is an exogenous change in household income for kth sector, total industry output change
by hk.
Input-Output Analysis and Linear Programming
The linear programming (LP) is applied to input-output analysis by Brink and McCarl
(1977) and they demonstrate how the output multiplier can be obtained from LP by
setting as
max i'X max i'X
(5) s.t. (I-A)X≤Y or s.t. (I-A)X+IS=Y,
X≥0 X, S≥0
where S is slack variables matrix. The problem is to maximize the value of the sum of
outputs from all industries under the constraint that the output from each industry does
not exceed the use of that output in final demand and as input to other industries. As
argued in Brink and McCarl (1977), the matrix (I - A) is the basis in LP formulation1. It
is easily understood because the optimal solution should be identical to the level of
production from input-output table and thus all elements in X are positive, which implies
elements in X are basic variables and thus (I - A) is basis.
Shadow price in LP formulation is defined as the expected rate of change in the
objective function (here, i’X ) when the right hand sides (here Y) are changed. In other
∂z 1
words, — = C BB 1, where z is the objective function, b is the right hand sides, CB is the
∂b
1 LP theory (Bazaraa, Jarvis and Sherali, 1990, p53; McCarl and Spreen, 2006, Chapter 3, pp3 reveals that
a solution to the LP problem will have a set of nonzero variables equal in number to the number of
constraints. Such a solution is called a basic (feasible) solution and the associated variables are commonly
called basic variables. The matrix containing the coefficients of the basic variables as they appear in the
constraints is called basic matrix or basis, which is n × n square matrix.
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