Difference between revisions of "Dynamic Input-Output Models"

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

↑ 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

[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

[1]

[2]

@@ Line 1: / Line 1: @@
 == 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<ref>Eurostat Manual of Supply, Use and Input-Output Tables, 2008 edition</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 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  ==
-The typical equations of the dynamic input-output model:
+Typical equations of a dynamic input-output model are:
-* X = AX + C + Dt
+* <math>X_t = A X_t + C_t + D_t</math>
-* D = BX - BXt t+1 t
+* <math>D_t = B (X_{t+1} - X_{t})</math>
-* Xt = AXt + Ct + BXt + 1 - BXt
+* <math>X_t = A X_t + C_t + B (X_{t+1} - X_{t})</math>
-* (I – A + B) Xt = Ct + BX
+* <math>(I - A + B) X_t = C_t + B X_{t+1}</math>
-The production of period t is defined:
+The total production vector X of period t is related to the production in period t+1 through equation:
-* X = (I – A + B)-1 (C + BX )
+:<math>
+X_t = (I - A + B)^{-1} (C_t + B X_{t+1} )
+</math>
-while the production of period t+1 is determined by:
+while the production of period t+1 is related to the production in period t through the inverse equation:
-* X =B-1[(I – A + B)X - C ]
+:<math>
+X_{t+1} = B^{-1}[(I - A + B) X_t - C_t]
+</math>
 Where:
-* Y = final demand
+* Y is the [[Final Demand]] which splits into C + D, where in turn
-* I = unit matrix
+** C is the exogenous final demand (consumption)
-* A = input coefficients for intermediates
+** D is the induced investment
-* (I-A)-1 = matrix of cumulative input coefficients (inverse)
+* 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).
-* B = input coefficients for capital
+* I is the unit (identity) matrix
-* C = exogenous final demand (consumption)
+* A are input coefficients for intermediate production ([[Technical Coefficient Matrix]])
-* D = induced investment
+* <math>(I - A)^{-1}</math>  is the matrix of cumulative input coefficients ([[Leontief Inverse Matrix]])
-* T = time index
+* 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.
-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 ==
-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 ==
+* [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 ==