Dynamic Input-Output Models

Definition

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

Dynamic (inter-temporal) frameworks can address phenomena such as stock accumulation (both physical and capital), technology changes and other such time-dependent developments.

Mathematically the equations of the standard IO model become finite difference equations involving one or more timepoints.

Formula

The typical equations of a dynamic input-output model are:

• X(t) = A X(t) + C(t) + D(t)
• D(t) = B (X(t+1) - X(t))
• X(t) = A X(t) + C(t) + B X(t+1) - B X(t)
• (I - A + B) X(t) = C(t) + B X(t+1)

The production of period t is defined:

$X(t) = (I - A + B)^{-1} (C(t) + B X(t) )$

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

$X(t+1) = B^{-1}[(I - A + B) X(t) - C ]$

Where:

• Y Final Demand splits into C + D where
  • C = exogenous final demand (consumption)
  • D = induced investment
• B = input coefficients for capital (the amount of sector i’s product (in dollars) held as capital stock for production of one dollar’s worth of output by sector j).
• I = unit matrix
• A = input coefficients for intermediate production (Technical Coefficient Matrix)
• (I - A)^{-1} = matrix of cumulative input coefficients (Leontief Inverse Matrix)
• t = time index of successive periods

This is a system of linear difference equations, since the values of the variables X are related to different periods of time. NB: In this example the coefficient matrices are assumed time invariant.

Issues and Challenges

• Practical problems relate to the matrix B of capital coefficients.

Further Resources

• Crash Course on Input-Output Model Mathematics