Dynamic Input-Output Models

From Open Risk Manual

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 [1], [2]

Dynamic (inter-temporal) frameworks can address phenomena such as stock accumulation (both physical and capital), technology changes and other time-dependent developments that cannot be represented in a stationary equilibrium model.

Mathematically the equations of the standard IO model become finite difference equations relating two or more timepoints.

Formula

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} - X_{t})
  • (I - A + B) X_t = C_t + B X_{t+1}


The total production vector X of period t is related to the production in period t+1 through equation:


X_t = (I - A + B)^{-1} (C_t + B X_{t+1} )

while the production of period t+1 is related to the production in period t through the inverse equation:


X_{t+1} = B^{-1}[(I - A + B) X_t - C_t]

Where:

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

If B is zero (no amounts held in capital stock), the equations reduce to the standard (stationary) model.

Issues and Challenges

  • Practical problems relate to the matrix B of capital coefficients. Only a few sectors produce capital goods. Therefore it can not be expected that matrix B has an inverse. There is a large literature on the singularity problem in the dynamic input-output model and many problems remain for empirical applications.

Further Resources

References

  1. Eurostat Manual of Supply, Use and Input-Output Tables, 2008 edition
  2. R.E. Miller and P.D. Blair, Input-Output Analysis: Foundations and Extensions, Second Edition, Cambridge University Press, 2009