📊 Static Leontief Model

Theory

Y = C + I + V + E (breakdown of final demand per sector)

• C = consumption (private households and government)
• I = gross fixed capital formation
• V = changes in inventories (can be negative)
• E = exports

If the dataset contains no E, it is calculated as a residual: E = Y − C − I − V. The small example datasets (ZIP) describe a closed economy with V = E = 0; the 5×5 example and the Destatis data contain all four components.

Input

Theory

Imports by product group

m_gj = imports of group g per unit of output of sector j (share of output)

Im_gj = m_gj · X_j    and    Im_j = Σ_g Im_gj

Example data: three groups (raw materials, intermediate products, services). Destatis: one group (imported intermediate inputs). At most three groups are evaluated.

Input

Theory

Cost structure per sector j (column of the input-output table)

X_j = Σ_i a_ijX_j + M_j + L_j + G_j + T_j

or in coefficients: Σ_i a_ij + m_j + l_j + g_j + t_j = 1

BWS_j = L_j + G_j + T_j

• Intermediate inputs = column sum of A times X_j
• M = imports (sum of the import groups from step 2)
• L = wages (compensation of employees)
• G = profits (gross operating surplus incl. depreciation)
• T = taxes less subsidies (can be negative)

Destatis: besides other taxes on production, T also contains taxes on products; L + G + T is therefore somewhat larger than gross value added in the national accounts.

Input

Theory

Price model (dual Leontief model)

• Quantity model: X = (E−A)⁻¹ · Y
• Price model: p^T = p^T · A + v^T  ⇒  p^T = v^T · (E−A)⁻¹

p = price index per sector (base 1), v = primary input costs per unit of output. From the databox, v = share of gross value added is used. Since imported intermediate inputs are not included, Σ_i a_ij + v_j < 1 and therefore p < 1. With v = value-added share plus import share, p = 1. For cost changes, Δp^T = Δv^T · (E−A)⁻¹ (e.g. a wage increase in one sector).

Input

Theory

Satellite accounts: environment and employment

f_j = indicator per unit of output of sector j

• Direct: f_j · X_j
• Indirect (first supply stage): Σ_i f_i · a_ij · X_j
• Total: direct + indirect

Total intensity over all supply stages: f^T · (E−A)⁻¹

Example data: CO2 (t), energy (MWh) and employment (persons employed) per unit of output. Destatis: persons employed per EUR million of output. At most three indicators are evaluated.

Input

Theory

Variation of individual coefficients a_ij by ±Δ% → effect on output X

Elasticity: ε = (ΔX/X) / (Δa/a)

Parameter

Theory

Variation of the final use components → effect on total output

Parameter

Theory

Variation of the import coefficients → effect on domestic production

Parameter

Theory

Variation of value added v → effect on price vector p

Parameter

Theory

Variation of the emission coefficients f → effect on environmental burden

Parameter