Difference between revisions of "Dynamic Input-Output Models"
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* X(t) = A X(t) + C(t) + D(t) | * X(t) = A X(t) + C(t) + D(t) | ||
* 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) = A X(t) + C(t) + B (X(t+1) -X(t)) |
* (I - A + B) X(t) = C(t) + B X(t+1) | * (I - A + B) X(t) = C(t) + B X(t+1) | ||
| − | The production of period t is | + | The production 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 | + | while the production of period t+1 is related to the production in period t through equation: |
:<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> | ||
| Line 29: | Line 29: | ||
** C = exogenous final demand (consumption) | ** C = exogenous final demand (consumption) | ||
** D = induced investment | ** D = induced investment | ||
| − | * B = input coefficients for capital (the amount of sector | + | * 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 | * I = unit matrix | ||
* A = input coefficients for intermediate production ([[Technical Coefficient Matrix]]) | * A = input coefficients for intermediate production ([[Technical Coefficient Matrix]]) | ||
Revision as of 11:45, 19 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], [2]
Dynamic (inter-temporal) frameworks can address phenomena such as stock accumulation (both physical and capital), technology changes and other such time-dependent developments.
Mathematically the equations of the standard IO model become finite difference equations involving one or more timepoints.
Formula
The 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 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.
