Dynamic Input-Output Models

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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:

$X_{t}=AX_{t}+C_{t}+D_{t}$
$D_{t}=B(X_{t+1}-X_{t})$
$X_{t}=AX_{t}+C_{t}+B(X_{t+1}-X_{t})$
$(I-A+B)X_{t}=C_{t}+BX_{t+1}$

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}+BX_{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

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

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[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

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