Price re-interpretations of the basic IO quantity models result in the ultimate input-output equations

Figures & data

Table 1. Input–output table with macro totals.

	Industries	Final demand	Total output
Industries	$z_{\mathrm{ij}}\in \mathbf{\text{Z}}$	$y_{\mathrm{iq}}\in \mathbf{\text{Y}}$	$x_i\in \mathbf{\text{x}}$
Primary supply	$v_{\mathrm{pj}}\in \mathbf{\text{V}}$	$r_{\mathrm{pq}}\in \mathbf{\text{R}}$	M Y
Total input	$x_j\in {\mathbf{\text{x}}}^{\mathrm{\prime }}$	C I G E

Figure 1. Common causal structure of the demand-driven IO price and quantity model.

Figure 2. Common causal structure of the supply-driven IO price and quantity model.

Table

Demand-driven quantity & cost-push price model:	Supply-driven quantity & revenue-pull price model:
For the individual firm:
given demand for its single homogeneous output i.e. perfect substitution among all outputs full complementarity of all inputs (fixed input ratios) cost minimization at given input prices derived demand for inputs (backward linkages) full competition, i.e. forward passing on of all input price changes into the single output price	given supply of its single homogeneous input, i.e. perfect substitution among all inputs perfect jointness of all outputs (fixed output ratios) revenue maximization at given output prices derived supply of outputs (forward linkages) full competition, i.e. backward passing on of all output price changes into the single input price
For the economy as a whole:
exogenous demand for final outputs per industry endogenous demand for all inputs per industry perfectly elastic supply of all primary inputs, i.e. exogenous primary input prices endogenous total output prices and quantities	exogenous supply of primary inputs per industry endogenous supply of all outputs per industry perfectly elastic demand for all final outputs, i.e. exogenous final output prices endogenous total input prices and quantities
Solution of the two Leontief models:	Solution of the two Ghosh models:
$\mathbf{v}=\mathbf{C}(\mathbf{I}-\mathbf{A}{)}^{-1}\mathbf{y}$, with $\mathbf{y}=\mathbf{Y}\mathbf{i}$ and $\mathbf{v}=\mathbf{V}\mathbf{i}$ ${{\mathbf{p}}^{\mathrm{\prime }}}_y={\mathbf{p}}^{\mathrm{\prime }}={{\mathbf{p}}^{\mathrm{\prime }}}_v\mathbf{C}(\mathbf{I}-\mathbf{A}{)}^{-1}$	${\mathbf{y}}^{\mathrm{\prime }}={\mathbf{v}}^{\mathrm{\prime }}(\mathbf{I}-\mathbf{B}{)}^{-1}\mathbf{D}$, with ${\mathbf{y}}^{\mathrm{\prime }}={\mathbf{i}}^{\mathrm{\prime }}\mathbf{Y}$ and ${\mathbf{v}}^{\mathrm{\prime }}={\mathbf{i}}^{\mathrm{\prime }}\mathbf{V}$ ${\mathbf{p}}_{v^{\mathrm{\prime }}}=\mathbf{p}=(\mathbf{I}-\mathbf{B}{)}^{-1}\mathbf{D}{\mathbf{p}}_{y^{\mathrm{\prime }}}$

Source: Oosterhaven (1989 and 2022, Ch. 7).

Oosterhaven, J. (1989). The supply-driven input–output model: A new interpretation but still implausible. Journal of Regional Science, 29(3), 459–465. https://doi.org/10.1111/j.1467-9787.1989.tb01391.x

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Oosterhaven, J. (2022). Rethinking input–output analysis, a spatial perspective, Second edition. Advances in Spatial Science. Springer.

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