A non-possibility theorem for joint stability in interindustry models

Abstract

Joint-stability in interindustry modeling relates to the mutual simultaneous consistency of the demand-driven and supply-driven models of Leontief and Ghosh, respectively. Previous work has claimed joint-stability to be an acceptable assumption from the empirical viewpoint, provided only small changes in exogenous variables are considered. We show in this note, however, that the issue has deeper theoretical roots, offering a mathematical demonstration that shows the impossibility of mutual consistency between demand-driven and supply-driven models. Mutual consistency would entail that coefficient matrices under both model versions would remain constant when the economic equilibrium changes following an external shock.

Introduction

Interindustry models are essentially of two types. The demand-driven model of Leontief [1] takes final demand as external with gross outputs and primary factors use responding to accommodate demand while keeping the demand-supply balance in check. In contrast, Ghosh [2] takes primary factors as external whereas gross inputs and final demand adjust to the availability of primary factors in production. Clearly, these two models emphasize different driving forces while attempting to determine total activity levels in both cases. In the Leontief version, activity levels refer to total or gross outputs, as seen from the use perspective. In Ghosh, they refer to total inputs, or the resource perspective. We know from basic National Accounting rules that gross outputs and gross inputs will necessarily coincide in equilibrium. It is therefore from this accounting connection that the two competing models naturally arise. It is also known that the behavioral information of both models is not independent. Equilibrium changes in the demand-driven model will modify the underlying coefficient matrix in the supply-driven version, and vice versa. Thus, mutually compatible coefficient matrices, in the sense that they would both remain unaltered when facing external shocks, would guarantee quantitative applications with a sound theoretical basis and a common accounting platform. This constancy of the technical and allocation coefficients is the property known as joint stability in the literature.

Previous research by Chen and Rose [3], [4], Bon [5], and Rose and Allison [6], among others, has correctly identified a version of the joint-stability condition but has not taken it to its theoretical limit. In fact, it has been argued the condition to be sufficiently acceptable for empirical quantitative work as long as only small changes in external parameters are considered. We consider this conclusion to be faulty and prove so by formally showing its theoretical unsuitability, that is to say, any external change, however small, would never keep joint-stability. In Section 2 we introduce the elements of the discussion and redefine the joint-stability condition. Section 3 explores the implications of the condition and presents the main theoretical result showing the condition to be contradictory with basic axioms. Section 4 summarizes.

An n-sector interindustry economy $\mathcal{E}(n)$ is characterized by a n × n matrix of bilateral aggregate flows Z, a (column) n vector of final demands f and a (row) n vector of primary factors use ${\mathbf{v}}^{\prime }$, or value-added. In compact expression we may represent this economy by $\mathcal{E}(n)=(\mathbf{Z}\text{,}\mathbf{f}\text{,}{\mathbf{v}}^{\prime })$. Matrix Z = (z_ij) contains interindustry exchanges between sectors i and j. The balance in the national accounting identities ensures the following relationship holds true for all i:

$${\displaystyle \sum _{j=1}^n}{z}_{\mathit{ij}}+f_i={\displaystyle \sum _{j=1}^n}{z}_{\mathit{ji}}+v_i=x_i\text{,}$$

where x_i stand for benchmark gross output (left-hand side) and gross input (rigth-hand side) level for sector i. We introduce now behavioral assumptions. From the perspective of production (i.e. inputs) we define a n × n technical coefficient matrix a_ij = [A]_ij by setting $a_{\mathit{ij}}=z_{\mathit{ij}}/x_j$. Similarly, from the perspective of distribution (i.e. outputs), we introduce a n × n allocation matrix b_ji = [B]_ji by taking $b_{\mathit{ji}}=z_{\mathit{ji}}/x_j$. Introducing these two definitions in expression (1) we transform it into two behavioral equations:

$${\displaystyle \sum _{j=1}^n}{a}_{\mathit{ij}}·x_j+f_i=x_i\text{,}$$

$${\displaystyle \sum _{j=1}^n}{b}_{\mathit{ji}}·x_j+v_i=x_i\text{.}$$

In compact matrix notation we can write (and solve) them as:

$$\mathbf{x}=\mathbf{A}·\mathbf{x}+\mathbf{f}=(\mathbf{I}-\mathbf{A}{)}^{-1}·\mathbf{f}\text{,}$$

$${\mathbf{x}}^{\prime }={\mathbf{x}}^{\prime }·\mathbf{B}+{\mathbf{v}}^{\prime }={\mathbf{v}}^{\prime }·(\mathbf{I}-\mathbf{B}{)}^{-1}\text{.}$$

The first of these two expressions is the basic Leontief quantity model and the second one corresponds to the quantity model of Ghosh.¹ In Leontief’s model total production is determined as a result of demand-driven (i.e. f) inputs adjustments whereas in Ghosh is a consequence of supply-driven (i.e. ${\mathbf{v}}^{\prime }$) allocation adjustments in output. If we use $\mathbf{X}$ as the diagonalized matrix version of the gross output vector x, we can easily check from the definitions of A and B that $\mathbf{A}=\mathbf{Z}·{\mathbf{X}}^{-1}$ and $\mathbf{B}={\mathbf{X}}^{-\mathbf{1}}·\mathbf{Z}$. It is therefore immediate that A and B are similar matrices through change of basis matrices $\mathbf{X}$ and ${\mathbf{X}}^{-1}$:

$$\mathbf{B}={\mathbf{X}}^{-1}·\mathbf{A}·\mathbf{X}\text{.}$$

It can quickly be seen that for the same bases, similarity of matrices B and A implies similarity of (I-A) and (I-B) and, provided B and A are invertible, similarity of ${\mathbf{B}}^{-1}$and ${\mathbf{A}}^{-1}$ as well. As a combined result, similarity on the Leontief and Ghosh inverses, if they exist, also follows for the same bases:

$$(\mathbf{I}-\mathbf{B}{)}^{-1}={\mathbf{X}}^{-1}·(\mathbf{I}-\mathbf{A}{)}^{-1}·\mathbf{X}\text{.}$$

Take now expression (5) and substitute into Eq. (3a):

$${\mathbf{x}}^{\prime }={\mathbf{v}}^{\prime }·(\mathbf{I}-\mathbf{B}{)}^{-1}={\mathbf{v}}^{\prime }·({\mathbf{X}}^{-1}·(\mathbf{I}-\mathbf{A}{)}^{-1}·\mathbf{X})\text{.}$$

For simplicity, let $\alpha$ denote now the coefficients of the Leontief inverse $\alpha =(\mathbf{I}-\mathbf{A}{)}^{-1}$ so that (6a) becomes:

$${\mathbf{x}}^{\prime }={\mathbf{v}}^{\prime }·({\mathbf{X}}^{-\mathbf{1}}·\mathit{\alpha }·\mathbf{X})\text{.}$$

Similarly, let $\mathit{\beta }=(\mathbf{I}-\mathbf{B}{)}^{-1}$ stand for the Ghosh inverse matrix. If we now expand (6b) and write the equivalent algebraic relationship, we obtain

$$x_j={\displaystyle \sum _{i=1}^n}{v}_i·x_j·{\alpha }_{\mathit{ij}}·\frac{1}{x_i}={\displaystyle \sum _{i=1}^n}{v}_i·{\alpha }_{\mathit{ij}}·\left(\frac{x_j}{x_i}\right)={\displaystyle \sum _{i=1}^n}{v}_i·{\alpha }_{\mathit{ij}}·{\gamma }_i^j\text{,}$$

where ${\gamma }_i^j$ represents the output ratio between sectors j and i , i.e. ${\gamma }_i^j=x_j/x_i$. Take the partial derivatives first in expression (3a) and then in (7) to obtain:

$$\frac{\partial x_j}{\partial v_i}={\beta }_{\mathit{ij}}={\alpha }_{\mathit{ij}}·{\gamma }_i^j\text{.}$$

Both interindustry models will appear to be simultaneously equivalent in their partial effects, or jointly stable, provided the output ratios ${\gamma }_i^j=x_j/x_i$ remain unaltered after a change in sector i’s value-added. Should these ratios change following an external shock in value-added, (8) would imply that constancy of coefficients does not follow. Notice that this would be possible only if in the new equilibrium output quantities do not change or changes are proportional everywhere, i.e. a balanced growth situation. This is the same type of conclusion reached in [3], [4], [5], [6] but we have used the coefficients of the inverse matrices, $\mathit{\alpha }$ and $\mathit{\beta }$, instead of the direct input and allocation coefficients of matrices A and B. This novel presentation will allow us now to further explore the implications of the constancy of the output ratios.