Difference between revisions of "Dynamic Input-Output Models"
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The typical equations of a dynamic input-output model are: | The typical equations of a dynamic input-output model are: | ||
* X(t) = A X(t) + C(t) + D(t) | * X(t) = A X(t) + C(t) + D(t) | ||
| − | * D(t) = B X(t+1) - | + | * D(t) = B (X(t+1) - X(t)) |
* X(t) = A X(t) + C(t) + B X(t+1) - B X(t) | * X(t) = A X(t) + C(t) + B X(t+1) - B X(t) | ||
* (I - A + B) X(t) = C(t) + B X(t+1) | * (I - A + B) X(t) = C(t) + B X(t+1) | ||
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X(t) = (I - A + B)^{-1} (C(t) + B X(t) ) | X(t) = (I - A + B)^{-1} (C(t) + B X(t) ) | ||
</math> | </math> | ||
| − | |||
while the production of period t+1 is determined by: | while the production of period t+1 is determined by: | ||
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X(t+1) = B^{-1}[(I - A + B) X(t) - C ] | X(t+1) = B^{-1}[(I - A + B) X(t) - C ] | ||
</math> | </math> | ||
| − | |||
Where: | Where: | ||
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** 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 i’s 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 | + | * A = input coefficients for intermediate production ([[Technical Coefficient Matrix]]) |
| − | * (I - A)^{-1} = matrix of cumulative input coefficients ( | + | * (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 are related to different periods of time. | + | 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 == | == Issues and Challenges == | ||
Revision as of 11:41, 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) - B X(t)
- (I - A + B) X(t) = C(t) + B X(t+1)
The production of period t is defined:
while the production of period t+1 is determined by:
![X(t+1) = B^{-1}[(I - A + B) X(t) - C ]](https://www.openriskmanual.org/wiki/images/mathdata/6/2/b/62b74b1adcd8c7895900cf3857432f2a.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’s 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.
