June 2014 Joji tokui (Shinshu University and rieti), Tsutomu miyagawa (Gakushuin University and rieti), Kazuyasu kawasaki
Appendix 2: The Forward Linkage with the Bottleneck Effect in the First-Stage
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Appendix 2: The Forward Linkage with the Bottleneck Effect in the First-Stage When we assume the strong complementarity in parts input in manufacturing, the first-stage forward linkage effect can be replaced from (A-1) to the following. (A-2)  ,
where M and N in the subscripts stand for manufacturing and non-manufacturing respectively. Here, we assume Leontief-type production function in the first-stage input-output only for the input from manufacturing to manufacturing. This means that the input sector with the maximum rate of decline is the bottleneck and force all the other input to fall at the same rate. (1) The first-stage forward linkage effect, in addition to the direct damage When we compute the first-stage forward linkage effect in addition to the direct damage, we use the following three-step procedure. Step 1: Based on (A-2), we compute the first-order spill-over effect on manufacturing, that is, .
We compute Step 2: For non-manufacturing, we use (A-1), that is,  ,
which is the same as, 
Step 3: We add the first-stage forward linkage effect for both manufacturing and non-manufacturing to the direct damage. (2) The total forward linkage effect, in addition to the bottleneck first-stage effect and the direct damage The total forward linkage matrix G in the equation (2) can be written as, 
When we assume that bottleneck effect as in (A-2) occur only in the matrix corresponding to the first-stage, which is denoted by  .
Using  .
By applying the method of transforming the change in terms of factor income into the change in terms of outputs with the diagonal elements of the matrix G, we obtain the followings, (A-3)  .
Step 1 & 2: Since the part where we multiply  in the second term of the right-hand side of the equation (A-3), that is , is nothing but the first-stage forward linkage with bottleneck, which we know above. That is, for non-manufacturing,
 ,
and for manufacturing, using 
we compute Step 3: Substituting these into the equation (A-3) above to carry out the rest of the computation, we get the total forward linkage effect with first-stage bottleneck. That is, 
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for each manufacturing sector and multiply them by the output for each sector Xi and obtain the change in the output after the first-stage forward linkage, 
. 
above, we can rewrite the equation (2) as,
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