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Format: matrix A = n rows × n values, vector Y = n values (row or column). Comma, semicolon or tab; header row and sector names optional.
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Dynamic Leontief model
Formula: X(t+1) = (E-A)⁻¹ · [Y(t) + B·ΔK(t)]
B = capital coefficient matrix, ΔK(t) = investment vector in period t
1. Enter capital matrix B
Capital coefficients describe the capital requirement per unit of output
📁 Upload matrix B (CSV)
2. Simulation parameters
Simulation results
Detailed time series
Investment path
📊 Dynamic Leontief model (simple approach)
Recursive formulation without sector partitioning
xt+1 = B-1[(E+B-A)xt - ct]
⚠️ Prerequisite: the static model must be calculated (tab "Input")
1. Capital coefficient matrix B
Matrix B (n×n) indicates how much capital is required for production
Everything that is not given is treated in the model as investment demand. For data with exports and inventories (e.g. Destatis), C + V + E is correct; for closed examples (V = E = 0) both variants are the same.
Suggestion: C(0) ≈ 85% of Y from the static model (Y = C + I; the remaining ~15% are left as room for investment/capital formation – with C(0) = Y there is no room left for investment, which leads to negative output faster). The button "📊 Suggestion from Y" takes this value; with "Take from static model" for X(0) it is also suggested automatically.
C(t) = C(0) + Δ · t (linear, more stable than exponential)
C(t) = C(0) · (1 + g/100)^t (exponential, often unstable)
Manual input. Note: the simulation stops automatically earlier as soon as negative output occurs in a sector.
3. Start simulation
Results
📊 Dynamic Leontief model (DynModell3 approach)
Recursive formulation with partitioning into investment and consumer goods sectors
xt+1 = [E + (Bd)-1(E-A)] · xt - (Bd)-1 · ctd
⚠️ Prerequisite: the static model must be calculated (tab "Input")
1. Define sector structure
Partitioning of the sectors into investment goods and consumer goods sectors
Sectors 1 to r produce investment goods, sectors r+1 to n produce consumer goods
Partitioning:
2. Investment coefficient matrix BI
Matrix of the capital coefficients for investment goods sectors (r×r)
Everything that is not given is treated in the model as investment demand. For data with exports and inventories (e.g. Destatis), C + V + E is correct; for closed examples (V = E = 0) both variants are the same.
C(t) = C(0) + Δ · t (linear, more stable than exponential)
C(t) = C(0) · (1 + g/100)^t (exponential, often unstable)
⚠️ Important: Y(0) = C(0) + I(0) for the static model. When the matrix size is changed
(tab "Input") this field is cleared automatically, since old values no longer fit.
Note: check whether Y = C(0) + I(0) holds!
The simulation stops automatically as soon as a sector shows negative output.
4. Start simulation
Status:
Simulation results
Comparison: standard vs. DynModell3
Export results
Download the calculated matrices and vectors as CSV files.
Leontief model
Inverse Matrix (E-A)⁻¹
Output vector X
Dynamic model 1
Simulation results
Investment path
Matrix B (capital coefficients)
Dynamic model 2
Output time series X(t)
Consumption time series C(t)
Matrix B (capital coefficients)
Complete export
Dynamic model 3
Matrices
Matrix BI
Matrix Bd
(Bd)-1
Transition matrix T
Time series
Output X(t)
Consumption C(t)
Final demand Y
Complete export
All simulation data
Contains: all matrices, time series and parameters