Interregional Input–Output Models

Abstract

This chapter presents and critically evaluates the economic assumptions and applicability of a series of regional and interregional interindustry models. It begins with the demand-driven, single-region Leontief quantity model and its cost-push price dual. Then Section 45.4 discusses the ideal, full information, interregional input–output model with interregional spillover and feedback effects at length, and compares it with the requirements and assumptions of more limited information, multiregional input–output models. Section 45.5 discusses how to construct and add an interregional consumption function to obtain the type II interregional interindustry model. Section 45.6 outlines further extensions, all through to the most complex price-quantity interacting interregional demo-economic model LINE. Finally, an Appendix presents the microeconomic foundation for the Leontief model and compares it with the alternative supply-driven quantity model and its demand-pull price dual.

Appendix: The Microeconomic Foundation of the Leontief and the Ghosh IO model

The basic Leontief price and quantity model for a closed economy, introduced in Sects. 45.2 and 45.3, may be derived from microeconomics by assuming that all firms in each industry sell that industry’s single homogeneous output under full competition, while they minimize their cost at given prices under a Walras-Leontief production function:

$$ {x_j}=\min ({z_{ij }}/{a_{ij }},\forall i;{v_{pj }}/{c_{pj }},\forall p) $$

(45.22)

This results in a perfectly elastic supply of that single homogeneous output and a perfectly inelastic demand for intermediate and primary inputs, z _ij and c _pj , under fixed input ratios, a _ij and c _pj . Consequently, any change in the exogenous primary input prices is entirely and precisely passed on to all intermediate and final markets for the output of that firm. The left-hand side of Fig. 45.8 summarizes these individual firm assumptions and adds the assumptions about what is determined exogenously and what follows endogenously for the economy as a whole.

Fig. 45.8

Assumptions of the basic Leontief and Ghosh models for market economies

The interregional model extension, introduced in Sect. 45.4, adds trade coefficients to the basic Leontief model for a closed economy. The theoretical foundation for assuming trade coefficients to be fixed is less convincing than that for the technical coefficients by means of Eq. (45.22). It may be assumed that the output of, for example, agriculture is a different product in each different region. The trade coefficients will then have a technical character and will be fixed for the same reason. As each cell then relates to different goods, this assumption fits best with the “ideal,” full information, interregional IO model. It may also be assumed that the products of, for example, agriculture in different regions are close substitutes for each other. The trade coefficients will then be fixed only for as long as the relative prices of agricultural outputs from different regions remain unchanged. As relative prices will influence all trade coefficients along a row of the IO table in the same manner, this assumption fits best with the limited information, multiregional IO model.

The left-hand side of Fig. 45.9 shows the implications of the above assumptions for the working of, for example, an individual intermediate input market. The vertical demand for these inputs is determined by the Leontief quantity model, whereas the horizontal supply of these inputs is determined by the Leontief price model. Any change in the demand of the purchasers is matched exactly by a corresponding change in its supply, without any change in the price asked by the suppliers. Hence, demand drives the quantity model. On the other side of the market, any change in the price asked by the suppliers is accepted by its purchasers, without any effect on their demand for this input. Hence, cost pushes the price model. Clearly, in the short run, this is not a realistic model unless there is excess capacity on all relevant primary input markets, whereas, in the long run, this model is only realistic if the relative prices of the primary inputs do not change.

Fig. 45.9

The functioning of markets in the basic two input–output models. (a) The Leontief IO model, (b) The Ghosh IO model

The obvious follow-up question is whether the alternative IO quantity model of Ghosh (1958), and its dual price model (Oosterhaven 1996), offers a more plausible alternative. Ghosh developed his alternative IO model for the essentially centrally planned Indian economy of that time. Here we interpret the Ghosh model as a model for a market economy. As such it represents the pure opposite of the Leontief model, as can be seen by comparing the right- and left-hand side of Figs. 45.8 and 45.9.

In the Ghosh quantity model, the homogeneity assumption is made for all inputs along the columns of the IOT, instead of for all outputs along the rows of the IOT, as in the Leontief model. This implies that all inputs are perfect substitutes for each other. Hence, factories may run without labor, and cars may run without gasoline. Next, the Ghosh model assumes perfect complementarity of the outputs along the rows of the IO table, which is technically plausible for chemical industries, but which has to be based on a marketing desire to service all markets with the same constant market share for other industries. This is only possible if this supply of outputs is confronted with a perfectly elastic demand for them. Hence, supply drives the Ghosh quantity model. See the right-hand side of Fig. 45.8 for the remaining assumptions.

The mathematics of the Ghosh quantity model is far simpler than its economics, if only because it is the pure opposite of the Leontief model. Its solution reads as follows (see Oosterhaven 1996 for details):

$$ {\mathbf x}^\prime={\mathbf v}^\prime{{\left( {{\mathbf I}-{\mathbf B}} \right)}^{-1 }}={\mathbf v}^\prime{\mathbf G}\ \quad{\rm and}\quad {\mathbf i}^{\prime}\mathbf{Y}=\mathbf{v}^\prime\mathbf{G}\;\mathbf{D} $$

(45.23)

where B = N × N-matrix of intermediate output coefficients, D = N × Q-matrix of final output coefficients, and G = the so-called Ghosh inverse. In contrast to the Leontief inverse, which may be used as a measure of the backward linkages of each sector with its direct and indirect suppliers along the columns of the IOT, the Ghosh inverse provides an indication of each sector’s direct and indirect forward linkages with its customers, along the rows of the IOT. When used to measure forward linkages, in causal terms, the Ghosh model is best interpreted as a cost-push IO price model measured in values, instead of in prices as in Sect. 45.3 (see Dietzenbacher 1997).

The price version of the Ghosh model, which is called the demand-pull IO price model, is the pure opposite of the Leontief price model. Its solution reads as (see Oosterhaven 1996 for details)

$$ \mathbf{p} = {{\left( {\mathbf{I}-\mathbf{B}} \right)}^{-1 }}\mathbf{D}\ {{\mathbf{p}}_{\mathbf{y}}}=\mathbf{G}\ \mathbf{D}\ {{\mathbf{p}}_{\mathbf{y}}} $$

(45.24)

where p _y = Q-vector of (index) prices for total final use per category. As opposed to the cost-push model, where p refers to the price for each sector’s single homogeneous output, p in Eq. (45.24) relates to the price for each industry’s single homogeneous input. Furthermore, as opposed to the cost-push model, where primary input was homogeneous across the rows, here final use is homogeneous across each column of the IO table. This assumption implies that not only firms but also consumers may drive cars without gasoline and run home appliances without electricity. See Fig. 45.8 for the remaining assumptions.

Finally, each IO market in the Ghosh model thus functions as in the right-hand side of Fig. 45.9. Prices and quantities move independently. Demand is perfectly price elastic. This means that there is infinite demand at the going market price, which is a good description of the functioning of the butter mountains and the milk lakes of the old common agricultural policy of the EU. Supply, on the other hand, is perfectly inelastic to price changes. Clearly, the Ghosh model does not offer a plausible alternative to the Leontief model, but studying it does enlarge our understanding of the nature of the Leontief model.