Difference between revisions of "Dynamic Input-Output Models"
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== Definition == | == 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<ref>Eurostat Manual of Supply, Use and Input-Output Tables, 2008 edition</ref> | + | '''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 <ref>Eurostat Manual of Supply, Use and Input-Output Tables, 2008 edition</ref>, <ref>R.E. Miller and P.D. Blair, Input-Output Analysis: Foundations and Extensions, Second Edition, Cambridge University Press, 2009</ref> |
+ | |||
+ | 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 == | == Formula == | ||
− | The typical equations of | + | The typical equations of a dynamic input-output model are: |
* X = A X + C + Dt | * X = A X + C + Dt | ||
* D = B X - B Xt t+1 t | * D = B X - B Xt t+1 t | ||
− | * | + | * X(t) = A X(t) + C + B (t) + 1 - BXt |
− | * (I - A + B) | + | * (I - A + B) X(t) = C(t) + B X(t+1) |
The production of period t is defined: | The production of period t is defined: | ||
− | + | ||
+ | :<math> | ||
+ | X(t) = (I - A + B)^{-1} (C + B X(t) ) | ||
+ | </math> | ||
while the production of period t+1 is determined by: | while the production of period t+1 is determined by: | ||
− | + | :<math> | |
+ | X(t+1) = B^{-1}[(I - A + B) X(t) - C ] | ||
+ | </math> | ||
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* Y = [[Final Demand]] | * Y = [[Final Demand]] | ||
* I = unit matrix | * I = unit matrix | ||
− | * A = input coefficients for | + | * A = input coefficients for intermediate production |
− | * (I-A)-1 = matrix of cumulative input coefficients (inverse) | + | * (I - A)^{-1} = matrix of cumulative input coefficients (inverse) |
− | * B = input coefficients for capital | + | * 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). |
* C = exogenous final demand (consumption) | * C = exogenous final demand (consumption) | ||
* D = induced investment | * D = induced investment |
Revision as of 10:52, 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 = A X + C + Dt
- D = B X - B Xt t+1 t
- X(t) = A X(t) + C + B (t) + 1 - BXt
- (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:
Where:
- Y = Final Demand
- I = unit matrix
- A = input coefficients for intermediate production
- (I - A)^{-1} = matrix of cumulative input coefficients (inverse)
- 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).
- 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.