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> | + | * <math>X_t = A X_t + C_t + B (X_{t+1} - X_{t}) |
* <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> | ||
Revision as of 16:08, 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:
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.
Issues and Challenges
- Practical problems relate to the matrix B of capital coefficients.