Dynamic Input-Output Models
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:
- D(t) = B (X(t+1) - X(t))
- X(t) = A X(t) + C(t) + B (X(t+1) -X(t))
- (I - A + B) X(t) = C(t) + B X(t+1)
The production 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 equation:
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 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.