Literature: P. Fleissner, W. Böhme, H.-U. Brautzsch, J. Höhne, J. Siassi, K. Stark: Input-Output-Analyse. Eine Einführung in Theorie und Anwendungen. Springer-Verlag, Vienna, New York 1993.
1 · Define structure
Manage databox ↗Flow tables (goods × sectors); static datasets under "More" as n goods, 1 sector.
CSV format (semicolon-separated): row 1 "n=…;m=…", row 2 column headers, then n·m rows of the
intermediate flow matrix, last row "Y;…" for final demand. Labels G1-B1 = good 1 from sector (Bereich) 1.
📦 Save in databox
Saves the current table (after "Calculate") with the complete four-dimensional structure
(n, m, Z, Y) as well as matrix A and Y in the central databox.
2 · Intermediate flow matrix Xijkl and final demand Yik (model M1)
3 · Results M1 (four-dimensional approach)
Xik = Σ row(Xijkl) + Yik
a(ik)(jl) = Xijkl / Xjl
L = (I − A)−1
Continue to tab 2 to derive models M2–M9 from these data.
4 · Weight matrix W (share coefficients per supplying sector)
wijkl = Xijkl / Σk' Xij k'l — share of sector k in the total
delivery of good i to the use (j,l). Column sums per good equal 1.
5 · Forecast with changed final demand
Enter new final demand Y' — the resulting gross output X' = L · Y' is calculated,
assuming constant input coefficients A.
6 · Price valuation — Ap = ⟨p⟩A⟨p⟩⁻¹
One price pik per good and sector (G1-B1, G2-B1, …): goods from different sectors can
have different prices. Xpik = pik·Xik,
apijkl = aijkl·pik/pjl. Standard: 1.
Xp = ⟨p⟩·X (gross output in value units)
Zp = ⟨p⟩·Z (intermediate flow matrix in value units)
Ap = ⟨p⟩·A·⟨p⟩⁻¹ — price-valued input coefficients
Lp = (I − Ap)⁻¹
⚠ Uses the data from tab 1 — please define the structure there first and run "Calculate".
Assignment rules of the classification characteristics
Number of the approach
Assignment rule
Usual name of the model
M1
i,k → j,l
Four-dimensional approach
M2
i,k → j
—
M3
i,k → l
—
M4
i → j,l
Simplified scheme of the material-value balance
M5
i → j
Leontief model (product-by-product or functional representation)
M6
i → l
Rectangular input-output model
M7
k → j,l
—
M8
k → j
Rectangular input-output model
M9
k → l
Leontief model (industry-by-industry or institutional representation)
The assignment rule indicates which classification characteristics (good/sector) are retained on the input and
output side compared with the four-dimensional approach M1. M5 and M9 are the
classical (square) Leontief models; the others are rectangular.
Formation rules of the input coefficients
Every coefficient of a simpler model can be formed from aijkl (M1) by summation over the
dropped characteristic. If a characteristic is dropped on the output side, it must additionally be weighted with c or d
:
djl = Xjl / Xj· — share of sector l in the gross output of good j (Xj· = Σl Xjl)
cjl = Xjl / X·l — share of good j in the gross output of sector l (X·l = Σj Xjl)
M2/M3 apply c or d directly to M1; M5/M6 and M8/M9 build on the models M4 or M7, which have already been
simplified (on the input side). Arithmetically this corresponds exactly to forming
Sum(Xijkl)/Sum(Xjl) over the dropped indices — both ways give the same values.
⚠ Uses the data from tab 1 — please define the structure there first and run "Calculate".
General mechanism
The goods indices are split into two disjoint subsets N₁ and N₂ (N₁∪N₂=N, N₁∩N₂=∅).
Intermediate inputs flowing between the two groups are separated out of the respective partial table:
deliveries to the other group increase final demand (Y*), purchases from the
other group increase the primary input (P*) of the respective partial table.
Y*ik = Yik + ΣlΣj∈N₂ Xijkl for i∈N₁,k∈M
P*jl = ΣkΣi∈N₂ Xijkl for j∈N₁,l∈M
Symmetrically for partial table 2 (N₂) with the roles of N₁ and N₂ swapped. Each partial table is
then a complete Leontief system that can be solved on its own, X = (I−A)⁻¹Y*.
The basis is the table currently entered or loaded in tab 1. Assign goods
to a group N₁ — the rest automatically forms N₂:
Scheme: simplified make-use scheme (empty)
Structure: rows = goods, columns = sectors, plus final demand y
and gross output x. No values yet.
Define structure
Use-Matrix U (good × sector): use of a good as intermediate input in the respective sector.
Make-Matrix V (sector × good): production of a good in the respective sector. Structure independent
of tabs 1–3 — own number of goods/sectors.
U: row = good i, column = sector l. V: row = sector k, column = good i. y: final demand
by good. Balance: row sum(U)+y = column sum(V).
Use matrix U (n × m) + final demand y
Make-Matrix V (m × n)
Industry technology: each sector has a uniform input structure, regardless of which
good it produces. Commodity technology: each good is produced with the same input structure, regardless
of the sector in which it is produced (requires n = m).
Derived matrices
x (output by good)
g (output by sector)
D = V·⟨x⟩⁻¹ — market share matrix
B = U·⟨g⟩⁻¹ — use coefficients
C = V'·⟨g⟩⁻¹ — production structure
A (good × good)
aij: input of good i required to produce one unit of good j.
A (sector × sector)
akl: input from sector k required to produce one unit of the output of sector l.
Lii = (I−Aii)⁻¹, Xii = Lii·y
lij: total (direct + indirect) input of good i required to meet one unit of final demand for good j.
Lll = (I−All)⁻¹, y' = D·y, Xll = Lll·y'
lkl: total (direct + indirect) input from sector k required to meet one unit of final demand for sector l.
Define structure
Own structure, independent of tab 1. Starting point for the input-output scheme variant 1 —
preliminary stage of the material-value balance. If desired, the current data from
tab 1 can also be taken over.
Intermediate flow matrix and final demand
Prices (for sector rows Q3/Q4 and the value representation)
One price pik per good and sector (G1-B1, G2-B1, …). The sector rows are formed in value terms with these prices:
Sectork[j,l] = Σi pik · Xijkl (over all goods i, not only N₁).
Assign goods to N₁ — the rest forms N₂ (residual production):
Scheme A — input-output scheme variant 1 (empty)
Four quadrants: Q1 (A1x) = good-related intermediate inputs of N₁; Q2 (P2x) = deliveries from N₁ to N₂ aggregated
by sector; Q3 (Q2x) = deliveries from N₂ to N₁ aggregated by sector;
Q4 (A2x) = sector-related intermediate inputs from the full table (analogous to model M9).
In addition: residual production (aggregated delivery to N₂), Y (final demand), X (gross output).
Calculation
Table A (N₁ rows): for each sector k completely in physical units — Q1 (N₁→N₁) plus for each sector l
a "residual production" column (N₁→N₂, aggregated). Table B (material-value balance):
the same rows, but summed over the sectors k (M4 principle). The associated sector rows
(Q3/Q4) are in value terms and require price data — shown here as example values only.
Formula 1: Res.prod.ikl = Σj∈N₂ Xijkl — delivery from (i,k)∈N₁ to N₂, per sector l
Formula 2: Xi (table B) = Σk Xik (table A) — aggregation over sectors
aijkl = Xijkl/Xjl: input coefficient as in the four-dimensional model (tab 1).
ai,Rl = Res.prod.ikl / X·l: share of residual production in the intermediate inputs of sector l.