Dynamic Input-Output Models

Definition

Dynamic Input-Output Models is a category of various possible generalization of the basic Input-Ouput Model that allow accounting for more sophisticated temporal behavior

Formula

The typical equations of the dynamic input-output model:

• X = AX + C + Dt
• D = BX - BXt t+1 t
• Xt = AXt + Ct + BXt + 1 - BXt
• (I – A + B) Xt = Ct + BX

The production of period t is defined:

• X = (I – A + B)-1 (C + BX )

while the production of period t+1 is determined by:

• X =B-1[(I – A + B)X - C ]

Where:

• Y = final demand
• I = unit matrix
• A = input coefficients for intermediates
• (I-A)-1 = matrix of cumulative input coefficients (inverse)
• B = input coefficients for capital
• C = exogenous final demand (consumption)
• D = induced investment
• T = time index

This is a system of linear difference equations, since the values of the variables are related to different periods of time. Consumption is expected to grow at the annual rate (1+m)t.

Issues and Challenges

Practical problems relate to the matrix B of capital coefficients.