Input-Output Analysis, Linear Programming and Modified Multipliers

where I is the n × n identity matrix. The (I - A) matrix is called the Leontief matrix and
(I - A)^-1 is called the Leontief inverse matrix which shows the total-requirements matrix
for the economy. Equation (1) can be interpreted as∆X = (I -A)^-1∆Y , which means
changes in total industry output are predicted using the Leontief inverse matrix. Thus the
column sum of (I - A)^-1 is interpreted as the total changes in output from the changes in
final demand, which is called output multiplier

(2) α = i'(I - A)^-1,
where α is the output multiplier column vector and i is an n × 1 column vector of ones.
Thus kth element in α implies there is exogenous change in final demand for kth sector
total industry output change by αk. Likewise, the employment multiplier can be defined
as follows
(3) e' = i'N(I - A)^-1,
where N is the matrix with diagonal of n₁,n₂,..., n_nand off diagonal all zeros, where

Employment_i

n_i =----—------i- (i = 1, 2, ... n). Hence, the kth element in e implies there is an

Output_i

exogenous change in employment for kth sector, total industry output change by ek.

Similarly, the income multiplier can be defined as

(4)     h' = i'H(I - A)^-1,
where H is the symmetric matrix with diagonal of h1, h2, ., hn and off diagonal all zeros,

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