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 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.