Difference between revisions of "Dynamic Input-Output Models"
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== Formula == | == Formula == | ||
The typical equations of the dynamic input-output model: | The typical equations of the dynamic input-output model: | ||
| − | * X = | + | * X = A X + C + Dt |
| − | * D = | + | * D = B X - B Xt t+1 t |
| − | * Xt = | + | * Xt = A Xt + Ct + BXt + 1 - BXt |
| − | * (I | + | * (I - A + B) Xt = Ct + BX |
The production of period t is defined: | The production of period t is defined: | ||
| − | * X = (I | + | * X = (I - A + B)-1 (C + BX ) |
while the production of period t+1 is determined by: | while the production of period t+1 is determined by: | ||
| − | * X =B-1[(I | + | * X =B-1[(I - A + B)X - C ] |
Where: | Where: | ||
| − | * Y = | + | * Y = [[Final Demand]] |
* I = unit matrix | * I = unit matrix | ||
* A = input coefficients for intermediates | * A = input coefficients for intermediates | ||
| Line 32: | Line 32: | ||
== 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. |
| + | |||
| + | == Further Resources == | ||
| + | * [https://www.openriskacademy.com/mod/page/view.php?id=800 Crash Course on Input-Output Model Mathematics] | ||
== References == | == References == | ||
Revision as of 13:17, 18 September 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]
Formula
The typical equations of the dynamic input-output model:
- X = A X + C + Dt
- D = B X - B Xt t+1 t
- Xt = A Xt + Ct + BXt + 1 - BXt
- (I - A + B) Xt = Ct + BX
The production of period t is defined:
- X = (I - A + B)-1 (C + BX )
while the production of period t+1 is determined by:
- X =B-1[(I - A + B)X - C ]
Where:
- Y = Final Demand
- I = unit matrix
- A = input coefficients for intermediates
- (I-A)-1 = matrix of cumulative input coefficients (inverse)
- B = input coefficients for capital
- 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 are related to different periods of time. Consumption is expected to grow at the annual rate (1+m)t.
Issues and Challenges
- Practical problems relate to the matrix B of capital coefficients.
Further Resources
References
- ↑ Eurostat Manual of Supply, Use and Input-Output Tables, 2008 edition
