Difference between revisions of "Dynamic Input-Output Models"
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== Formula == | == Formula == | ||
The typical equations of a dynamic input-output model are: | The typical equations of a dynamic input-output model are: | ||
− | * X = A X + C + | + | * X(t) = A X(t) + C(t) + D(t) |
− | * D = B X | + | * D(t) = B X(t+1) - B X(t) |
− | * X(t) = A X(t) + C + B (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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:<math> | :<math> | ||
− | X(t) = (I - A + B)^{-1} (C + B X(t) ) | + | X(t) = (I - A + B)^{-1} (C(t) + B X(t) ) |
</math> | </math> | ||
Revision as of 11:34, 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) - 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)
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.