# 2  An Input-Output Approach

This section presents the framework of analysis used to compute minimum disruption changes. Subsection 2.1 derives an expression for total carbon dioxide emissions, using an input-output approach. Subsection 2.2 derives the minimum disruption changes to final demands necessary to achieve a required rate of reduction in total emissions, and subject to growth and employment constraints. Finally, subsection 2.3 examines the changes in the fuel use coefficients required to achieve a target carbon dioxide emissions reduction.

## 2.1  Total Carbon Dioxide Emissions

Consider increasing the final consumption of a good by \$1. The problem is to evaluate how much carbon dioxide this would involve. This increase in the final demand by \$1 involves a larger increase in the gross, or total output, of the good - as well as requiring increases in the outputs of other goods. This is because intermediate goods, including the particular good of interest, are needed in the production process. The extent to which there is an increase in carbon dioxide depends also on the intermediate requirements of all goods which are themselves intermediate requirements for the particular good. Indeed, the sequence of intermediate requirements continues until it ‘works itself out’, that is, the additional amounts needed become negligible. This is in fact a standard multiplier process. It can be set out formally as follows.

An industry’s gross output derives from both intermediate output which serves as input to other industries and final demand. Let denote the value of output flowing from industry to industry and let denote the value of final demand, by consumers, for the output of industry . The value of an industry’s gross output, , may therefore be expressed as the sum of intermediate and final demands:

(1)

The direct requirement co-efficient, , measures the value of output from industry directly required to produce \$1 worth of output in industry. Hence:

(2)

Using (2) to write and substituting the resulting expression into equation gives gross output as:

(3)

Let and denote the n-element vectors of and respectively. Further, let denote the matrix of the direct requirement coefficients, . These definitions enable the system of equations described in equation (3) to be expressed in matrix notation as:

(4)

Continuous substitution for on the right-hand side of equation (4) produces the following geometric sequence:

(5)

If the condition is satisfied, the system is productive and the non-negative solution is:[3]

(6)

and      is the matrix multiplier required.

Let denote thematrix of energy requirements (in PJs) for industries across fossil fuel types. Let denote the k-element vector of CO2 emissions (tonnes of carbon dioxide) per unit of energy (PJ) associated with each of the fossil fuels.

Multiplying the transpose of the vector by the transpose of thematrix gives the following row vector which contains the carbon dioxide emissions per unit of gross output from each industry:

(7)

Total carbon dioxide emissions, , can then be obtained by post-multiplying the above row vector by the column vector of gross output, :

(8)     >

The term in square brackets gives the row vector, , of the carbon dioxide intensities:

(9)     >

Equation (8) is used in determining the necessary structural changes to achieve a specified reduction in emissions. Proops et al (1993, pp.11-12) identified three main areas where a change in economic structure might give rise to reductions in carbon dioxide emissions. First, there are changes to final demands, . Second, there are changes to the efficiency of fuel use, . Third, changes to the structure of inter-industry trading, can be made.

The objective is to minimise the disruption to industries with regard to one of these variables while achieving a specified reduction in emissions. Disruption to any variable, say, in industry is measured in terms of the proportional change in that variable, . In specifying an objective function, Proops et al (1993, p.228) adopted a quadratic cost function, but it is useful to consider the more general form given by:

(10)

The term is simply a scaling factor which drops out in differentiation. This objective function assumes that there is an equal social cost associated with a 1 percentage point change in a certain variable, irrespective of the industry.[4]

#### Notes

• [3]This is given from the solution to the geometric matrix series , which must be non-negative given that all elements of are either zero or positive. For the system to be productive it is not merely sufficient for  to have a solution. The convergence requirement is equivalent to the Hawkin-Simons conditions.
• [4]It might be argued that there should be some weighting attached to the different industries, according to each industry's proportional contribution to the total level of an appropriate variable, such as aggregate employment. However, the method imposes constraints on such variables, so that further weighting is not necessary. Indeed, it can be shown that such further weighting is not possible if the weighting mechanism desired uses the same variable as that already accounted for in the constraint.
Page top