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 | + | 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 | + | Mathematically the equations of the standard IO model become finite difference equations relating two or more timepoints. |
== Formula == | == Formula == | ||
| − | + | Typical equations of a dynamic input-output model are: | |
| − | * 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) - X(t)) | * D(t) = B (X(t+1) - X(t)) | ||
* X(t) = A X(t) + C(t) + B (X(t+1) -X(t)) | * X(t) = A X(t) + C(t) + B (X(t+1) -X(t)) | ||
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== Further Resources == | == Further Resources == | ||
| − | * [https://www.openriskacademy.com/ | + | * [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 == | ||
Revision as of 16:01, 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:
-
- 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 production of period t is related to the production in period t+1 through equation:
while the production of period t+1 is related to the production in period t through equation:
![X(t+1) = B^{-1}[(I - A + B) X(t) - C(t) ]](https://www.openriskmanual.org/wiki/images/mathdata/7/6/6/7662a5915ec97bf10cdfc257800082e1.png)
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 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.
