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Show A^ and consider further H = (( ho0), the row vector of direct income payment, and V= (( Vji)), the row vector of water intake coefficients. In the following equations we see that: ( 20) H( I- A)"'= K and - 1 - ( 2i) V( I- A)" 1= M where K is the household income row vector (( Kp) and Mis the water row vector (( M^)). Expanding these equations we obtain: ( 22) Kj - ghoiAij = g. hi j and ( 23) Mj - JgVuAjj = jffl. ( j = 1 n). i- 1 J K' The ratio, J , is the input- output Type I income multiplier. The ratio h0j ( 24) EL = £[ Y_ Mj dW jth is the ( joint) marginal value of water in terms of income when the propagation effect is started in the j" 1 sector ( the first set of coefficients in Table 2). The Type II income multiplier is traditionally derived from the inverse of the ( n + 1) x ( n + 1) matrix with households as an endogenous sector. The coefficients of the bottom row of this matrix Ajj , ^ . are the set of direct, A* indirect and induced payments to the households. The ratio, - n + JJJ , is hoj defined as the Type II multiplier. It was noted and proven in Bradley and Gander [ 9] that the set of Type II multipliers bears a constant relationship to the set of Type I multipliers. This gives an alternative method of derivation for Type II multipliers. 17 |