Difference between revisions of "Dynamic Input-Output Models"
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* <math>X_t = A X_t + C_t + D_t</math> | * <math>X_t = A X_t + C_t + D_t</math> | ||
* <math>D_t = B (X_{t+1} - X_{t})</math> | * <math>D_t = B (X_{t+1} - X_{t})</math> | ||
| − | * <math>X_t = A X_t + C_t + B (X_{t+1} - X_{t}) | + | * <math>X_t = A X_t + C_t + B (X_{t+1} - X_{t})</math> |
* <math>(I - A + B) X_t = C_t + B X_{t+1}</math> | * <math>(I - A + B) X_t = C_t + B X_{t+1}</math> | ||
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** C is the exogenous final demand (consumption) | ** C is the exogenous final demand (consumption) | ||
** D is the induced investment | ** 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). | + | * 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 | * I is the unit (identity) matrix | ||
* A are input coefficients for intermediate production ([[Technical Coefficient Matrix]]) | * A are input coefficients for intermediate production ([[Technical Coefficient Matrix]]) | ||
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| − | 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 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 == | == 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 == | == Further Resources == | ||
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:
The total production vector X 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 the inverse equation:
![X_{t+1} = B^{-1}[(I - A + B) X_t - C_t]](https://www.openriskmanual.org/wiki/images/mathdata/5/6/a/56a91fc98a639bf79666c56d0ec49505.png)
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)
-
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.
