COMPARISON TO DEMAND-DRIVEN I-O MODELS
FUNDAMENTAL RELATIONSHIPS AND NOTATION
Demand-driven and supply-driven I-O models both rely on the same underlying data, which describe the relationships between industries and institutions in an economy. Each industry’s total output is the sum of the amount of its output that is used as an input by other industries (intermediate demand) and the amount of its output that is used to meet final demand (either as a final product used by local institutions (e.g., households, government, investment) or exported outside the region). Following the notation in Miller and Blair (2022), for a single industry i we have xi = ∑zij + fi where:
xi = the total output (value of production) of industry i
zij = the intermediate demand for industry I’s product by industry j
fi = the final demand for industry I’s product
Extending this to all n industries in an economy, we have X = Zi + f, where
X = vector of output by industry
Z = the direct requirements matrix
f = vector of final demands for each industry
i = vector of 1s (a summation vector)
DEMAND-DRIVEN I-O MODELS
In the standard I-O model, industry output is assumed to be demand-driven, meaning that industry output (value of production) is determined by final demand for goods and services. Such models are also sometimes referred to as demand-pull models since final demand pulls industries to produce output.
In the demand-driven I-O model, X = (I – A)-1f, where
X = vector of output by industry
A = matrix of technical coefficients
f = vector of final demands for each industry
I = identity matrix
Each element in A is calculated by dividing each input purchase value by the purchasing industry’s total output (i.e., by dividing each element in a column by that column’s total). The resulting coefficients are known as input coefficients or technical coefficients. As such, element aij represents industry j’s purchase from industry i as a proportion of industry j’s total output. The (I – A)-1 matrix is known as the Leontief Inverse, after the economist Wassily Leontief who won a Nobel Prize for his research on I-O models. The (I – A)-1 matrix is also sometimes called the output multiplier matrix. Multiplying this matrix by the vector of final demands yields the total industry output required by each industry to meet those final demands. Summing down the columns of the output multiplier matrix yields the summary (or total) output multipliers by industry.
The basic assumption of the demand-side approach is that the input distributions in A are stable; if the output of industry i were to double, then the purchases i makes from each of the industries that supply inputs to i will also double. Supplies are assumed to be unlimited and final demands are assumed to be the limiting factor (i.e., the driver).
SUPPLY-DRIVEN I-O MODELS
Supply-driven I-O models are an alternative to the standard I-O model in which industry output is assumed to be supply-driven, meaning that industry outputs determine the demand for goods and services. Such models are sometimes referred to as supply-push models since industry output pushes demand.
Supply-driven models are based on the same matrix of direct requirements that underpins the demand-driven model, but rather than dividing each element by its column total to get technical coefficients, each element is divided by its row total to get allocation coefficients. In the demand-driven model, the focal equation X = (I – A)-1f derives from a rearrangement of the equation X = AX + f, interpreted to mean that each industry’s output is the sum of the intermediate demand and final demand for its output.
The analogous equation in the supply-driven model is X = XB + v, interpreted to mean that each industry’s output is the sum of its intermediate expenditures and value added, with each element bij in B representing industry i’s sales to industry j as a proportion of industry i’s total output. Rearranging, we have X = (I – B’)-1v, with B being transposed as necessary for the matrix calculations. The (I – B’)-1 matrix has been called the input multiplier matrix, the supply multiplier matrix, or the Ghosh Inverse, named for the economist Ambica Ghosh who presented this alternative I-O model in 1958. Element gij represents the value of production that comes about in sector j per unit of primary input (value-added) use by sector i.
Multiplying this matrix by the vector of value added (primary inputs) yields the total industry output subsequently produced by each industry resulting from those primary inputs. Summing across the rows of the input multiplier matrix yields the summary (or total) input multipliers by industry, which represent the effect on total output throughout all industries of the economy that would be associated with a $1.00 change in the value added (primary inputs) of sector i.
The basic assumption of the supply-side approach is that the output distributions in B are stable; if the output of industry i were to double, then the sales from i to each of the industries that purchases from i will also double. Thus, instead of fixed input coefficients – an assumption of the demand-side model – fixed output coefficients are assumed in the supply-side model (Miller and Blair, 2022, pp. 290-291). In this model, it is demand that is assumed to be unlimited, and supplies are assumed to be the limiting factor (i.e., the driver).
USES AND LIMITATIONS
LIMITATIONS
The main issue with supply-side models concerns the concept of constant supply distribution patterns. Ghosh had in mind the context of a planned economy experiencing severe excess demand, with government-imposed restrictions on supply patterns, though this is not likely a very general situation in much of the modern world (Miller and Blair, 2009, p. 548). The problem is that increases in industry j’s purchases of primary inputs (value-added, v) are transmitted forward to output increases in all industries that buy from industry j, without any corresponding increases in those industries’ use of primary inputs. This is because the change in v is viewed as exogenous, which deviates from the notion of industry production functions where intermediate inputs and value-added are used in fixed proportions (Miller and Blair, 2022, p.295).
REINTERPRETATION AS A PRICE MODEL
To overcome the criticisms of the original view of the Ghosh model, Dietzenbacher (1997) proposed an alternative interpretation whereby the model is viewed not as a quantity model but as a price model. In this interpretation, the elements in the supply-driven model are viewed not as quantities, in which case elements in Δv are interpreted as changes in the amounts of primary inputs available to the economy and elements in Δx are interpreted as changes in quantities produced, but rather as values, in which case the elements in Δv reflect changes in the prices of primary inputs and elements in Δx reflect changes in the values of outputs.
While it is a stretch to think that producing a higher quantity of an input would guarantee the purchase of those additional units of the input, it is not hard to imagine that an increase in the value (price) of an input would cause an increase in the value of production (output) of the industries that purchase that input. In fact, all this requires is to assume that the purchasing industries would not pass the increased cost onto their own customers rather than take a hit to profit; under this assumption, it is then true by definition that output of other industries would increase, given that output is defined as the sum of intermediate expenditures (costs of inputs) and value-added – holding value-added constant, an increase in intermediate expenditures (thanks to the increase in value-added in the supplying industry) must increase output.
This reinterpretation has been called the Ghosh price (cost-push) model. Changes in primary input costs are transmitted throughout the economy as they are passed on by producers in the prices of their products that are purchased by other intermediate users, who in turn increase their prices accordingly, etc. In this case, we are assuming that all quantities are fixed and the model is used to assess the economy-wise effects of a change in primary input prices. In comparison, in the standard demand-driven model, all prices are fixed in an impact analysis and quantities change as a result of changes in the quantities of final demands.
HYPOTHETICAL EXTRACTION
One application where a quantity interpretation would be reasonably justified is that of hypothetical extraction (aka, contribution analysis), where the goal is to quantify the economy-wide impact if a particular industry were removed from that economy. A parallel to replacing column j in A with zeros to perform the standard hypothetical extraction would be to replace row j in B with zeroes.
One application of a supply-driven hypothetical extraction is in the context of disaster analysis. In the event of a natural disaster, a firm may shut down and its client businesses might lack an alternative supplier in the short-term, which would require those client businesses to shut down as well. For an example, see Rose and Wei (2012).
USING BACKWARD AND FORWARD LINKAGE MEASURES TO IDENTIFY KEY INDUSTRIES
Many studies that look to identify key industries by way of inter-industry linkages calculate both backward and forward linkages (generally in normalized form, taking into account the relative size of the industry) and then focus on those industries that score high on both measures (Miller and Blair, 2022, p. 305). In this way industries can be classified as 1) generally independent of other industries (low score on both linkage measures), 2) generally dependent on other industries (high score on both linkage measures), 3) generally dependent on interindustry supply (high score on backward linkage measure only), and 4) dependent on interindustry demand (high score on forward linkage measure only) (Miller and Blair, 2022, p. 306).
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REFERENCES
Dietzenbacher, E. 1997. “In Vindication of the Ghosh Model: A Reinterpretation as a Price Model,“ Journal of Regional Science, 37: 629-651.
Miller, R.E. and R.D. Blair. 2009. Input-Output Analysis: Foundations and Extensions, Second Edition. New York: Cambridge University Press.
Miller, R.E. and R.D. Blair. 2022. Input-Output Analysis: Foundations and Extensions, Third Edition. New York: Cambridge University Press.
Rose, A. and D. Wei. 2012. “Estimating the Economic Consequences of a Port Shutdown: The Special Role of Resilience.” Economic Systems Research, 25:2, 212-232.
Written September 3, 2025