Difference between revisions of "Dynamic Input-Output Models"

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'''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 <ref>Eurostat Manual of Supply, Use and Input-Output Tables, 2008 edition</ref>, <ref>R.E. Miller and P.D. Blair, Input-Output Analysis: Foundations and Extensions, Second Edition, Cambridge University Press, 2009</ref>
 
'''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 <ref>Eurostat Manual of Supply, Use and Input-Output Tables, 2008 edition</ref>, <ref>R.E. Miller and P.D. Blair, Input-Output Analysis: Foundations and Extensions, Second Edition, Cambridge University Press, 2009</ref>
  
−
Dynamic (inter-temporal) frameworks can address phenomena such as stock accumulation (both physical and capital), technology changes and other such time-dependent developments.
+
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 involving one or more timepoints.
+
Mathematically the equations of the standard IO model become finite difference equations relating two or more timepoints.
  
 
== Formula  ==
 
== Formula  ==
−
The typical equations of a dynamic input-output model are:
+
Typical equations of a dynamic input-output model are:
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* X(t) = A X(t) + C(t) + D(t)
+
* <math>X_t = A X_t + C_t + D_t</math>
−
* D(t) = B X(t+1) - B X(t)
+
* <math>D_t = B (X_{t+1} - X_{t})</math>
−
* X(t) = A X(t) + C(t) + B X(t+1) - B X(t)
+
* <math>X_t = A X_t + C_t + B (X_{t+1} - X_{t})</math>
−
* (I - A + B) X(t) = C(t) + B X(t+1)
+
* <math>(I - A + B) X_t = C_t + B X_{t+1}</math>
  
  
−
The production of period t is defined:
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The total production vector X of period t is related to the production in period t+1 through equation:
  
 
:<math>
 
:<math>
−
X(t) = (I - A + B)^{-1} (C(t) + B X(t) )
+
X_t = (I - A + B)^{-1} (C_t + B X_{t+1} )
 
</math>
 
</math>
  
 +
while the production of period t+1 is related to the production in period t through the inverse equation:
  
−
while the production of period t+1 is determined by:
 
 
:<math>
 
:<math>
−
X(t+1) = B^{-1}[(I - A + B) X(t) - C ]
+
X_{t+1} = B^{-1}[(I - A + B) X_t - C_t]
 
</math>
 
</math>
  
 +
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]])
 +
* <math>(I - A)^{-1}</math>  is the matrix of cumulative input coefficients ([[Leontief Inverse Matrix]])
 +
* t is the time index of successive periods
  
−
Where:
 
−
* Y = [[Final Demand]]
 
−
* I = unit matrix
 
−
* A = input coefficients for intermediate production
 
−
* (I - A)^{-1} = matrix of cumulative input coefficients (inverse)
 
−
* 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).
 
−
* 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 X are related to different periods of time. NB: In this example the coefficient matrices are assumed time invariant.
  
−
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.
+
If B is zero (no amounts held in capital stock), the equations reduce to the standard (stationary) model.
  
 
== Issues and Challenges ==
 
== Issues and Challenges ==
−
* Practical problems relate to the matrix B of capital coefficients.
+
* 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 ==
 
== Further Resources ==
−
* [https://www.openriskacademy.com/mod/page/view.php?id=800 Crash Course on Input-Output Model Mathematics]
+
* [https://www.openriskacademy.com/course/view.php?id=70 Crash Course on Input-Output Model Mathematics]
 +
* [https://www.openriskacademy.com/course/view.php?id=64 Introduction to Input-Output Models using Python]
  
 
== References ==
 
== References ==

Latest revision as of 16:39, 16 November 2023

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
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