Input-Output Analysis, Linear Programming and Modified Multipliers

objective function coefficients for basic variables and B is the basis (McCarl and Spreen,
2006, Chapter 3, p12). Shadow price for the LP formulation in equation (5) is given by

(6)     CbB-1 = i'(I - A)^-1.

Obviously, shadow price in equation (6) is identical to output multipliers in
equation (2) as shown in Brink and McCarl (1977). Using the similar logic the
employment and income multipliers are derived from the following models,

max     n'X            max     h'X

(7) s.t. (I-A)X≤Y and s.t.   (I-A)X≤Y,

X≥0                  X≥0

wheren′ = [n₁,n₂,...,n_n] and h′ = [h₁, h₂,..., h_n] . Shadow prices from these models are
given by n'(I - A)^-1 ≡ i'N(I - A)^-1 and h‘(I - A)^-1 = i'H(I - A)^-1, which are identical to
employment and income multipliers in equations (3) and (4), respectively.

Modified Multipliers using LP

As alluded in introduction, the LP approach is attractive because it allows us to study the
effects of exogenous capacity limitations in some industries, for example limiting
production to reduce greenhouse gas emissions from power generation sector, or
government’s ban on the cattle production due to the food safety issues. We suggest that
shadow prices from the restricted LP model with the additional exogenous capacity
limitations provide the modified output, employment and income multipliers. It can be
argued that these modified multipliers are crucial for the further policy or regional impact
analysis.

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