# 2.2  Disruptions to Final Demands

Consider first the problem of minimising the disruption to final demand. To impose no more than a constraint on the amount of emissions reduction is obviously not a case that should be considered seriously. In particular, the required final demand changes would all be negative. However, this case serves to introduce the basic approach adopted.[5]

Total emissions, when written in algebraic as opposed to matrix form are equal to , so that:

(11)

If is the required proportional change in total carbon dioxide emissions , the constraint can be written as:

(12)

where is 's share of emissions, and denotes the proportional change in final demand for industry . The Lagrangean for this problem is given by:

(13)

Differentiation gives the set of first-order conditions:

(14)

Multiplying equation by , adding over all industries, and solving for gives:

(15)

Substituting this result into the first-order condition gives the solution for the required proportional reduction in output of:

(16)

This result shows that the larger is , the smaller is the dispersion in the required rates of change. Therefore, increasing the power ultimately leads toward an equalisation of the proportional changes. Furthermore, when additional constraints are imposed, the first-order conditions cannot be solved explicitly. For this reason, the quadratic form is retained in this study, and the substitution of gives the result, as in Proops et al (1993, p.144), that:

(17)

## Carbon Dioxide and GDP Growth Targets

It is appropriate to include a constraint on GDP growth in addition to the constraint on the level of carbon dioxide emissions reduction. This constraint can be written as:

(18)

where represents the desired rate of growth, expressed as a weighted sum of the changes in final demands once again, with each weight being the proportion of that industry’s contribution to total GDP, that is . The Lagrangean for this problem is:

(19)

Differentiating with respect to each of the gives rise to the first-order conditions:

(20)

along with the two constraints relating to and . Using from the first-order conditions and substituting into the constraints gives the resulting two simultaneous equations:

(21)

If the determinant of this matrix is written as , the solutions for the Lagrange multipliers are:

(22)

The resulting multipliers can be substituted into the first-order conditions to solve for the s; see also Proops et al (1993, pp.234-235).[6]

## Carbon Dioxide, GDP and Employment Targets

An additional constraint concerns the rate of growth in employment, . This is expressed as:

(23)

where the weights are the levels of employment in each industry as a proportion of total employment. Minimising the disruption to final demands subject to all three constraints simultaneously, involves the Lagrangean:

(24)

In this case there are three Lagrangean multipliers, so that a set of three linear equations can be solved using matrix methods. The procedure is a simple extension of that described above; see also Proops et al (1993, pp.238-9).

# 2.3  Disruptions to Fuel-use Coefficients

As an alternative to minimising changes to the vector of final demands, consider minimising the change in fuel-use coefficients subject to a carbon dioxide emissions-reduction target. The direct fuel-use coefficients are embodied in the matrix, . The objective is to minimise:

(25)

where represents the proportional change in the production fuel requirement per unit of total demand of fuel in industry . The change is minimised subject to the constraint that a target proportional reduction, , in carbon dioxide emissions, attributable to changes in the production fuel-use coefficients, is achieved. Given that total emissions are , differentiation gives:

(26)

This can be rewritten as:

(27)

Hence is a weighted row and column sum of the production fuel-use-coefficients, with each weight given by , that is the proportional contribution to emissions of fuel in industry . The Lagrangean is therefore:

(28)

Following Proops et al (1993, pp.241, 144), solving this yields:

(29)

### Notes

• [5]Allowing for these factors handles the problem of weighting discussed above, as doing so implicitly attaches weights according to each industry's contribution to the total level of the constraint variable in question.
• [6]If weights equal to the proportional contribution of each industry to total GDP are attached to , the Lagrangean multipliers are not identified. That is, the constraints on the carbon dioxide emission target and the rate of growth of GDP, and , become equal to the sums of the Lagrangean multipliers, thus illustrating how additional weighting is not appropriate.
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