The additional exogenous capacity limitations can be represented as DX ≤ Z ,
where D is the m × n design matrix to impose restrictions on industries. Note that m is the
number of industries restricted and n is the number of industries in the economy. The
elements of matrix D are zero or one (or it could be other values) and one indicates
restriction is imposed. Z is the capacity limitations vector and its dimension is m × 1.
The equation (5) is now
max i'X
s.t. (In -A)X+InS1 =Y
DX + IkS2 =Z
X, S1, S2 ≥0
or
max i’X
s.t.
(In -A)
In
0
Im
S1
S2
X,S1,S2≥0
Note that In is n × n and Im is m × m identity matrices, and S1 and S2 are slack variables
correspondingly. In this formulation the matrix (In - A) is not the basis anymore because
of additional constraints and in turn, the shadow prices are different from those of LP
formulation in equation (5). This fact implies that the output multiplies with additional
constraints cannot be the same as multipliers from the input-output analysis. Because 0 ≤
Z ≤ X by construction, the slack variables for restricted industries should be nonzero and
they come into the basic variables. The (n + m) × (n + m) basis of the problem in equation
(8) is given by
where F = {-D(In - A)-1D’}-1 . Thus, the modified output multipliers (for sectors) are
obtained by
(9)
B=
I-A
D
D’
0
and B-1
(In - A)-1{In + D’FD(In - A)-1} -(In - A)-1D’F
-FD(In-A)-1 F
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