize the essence of the input- output model. This summary will also show the derivation of the components of ( 15). Let A = (( ajj)) be the n x n matrix of direct purchase coefficients; i. e., a « [ j is the proportion of the total jtn sector input that is purchased from the i^ sector. Also let D = (( Di)) be the column vector of final demands and X - (( Xj)) be the column vector of total outputs. ( For each processing sector total output is equal to total input.) Then the following is an identity: ( 17) AX + D = X From this we get: U8) ( l- A) X= D and ( 19) X = ( l- A) D. Let the coefficient of the direct and indirect purchase matrix ( |~ A ) be 16 |