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The Official Journal of the Pan-Pacific Association of Input-Output Studies (PAPAIOS)

Table 4 Expressions for the satellite multiplier for models A–D

From: Analyzing the effects of the choice of model in the context of marginal changes in final demand

Model Formula for impact analysis in terms of \({\mathbf{\Delta h}}\) Formula for impact analysis in terms of \({\mathbf{\Delta d}}\)
A (PTA-p*p)   \(\Delta {\mathbf{e}}^{{\left( {\text{A}} \right)}} = {\mathbf{W}}\left( {{\mathbf{V}}^{\text{T}} - {\mathbf{U}}} \right)^{ - 1} \Delta {\mathbf{d}}\)
B (ITA-p*p)   \(\Delta {\mathbf{e}}^{{\left( {\text{B}} \right)}} = {\mathbf{W}}\left( {{\hat{\mathbf{q}}\mathbf{V}}^{ - 1} {\hat{\mathbf{g}}} - {\mathbf{U}}} \right)^{ - 1} \Delta {\mathbf{d}}\)
C (ISA-i*i) \(\Delta {\mathbf{w}}^{{\left( {\text{C}} \right)}} = {\mathbf{W}}\left( {{\mathbf{V}}^{\text{T}} - {\mathbf{U}}} \right)^{ - 1} {\hat{\mathbf{g}}\mathbf{V}}^{{ - {\text{T}}}} \Delta {\mathbf{h}}\) \(\Delta {\mathbf{w}}^{{\left( {\text{C}} \right)}} = {\mathbf{W}}\left( {{\mathbf{V}}^{\text{T}} - {\mathbf{U}}} \right)^{ - 1} \Delta {\mathbf{d}}\)
D (PSA-i*i) \(\Delta {\mathbf{w}}^{{\left( {\text{D}} \right)}} = {\mathbf{W}}\left( {{\hat{\mathbf{q}}\mathbf{V}}^{ - 1} {\hat{\mathbf{g}}} - {\mathbf{U}}} \right)^{ - 1} {\mathbf{V}}{\hat{\mathbf{q}}}^{ - 1} \Delta {\mathbf{h}}\) \(\Delta {\mathbf{w}}^{{\left( {\text{D}} \right)}} = {\mathbf{W}}\left( {{\hat{\mathbf{q}}\mathbf{V}}^{ - 1} {\hat{\mathbf{g}}} - {\mathbf{U}}} \right)^{ - 1} \Delta {\mathbf{d}}\)
  1. See “Appendix 1” for the proofs