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where C is total fossil fuel consumption, T is time and RP is the relative price of energy. A is a constant and a and b are parameters.

For all countries, elasticities (a and b) are negative, as expected. Hence, energy demand is inversely related to price and time. The 'time' coefficients vary between -5.7 per cent (France) and -1.6 per cent (Netherlands). The relative price elasticities are between 5 per cent (United Kingdom) and 16 per cent (United States and Belgium), with the average being around 13 per cent. However, the impact of price and time vary greatly across countries. Therefore, it appears that the cost of reducing fossil fuel consumption, even given a similar level of GDP growth, is very different among the G9 countries.

Next, some illustrative calculations are carried out to estimate the degree of annual price change that would be necessary in order to stabilize fossil fuel consumption given various GDP growth rates, based on the assumption that both time and price elasticities remain constant. This is shown in Table 8.7, and the calculations involved are summarized in Table 8.8. The results are highly sensitive to the GDP growth rate assumed. For growth of less than 2 per cent in GDP in the G9 as a whole, fossil fuel consumption would decrease

Table 8.7 Growth rate of the relative price of fossil fuels necessary to maintain fossil fuel consumption constant, given assumptions about real GDP growth


GDP growth rate (% p.a.) 2.0 2.5








- 3.3












- 2.4








-4.7 11.1



United Kingdom




United States

- -



Note: -, consumption is stable or decreasing at constant prices.

Note: -, consumption is stable or decreasing at constant prices.

Table 8.8 Calculations required to estimate price changes required to stabilize fossil fuel consumption How price variations are obtained in Table 8.7

The equilibrium relationship derived from the dynamic equation is the following:

The variations per annum of both fossil fuel consumption and relative price are derived from this equation, leading to

If the consumption is to remain constant from one year to the other, then A In(ct)= 0. Therefore the variations in price can be set equal to

For example consider the UK with GDP growth of 3 per cent per annum. The coefficient b (that with respect to relative price) is 0.045 and d (time coefficient) is -0.0258. This gives if fuel prices remained unchanged. Growth of 2 per cent leads to a tax being required only in the Netherlands.

In the event that trends continue Belgium, France and Japan show no increase in consumption at growth rates up to 3.5 per cent per annum. As discussed previously, this is thought to be due to a combination of increasing nuclear power and structural and technological change. With GDP growth at 3 per cent or more, prices in Germany, the Netherlands, Canada, the United Kingdom and Italy would also have to increase dramatically every year in order to maintain constant consumption. In several cases the required prices are far too high to be implemented, indicating the necessity for supply-side policies to complement price rises.


In this section some preliminary results for the United Kingdom which outline and illustrate the power of the endogenous technological progress model given in (8.1) and (8.2) will be presented. Taking the ECM model derived for the United Kingdom (given in Appendix 8A), the general form is

If Tt was simply a deterministic trend this would be a conventional model. Generating the trend as follows the model is much richer:

where rp, as before, is relative energy prices and m/y is the share of manufacturing output in total GDP. It is to be expected that an increase in energy prices increases the rate of growth of technological innovation and that as manufacturing in the United Kingdom declines the aggregate level of energy use also declines. The estimation of this model can be carried out using the Kalman filter where (8.7) is the measurement equation and (8.8) is the state equation. The formal estimation strategy is given below.

A standard state-space formulation is presented, with the appropriate Kalman filter equations for the univariate case, following Harvey (1987). Let

be the measurement equation where Yt is a measured variable, zt is the state vector of unobserved variables, 8 is a vector of parameters and et~NID(0, rt). The state equation is then given as

where y and Q are parameters, Wis extra observed variables (rp and m/y) and y~NID(0, Qt); Qt is sometimes referred to as the hyperparameters.

The appropriate Kalman filter prediction equations are then given by defining i as the best estimate of zt based on information up to t, and Pt as the covariance matrix of the estimate 4, and stating

Once the current observation on yt becomes available, these estimates can be updated using the following equations:

Jointly, equations (8.9)-(8.14) represent the Kalman filter equations. If the one-step-ahead prediction errors are then defined as

then the concentrated log-likelihood function can be shown to be proportional to log 1 = 1, log/, + N log I

where ft=a'Pt\t-1a+rt, N=T-k and k is the number of periods needed to derive estimates of the state vector; i.e. the likelihood function can be expressed as a function of the one-step-ahead prediction errors, suitably weighted.

Applying this estimation technique to the model outlined above for the United Kingdom yields the following results:

Measurement equation:

Ac, = 2.2828 — 0.t714(c/v),_( - 0.0077rp(_, + 0.8443Ac^,

Transition equations:

T= \.0022T,_i + 1.0055&., - 0.0573Arp, + 0.2366m/v, where m/y is the share of manufacturing output in total GDP.

Statistical diagnosis

Mean of residuals Sum of residuals Standard deviation Coefficient of skewness Coefficient of kurtosis Box-Jenkins normality test







The state-space representation does not differ very much from the ECM representation. One remarkable change is the greater weight attached to the share of consumption relative to GDP Apart from this variable, the other coefficients appear largely the same. However, the share of manufacturing output relative to GDP seems to play a non-negligible role in explaining the declining trend in energy consumption. This representation seems reasonable

Figure 8.16 Endogenous technical progress for the UK
Table 8.9 Autocorrelation function, Box-Pierce statistic and Ljung-Box statistic



















Box-Pierce statistic









Ljung-Box statistic









with respect to statistical criteria (see Statistical diagnosis). There are signs of non-normality, which was also the case with the ECM representation, although it was much stronger. The Box-Pierce statistic and the Ljung-Box statistic (Table 8.9) suggest some sign of fourth-order autocorrelation of the residuals, which may be due to the use of interpolated data.

Overall, the model seems promising. The path of the endogenous technological progress effect, which is graphed in Figure 8.16, shows a rapid increase in energy efficiency in the early 1980s as manufacturing declined as a share of total GDP and as energy prices were particularly high. As these two effects decline in the late 1980s the rate of increase in trend efficiency falls off dramatically. Clearly, projecting the future growth rate from the later levels will give a very different picture from using either the earlier or average growth rates.

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