Plim aippu p2P2i ffPpii

If p 0, the time-series estimate measures the short-run effect /;„,. If p 1 and v and x are cointegrated, then which measures the long-run effect. However, for values of p between zero and unity, yields a biased estimate of both the short-run and long-run effects, and for p>0 it lies somewhere between the two. Aggregating the distributed lag equations over time gives where = xiJT and -j = E ,-iy-1> xitIT. [n terms of the growth terms Arx=(xL q l;i -xu T-})/T, we now have where

For large T and assuming |p|<l. the covariances between the tenns in A;x, and will tend to zero, and the OLS estimate of /;, say in the cross-section regression will give a consistent estimate of the 'mean' long-run effect of xit onyt assuming that the micro coefficients /;,, and sit are independently distributed. More formally, we have piim <j?) = E(0w) = tfij where fy is the mean over the N groups of the coefficients of xi,t-J-.4

To illustrate the problem graphically, consider an extreme case. Suppose yit is the logarithm of real expenditure on energy and xit the logarithm of the relative price of energy. Suppose further that there is a J-curve effect in the sense that in the current period an increase in prices increases expenditure on energy

Figure 2.1 Time-series and cross-section estimates because consumers are locked into inflexible capital stocks. However, in the long run, after one period say, demand is elastic and they can adjust fully. Thus, assuming homogeneity of coefficients across groups, we have

The static time-series regression for each group is and the cross-section regression is

Tf nrirp« wpr^ serially uncorrrelated, i.e. p=0, all the time-series estii"2fpc positive, plim

("{) ~0t) > 0- and the cross-section estimates would be negative, plim as shown in

Figure 2.1.

Hsiao (1986:7, Figure 2.3) presents the same picture but interprets it differently, as the consequence of intercept heterogeneity. The probability limits of the time-series intercepts are plim (aj)=a0+fi]/(,, differing between groups and correlated with ■*/, the estimate of In this example, which admittedly is somewhat artificial, the intercept heterogeneity is induced by dynamic misspecification of the time-series model. This misspecification also induces the correlation between the time-series estimates of the intercept and the regressors, which causes the cross-section estimate to differ from the average of the time-series estimates.

Measurement error in the independent variable, which will produce a downward bias in the slope of a bivariate regression, can have similar effects. The size of the bias depends on the variance of the measurement error relative to the variance of the independent variable. In cross-section, the variance of the independent variables tends to be much larger than in time series; thus the measurement error bias is smaller. This is the essence of Friedman's explanation of the smaller estimate of the elasticity of

Figure 2.2 Non-linearity and the relationship between coefficients and regressors

Figure 2.2 Non-linearity and the relationship between coefficients and regressors consumption to income in time series than in cross-section. The true explanatory variable is permanent income, but actual income includes a transitory component viewed as a measurement error.

Figure 2.2 shows how non-linearity, not obvious from time series, can induce a relationship between the coefficients and the regressors. In this case, there is a positive relationship between the slopes and the level of the explanatory variable. Time-series estimates for countries with low average levels show a negative effect; for countries with moderate levels, no effect; and for countries with high levels, a positive effect. Such non-linearities would be apparent either from the cross-sections on the means or from the association between the coefficients and the regressors. In the case of energy such non-linearities may arise in the course of development as households and producers switch between non-commercial and commercial sources of energy.

Cross-sections and time series alone are each inherently unable to identify the effect of certain types of variables. In panels, it is common to have some variables which differ substantially across groups but vary little over time. In the time series estimated for each group, such effects will be picked up by the group-specific intercept, which will then be correlated with the relevant group-varying measure. An example might be the effect of population density on demand for gasoline; it is clearly important but in any one country it changes so slowly over time that time-series estimates will not be able to reveal its impact on gasoline demand. Other variables differ substantially over time but vary little over groups. In cross-sections estimated for each time period, such effects will be picked up by the year-specific intercept, which will be correlated with the relevant time-varying measure. An example is gasoline prices for a cross-section of individuals in a particular country. Variations between individuals in the price they face tend to be quite small or correlated with other relevant variables, e.g. isolated rural communities may face higher prices because of transport costs. A heterogeneous panel model which includes both types of variables may take the form y,., = at + Hrxh + y c, + + et where zi stands for the variable that differs across groups but not over time, and wt is the variable that varies over time but not across groups. Since zi is constant for each group, we cannot identify different group-

specific coefficients of z. However, for each group we can estimate

Then if we assume that ai=a+ni, where n is random and distributed independently of zi, we can estimate the regression a. = a + yh +1)¡

using the estimated ai for each group as dependent variable. This approach is widely used in the panel literature. See, for example, Hsiao (1986:50-2).

Because the traditional panel literature is concerned with cases where T is small, it has focused on heterogeneity of the intercepts. In cases where we have data fields, i.e. large N and T panels, the same type of procedure can be applied to the slope coefficients pi. In particular the assumption that the coefficients are random can be examined by regressions of the form

Under the null hypothesis that the coefficients are random b= 0 and the error //, contains two components:

the random differences between groups and the estimation error for a particular group. The variance of the first component is constant, but the variance of the second component differs between groups. This heteroscedasticity can be allowed for at the inference stage by weighting the estimates A- inversely by the estimate of their standard errors.


Most studies of energy demand are confined to OECD countries where the required data are more readily available. The books by Pindyck (1979) and Griffin (1979) represent good examples of the early work carried out on energy demand in OECD countries. A more recent review covering some of the developments over the past two decades is given by Watkins (1991), who also focuses on energy demand in the OECD countries. In this section we use a new data set not analysed previously, and examine the empirical importance of the issues discussed above for energy demand in ten Asian developing economies over the eighteen years 1973-90. The data, taken from the Asian Development Bank (1992), are described in a data appendix and are available on request. The dependent variable is E, the logarithm of final commercial energy consumption in tons of oil equivalent (t.o.e.). The explanatory variables are Y, the logarithm of per capita GDP in 1985 prices converted into US dollars using 1985 market exchange rates, and P, the logarithm of the average consumer price of energy in domestic currency per t.o.e. relative to the GDP deflator (1985=1), converted into dollars at 1985 exchange rates. Although it is real income and the relative energy price in domestic currency that influence demand, cross-section and pooled estimates require internationally comparable statistics—incomes and prices in a common currency—and there is an inevitable arbitrariness in the conversion to a common currency.5

The empirical results reported in this section can only be regarded as illustrative of the type of methodological issues involved, because of the high level of aggregation used. Residential, industrial and transportation demands for different types of fuel will differ systematically in ways that will bias the aggregate estimates. For instance, industrial customers can substitute between fuels in response to relative price changes more rapidly than can residential customers. However, the procedures illustrated here could be readily applied to energy demand equations, disaggregated by sectors and by fuel types. There may also be a problem of identification in those countries such as India where energy prices are controlled and prices may be adjusted (through changes in energy taxes) in response to variations in energy consumption. In such cases the estimated equations would be a mixture of demand and supply responses.

We estimated two dynamic specifications of the energy demand equations for each of the ten countries. The first specification, which we denote by Mj, regresses E on Y, P and E(-1), the lagged values of E. The second specification, which we denote by M2, adds the lagged values of Y and P to model M1. Table 2.1 gives the time-series estimates of M1 for each of the ten countries, together with some summary and diagnostic statistics.6 The short-term income elasticity estimates lie between 0.189 (Korea) and 1.109 (Bangladesh) and are statistically significant for seven of the ten countries. The short-term price elasticities are statistically significant in the case of five countries (Bangladesh, Indonesia, Korea, Philippines and Thailand), and have the correct negative signs except for India, Pakistan and Sri Lanka. The price elasticities for these three countries are estimated to be positive, but are not statistically significant. The estimates of the long-run elasticities together with their estimated standard errors are given in Table 2.2. The long-run income elasticities range from 0.65 (Sri Lanka) to 3.15 (Bangladesh) and the long-run price elasticities from 0.13 (Sri Lanka) to -0.68 (Korea). Thus there is considerable heterogeneity between these estimates, though the extreme values for Sri Lanka may reflect particular problems with data for that country, But considering the high level of aggregation and the rather doubtful quality of the underlying time-series observations at least for some of the countries, these are encouraging results and in general accord with the a priori theory. The individual country estimates for the long-run income elasticities are generally well determined and with a few exceptions do not differ significantly from unity. By contrast the precision of the estimates of the long-run price elasticities are rather low and are statistically significant only in the case of Indonesia, Philippines and Thailand. Given the long lags that are usually involved between price changes and changes in energy demand these results accord reasonably well with the view that time-series estimates tend to underestimate price elasticities.

Table 2.3 presents alternative mean estimates of the income and price effects for both dynamic specifications M1 and M2. The first two columns

Table 2.1 Individual country estimates of dynamic energy demand equations for ten Asian developing countriesa (197490)

Specification M1 Summary statistics

Table 2.1 Individual country estimates of dynamic energy demand equations for ten Asian developing countriesa (197490)

Specification M1 Summary statistics



















Hong Kongc
















































Specification Mj Summary statistics











































Sri Lanka
























Notes: aThe dependent variable is E, the logarithm of per capita final energy consumption in t.o.e.; P is the logarithm of the relative price and Y is the logarithm of real per capita income. See the Data appendix for further details of the variables and the data sources. Ti^-iigures in parentheses are standard errors. R2 is the adjusted square of the multiple correlation coefficient, Rz is the regression's standard error, and/2Jt?(l) is the Lagrange multiplier test of residual serial correlation of order 1.

bSignificant at the 1 per cent level.

^Estimated over the period 1979-90.

Table 2.2 Individual country estimates of the long-run income and price elasticitiesa

Based on the dynamic specification(M1) Based on the cointegratingb relations

Table 2.2 Individual country estimates of the long-run income and price elasticitiesa

Based on the dynamic specification(M1) Based on the cointegratingb relations












Hong Kongc

















































Sri Lanka












Based on the dynamic specification(M1) Income Price

Based on the cointegratingb relations Income Price



Notes: aBased on the coefficient estimates reported in Table 2.1.

bThe cointegration relations are estimated by Johansen's approach and are reported when the hypothesis of cointegration between E, P and Y is not rejected. Estimates of the cointegrating vectors are only given in cases where the presence of cointegration amongst E, P and Y could not be rejected. See Table 2.5 below.

cThe estimates for Hong Kong are based on the sample period 1979-90.

dThe estimates of the cointegration relation in the case of Korea are obtained assuming a second-order vector autoregressive model for E, Y and p.

present the weighted mean group estimates using the Swamy (1971) procedure. The long-run elasticities are calculated from the means of the short-run coefficients (see (2.5)). In this application the differences between the two ways of calculating the long-run effects discussed in section 2.2 are not significant. The unweighted means of the country-specific long-run income and price elasticities reported in Table 2.1 are 1. 43 and -0.29, respectively. The long-run estimates calculated from the unweighted means of the short-run coefficients are 1.27 and -0.18, respectively. The second two columns give the pooled OLS estimates. These estimates graphically illustrate the theoretical points made above about the effect of parameter heterogeneity combined with serial correlation in the regressors on pooled estimates. The coefficient of the lagged dependent variable (in the case of both specifications) is very close to unity. In

Table 2.3 'Average' estimates of dynamic energy demand equations for ten Asian countries, 1974-90a

Mean group0


Pooled OLS estimates

Pooled fixed effect estimates





























































Long-run elasticity estimates


































log-likelihood values

Mean group0 estimates

Pooled OLS estimates

Pooled fixed effect estimates

M1 M2

M1 M2

M1 M2

Number of

40 60

4 6

13 15



Notes: aThe dependent variable is E, the logarithm of per capita final energy consumption in t.o.e.; P is the logarithm of relative price and Y is the logarithm of real per capita income. P and Y were converted to dollars using 1985 exchange rates. The figures in parentheses are the standard errors. bThese are the weighted Swamy (1971) estimates based on individual country coefficient estimates.

Notes: aThe dependent variable is E, the logarithm of per capita final energy consumption in t.o.e.; P is the logarithm of relative price and Y is the logarithm of real per capita income. P and Y were converted to dollars using 1985 exchange rates. The figures in parentheses are the standard errors. bThese are the weighted Swamy (1971) estimates based on individual country coefficient estimates.

the case of M2, the coefficients of current price and income variables are not very different from the mean group estimates; the same, however, cannot be said about the coefficients of lagged price and income variables. The last two columns in Table 2.3 give the pooled fixed effect estimates. Allowing for country-specific fixed effects reduces the problem somewhat, and the long-run estimates are not significantly different from the mean group estimates, but the short-run dynamics implied by the fixed effect estimator are highly misleading. The hypothesis that the slope coefficients are equal (imposed by the fixed effect estimator) is rejected for both specifications. For example, in the case of model M2 the total maximized log-likelihood value for the individual time-series regressions is 344, and for the fixed effect model 279,

Table 2.4 'Average' estimates of long-run income and price elasticities based on static energy demand equations for ten Asian countriesa (1974-90)

Estimates Income


Long-run elasiticities LLF


Mean group








Pooled OLS estimates







Pooled fixed effects








Aggregate time-series















Notes: aBased on static regressions of E on Y and P. For the variable definition and data sources see the notes to

Table 2.1. LLF is the maximized log-likelihood values. bSwamy's (1971) weighted estimates.

Notes: aBased on static regressions of E on Y and P. For the variable definition and data sources see the notes to

Table 2.1. LLF is the maximized log-likelihood values. bSwamy's (1971) weighted estimates.

giving a log-likelihood ratio statistic of 130, which is approximately distributed as a x2 variate with 45 degrees of freedom. This would imply the rejection of the null hypothesis of homogeneous slope coefficients even at the XA per cent level, critical value 72.12.

Table 2.4 gives elasticity estimates based on static regressions. Although the hypothesis that all the lagged variables are insignificant in the individual time series is rejected, the static estimates are of some interest. First, if the variables are I(1), which they seem to be (see Table 2.5), and were cointegrated, the static estimates could be interpreted as cointegrating regressions.7 Second, were the coefficients independent of the regressors, all the procedures would provide unbiased estimators of the mean effect in a static regression. The first row of this table gives the mean group estimate of the average of the individual time series. The mean group estimates based on static coefficients are slightly lower than those based on long-run estimates from the dynamic equations, but not significantly so (1.172 in Table 2.4 compared with the estimates 1.301 or 1.298 in Table 2.3). The second and the third rows in Table 2.4 give the pooled OLS estimates (with common intercepts) and the pooled fixed effect estimates that allow for individual country intercepts. The fourth row gives the aggregate time-series estimates based on the time-series regression of the average values over the ten countries, and the last row in Table 2.4 gives the cross-section estimates.

The coefficient for income does not differ significantly (using the appropriate standard errors in the Swamy procedure) between the different methods, while the coefficient of price does, the mean group, aggregate time-series and fixed effect methods producing rather low estimates of the long-run

Table 2.5 Tests of order of integration and cointegrationa

ADF(1) statistics (with trends)












Hong Kong



































Sri Lanka










Note: aThe augmented Dickey-Fuller (ADF) statistics are computed over the period 1975-90. The ADF statistics are not reported for Hong Kong where data are available only over the period 1979-90. The 95 per cent critical value of the Dickey-Fuller statistic is -3.73. Cointegration was tested using Johansen's (1988) procedure, based on a first-order vector autoregressive model in (E, Y, P), with a trend. See also the notes to Table 2.1.

Note: aThe augmented Dickey-Fuller (ADF) statistics are computed over the period 1975-90. The ADF statistics are not reported for Hong Kong where data are available only over the period 1979-90. The 95 per cent critical value of the Dickey-Fuller statistic is -3.73. Cointegration was tested using Johansen's (1988) procedure, based on a first-order vector autoregressive model in (E, Y, P), with a trend. See also the notes to Table 2.1.

price elasticity, and the cross-section method a large estimate. The pooled estimates lie between the fixed effect and the cross-section estimates, as expected. The difference arises because across countries there is a correlation between coefficients in individual equations and the average values of the regressors.

Although more work is required on the interesting substantive issues raised by this data set, the results are generally in accordance with the theoretical points raised in the earlier sections of the chapter concerning the possible significance of the alternative ways of measuring average long-run effects, namely the large biases possible in dynamic pooled estimators and the significance of the assumption that coefficient variation is uncorrelated with the exogenous variables.

These results are very suggestive in that, although the data are highly aggregated and of doubtful quality, systematic patterns appear from the comparative analysis. Price and income are generally significant for these economies. The average long-run income elasticity from time-series estimates for individual countries is greater (though not significantly greater) than unity. The average long-run price elasticity is and significantly different from zero (see Table 2.3). In cross-section, the income elasticity is almost exactly unity and the price elasticity is about a half, although it is rather poorly estimated (see Table 2.4). This higher price effect is consistent with both the folk wisdom that time-series estimates do not capture the slow adjustment to prices and our theoretical results which indicate that cross-section estimates are likely to be more robust to dynamic misspecification than the time-series estimates.

The pooled estimates showed the features our theoretical discussion predicted: coefficients of the lagged dependent variable tended towards unity; coefficients of independent variables tended towards zero if only current values were included; if lagged values were included, the coefficient of the current independent variable was close to its true value (as indicated by the average of the time-series estimates) but the coefficient of the lagged independent variable tended to become equal but opposite in sign to the coefficient of the current independent variable (see Table 2.3). In this application, OLS on the pooled regression gave ridiculous estimates of the long-run effects. When the intercepts were allowed to differ across countries the estimates were not unreasonable, though the estimates of the short-run coefficients were clearly implausible (see Table 2.3). In this example, the biases in the fixed effect estimator seem to have cancelled out, but there is no reason to assume that this will usually happen. The relatively good performance of the fixed effect long-run estimator may also reflect the similarity of the income elasticities across countries in this application.


The main conclusion of this chapter is that the random coefficient argument of Zellner (1969), which seems to have been widely accepted in the literature, does not extend to dynamic models. Thus aggregating over time or pooling dynamic heterogeneous panels can produce very misleading estimates.

The lesson for applied work is that, when large T panels are available, time-series regressions for individual relations should be estimated and the estimates should be averaged over groups and their standard errors calculated explicitly. With modern computational facilities this is very easy to do. The hypothesis of homogeneity, that slope coefficients are the same across group equations, can then be tested. Our experience is that the homogeneity hypothesis is regularly rejected, even when the size of the test is adjusted to take account of the large number of observations. The average of the time-series estimates for individual groups, which we have referred to as the mean group estimates, can then be compared with the cross-section estimates based on long time averages and the possibility of correlation between the coefficients and the regressors can be investigated.

The cross-section regression has the advantages that it can utilize the large variations that usually exist in regressors across groups; allows estimation of average long-run effects for group-specific variables which do not vary over time; and tends to be relatively robust to measurement errors and dynamic misspecifications when compared with the long-run estimates obtained from the individual group estimates. Conversely, cross-section regressions will not pick up effects that are common to countries but vary over time, and will not give estimates of the country-specific effects which differ from the average. The more coefficients differ between countries the larger is likely to be the standard error of the cross-section estimates. If the misspecification induces systematic patterns between the estimated time-series coefficients and the exogenous variables, the group mean estimates will differ from the cross- section estimates, and investigating the pattern may provide us with a clue as to how to make improvements in our specifications.

The empirical analysis of energy demand reported in the chapter could only be indicative because of the high level of aggregation and the poor quality of the data; but it did illustrate the practical importance of the methodological issues discussed. Very different estimates of the average long-run price elasticity were obtained from different procedures, with the cross-section estimate being the largest and dynamic pooled estimators showing very large biases relative to average time-series estimates. Despite the large differences between the different estimates of the long-run income and price elasticities, our results and analysis suggest a long-run income elasticity of slightly larger than unity, and a long-run price elasticity of around for the ten Asian developing economies. Detailed and disaggregated analysis of energy demand in these economies is needed, however, before a more reliable picture can be presented.


The series used were annual observations over the period 1973-90. The variables are:

GDP at 1985 prices (billion of domestic currency units)

GDP deflator (1985=1)

Population (million persons)

Final energy consumption (thousand tons of oil equivalent). This is the energy made available to the consumer before its final utilization or the energy consumed by the final user for all energy purposes. It excludes all energy lost in the transformation of primary to secondary energy, energy used within the transformation industries and energy lost during the transformation process. A standard ton of oil equivalent of 10 Gcal is assumed.

Consumer price of energy (in domestic currency per ton of oil equivalent)

In the regressions the dependent variable is E, the logarithm of final commercial energy consumption per capita in tons of oil equivalent (t.o.e.).

The explanatory variables are:

Y, the logarithm of per capita GDP in 1985 prices converted into US dollars using 1985 market exchange rates

P, the logarithm of the average consumer price of energy in domestic currency per t.o.e. relative to the GDP deflator (1985=1) converted into dollars at 1985 exchange rates

Mean values of real per capita income in 1985 dollars, the relative price of energy in thousands of 1985 dollars, per capita final energy consumption in t.o.e. and the average rate of change of these variables for each country are given in Table 2A.

Most of the data were taken from Asian Development Bank (ADB) (1992). In a number of cases the exchange rate, GDP and price data given in ADB did not agree with those published in International Financial Statistics, in which case the IMF data were used. There were also a number of cases where data in the ADB summary tables did not agree with the detailed overall energy balance tables, in which cases the latter were used.

Data for Hong Kong on energy consumption and prices were not available for 1973-78. Except for the estimates in Table 2.1, the other parameter estimates are based on interpolated data for these years for Hong Kong. For the sample where the data were available, the logarithm of per capita Hong Kong energy consumption was regressed on the logarithm of Hong Kong output and the logarithm of average energy consumption by the other nine countries. This equation was then used to predict Hong Kong energy consumption for 1973-78. Similarly, the logarithm of Hong Kong dollar price was regressed on the average dollar price for the other nine countries and this was used to predict Hong Kong energy price for the period 1973-8. The regression using the interpolated data was very similar to the regression reported in Table 2.1 using actual data over the shorter period.

Table 2A Country mean growth rates and per capita levelsa

Per capita energy use Per capita GDP Relative price of energy

Growth (%p.a.) Level (t.o.e.) Growth (%p.a.) Level (1000 1985 Growth (%p.a.) Level (1985$per

Growth (%p.a.) Level (t.o.e.) Growth (%p.a.) Level (1000 1985 Growth (%p.a.) Level (1985$per








Hong Kong

















































Sri Lanka














Note: aSee the Data appendix for definitions and statistical sources.

Note: aSee the Data appendix for definitions and statistical sources.


1 An earlier version of this paper was presented at the Conference on Estimating Long-run Energy Elasticities, Robinson College, Cambridge, September 1992. It draws heavily on Pesaran and Smith (1992) which contains a more technical discussion. We are grateful to Victoria Saporta for helping us with the compilation of the data analysed in the paper. The first author wishes to acknowledge partial financial support from the ESRC and the Isaac Newton Trust of Trinity College, Cambridge.

2 See Engle and Granger (1991) for a discussion of integration and cointegration properties of time series and their consequences for econometric analysis.

3 The assumption thatp is the same across groups is clearly restrictive and is made here for expositional simplicity. The same results, however, follow when p is allowed to vary across the groups.

4 Recall that by assumption E(Pj)=p, j=0, 1, ..., p.

5 One can also use purchasing power parity conversion factors as in Pindyck (1979), but given the illustrative nature of the present application we have not followed this route.

6 To save space the details of the estimation results for model M2 are not reported here, but alternative 'average' estimates of the parameter estimates based on M2 are given in Table 2.3.

7 Using Johansen's (1988) maximum likelihood procedure we were not able to reject the hypothesis that E, Y and P are cointegrated at the 10 per cent level of significance in the case of India, Indonesia, Korea, Philippines, Sri Lanka and Thailand. Because of data limitations, the test was not applied to Hong Kong. See Table 2.5. The Johansen test statistics were computed using the Microfit 3.0 package (see Pesaran and Pesaran 1991).

Chapter 3

A survey of international energy elasticities

Jago Atkinson and Neil Manning


This chapter considers evidence on energy elasticities gathered from a large number of studies published over the past twenty years or so. The emphasis of the survey is international and both multi-country and single-economy studies are included. The survey classifies this rather extensive literature in terms of the methodology used. The chapter also presents some estimates of long-run energy elasticities for the industrial sector in a group of fifteen countries over the period from 1960 to 1989. Estimation makes use of cointegration techniques and the resultant estimates are compared and contrasted with the results from previous studies discussed in the initial section of the chapter. The chapter further considers the reasons for differences in estimates of elasticities and considers further areas of potential study.


This chapter surveys the literature on international energy elasticities, an area to which economists have paid rather less attention than one may suspect. Indeed, following the oil crisis of the early 1970s there have been numerous studies on energy elasticities at the national level but rather fewer at the international level. However, given that the direction and strength of international energy elasticities is particularly relevant to the current debate on the use of 'carbon taxes' to address the problem of global warming, evidence on international energy elasticities is of crucial interest.

As such, this review aims to summarize both the methodology employed and the results obtained in the literature on international energy elasticities. The chapter attempts to integrate those studies which are purely international in focus with some of the more influential single-country studies. The majority of the studies reviewed examine substitution possibilities between aggregate energy and both capital and labour; rather fewer consider substitution possibilities between individual fuels.

The chapter is in three sections. Section 3.1 provides some summary background on the theory of factor demands and on the econometric techniques which are commonly applied. The major part of the literature review is presented in Section 3.2 which surveys energy demand studies grouped according to methodology. Section 3.2.1 considers studies based on the translog specification, the theory of which is briefly outlined in Section 3.1.2. Sections 3.2.2 and 3.2.3 consider studies based on Cobb-Douglas and constant elasticity of substitution (CES) production functions, while Section 3.2.4 considers more ad hoc studies. And finally, in Section 3.3, the chapter presents the results of applying fairly standard models to an international data set from fifteen countries spanning thirty years. In particular, the time-series properties of the data are considered in Sections 3.3.1 and 3.3.2 together with evidence from the application of cointegration techniques.


Energy elasticities estimated in the empirical literature

Several concepts of elasticity are commonly used in the empirical literature on energy demand. The usual price, income and cross-price elasticities of demand, denoted n, are familiar enough and will not be discussed. In a two-factor model the elasticity of substitution is defined as dln(A7L)

and is the elasticity of the ratio of the factors with respect to the marginal rate of substitution between them. In effect, (3.1) provides an index of the cost-minimizing factor input proportions to changes in relative factor prices (see McFadden 1978). The Allen-Uzawa elasticity of substitution (AES) is given as alj-CCijIC,Cj (3 2)

where the partial derivative C=cC cPr For the translog function, the own-price AES is given as b& + (Sf - S,

for factor i and the cross-price AES is

where i±j. The AES holds total output and all factor prices fixed and so substitution can take place between i and an input other thanj (see McFadden 1978). The price elasticities of demand for factor inputs in the translog model are related to the AES, and can all vary with the value of the cost shares.

The full elasticity of substitution (FES) (see Kang and Brown 1981) measures substitution between two factors when the ratio of marginal products of factor i and any factor other than j are held fixed. Theoretically, this implies that there is no substitution between factor i and any factor other than j. The FES is related to the AES as follows:

As noted in Section 3.1.2, various separability assumptions are employed in many of the translog-based studies. Berndt and Wood (1979) analyse engineering and econometric interpretations of energy/capital complementarity and consider utilized capital. They set up a higher stage or master production function as follows:

with an associated master cost function

A two-input linearly homogeneous weakly separable sub-function using capital and energy is specified as

yielding utilized capital K*. This implies that the optimal E/K ratios within the sub-function (3.9) only depend upon the prices of capital (PK) and energy (PE). There is a similar utilized capital cost sub-function related to the master cost function. The utilized capital cost sub-function is as follows:

where Ck*=PkK+PeE. The same procedure is used on an equivalent production sub-function, L*=f2(L, M,). The master production function can therefore be written as

and the master cost function is

where PK* and PL* are unit costs obtained from the cost sub-functions PK* = Ck*/K* and Pl*=Cl*/L*.

This structure can be used to estimate gross, net and scale elasticities. Gross elasticities, which are standard price elasticities of demand when referring to a sub-function, are evaluated when sub-function output is held fixed as factor prices change. Net elasticities allow for changes in sub-function output to occur along the 'expansion path' when the unit costs PK* or PL* change and total output is assumed constant. The scale elasticity is the difference between the gross elasticity and the net elasticity. IfXm is a positive, strictly quasi-concave homothetic production sub-function, the net cross-price elasticity is calculated as follows:

Vij = V*ij+SjrnVm„, where n*ij is the gross elasticity, Sjn and j is the cost share of the jth input in the total cost of producing Xm nmm is the own-price elasticity of demand for Xm when output at the level of the master production function is held fixed. Prywes (1986) also considers the gross and net elasticities but refers to them as engineering and economic elasticities. The AES and the gross price elasticities are estimated as usual.

Pindyck (1979a) defines partial and total elasticities for individual fuels in two-stage models with a separable (fuel) lower stage. These are related to the above net and gross elasticities. The standard own- and cross-price elasticities for fuels are 'partial' as the total quantity of energy consumed is held constant in that they only account for substitution between fuels. Total elasticities for fuels allow the total quantity of energy consumed to vary. They can be derived from the translog share equations as follows:

where V** is the total own-price elasticity for fuel /' and t]EE is the own-price elasticity of aggregate energy; .S',is the share of fuel i in total energy expenditure. The total cross-price elasticity for each fuel is as follows:

If the energy cost sub-function is homothetic, then 7ho"' ~ Veq where >jEQ is the elasticity of energy with respect to total output.

Pindyck also calculates the elasticity of the average cost of production with respect to the price of energy (>lHCl._j and with respect to fuel prices (;/ac;). As the energy cost function is homothetic:

The cost functions employed in the literature

Neoclassical economists have employed production functions in empirical research for many years. The well-established Cobb-Douglas production function and constant elasticity of substitution (CES) production function are often employed, and therefore the theoretical basis of the functions will not be discussed at length. The functions, moreover, are highly restrictive and as a consequence the more flexible translogarithmic production function has been more commonly employed of late. In the translogarithmic function, which is a second-order approximation to an arbitrary production function (see Christensen et al. 1971) no restrictions are placed on the Allen elasticities of substitution (see the previous section). The basic function is ln0 = an + X + X I a^lnX,)(lnXj)

where Q is gross output, the Xi are factor inputs, the ai are first-order parameters and the aij- are second-order parameters (see Fuss et al. (1978) for a survey of alternative functional forms). Given duality between cost and production and assuming exogeneity of output and factor prices, a twice-differentiable non-homothetic translog cost function, the dual to (3.17), may be written as loC=fcfl+X b, In P, + ^ X X ¿„(In p.)(in Pj)

where C is total cost, the Pi are factor prices, the bi are first-order parameters and the by are second-order parameters. The above function is homothetic if the biQ terms are zero and has constant returns to scale if bQQ=0 and bQ= 1 (see Berndt (1991) for a summary). Additionally, the symmetry restrictions imply that bi.=b1i for ijj. The 'traditional' static translog model (3.18) can be logarithmically differentiated to yield the estimated factor share equations via Shephard's lemma:

where 'adding up' implies Ei bi=1; Ei bj=Ej bij=0 since the shares, by definition, must sum to unity. The translog model provides three tests of the theory of demand. First, the assumption of homogeneity of degree one in input prices can be tested through the symmetry conditions. Second, non-negative input levels, and thus non-negative cost shares, must exist, and third, the translog function must be concave in input prices. Concavity of the cost function is satisfied if the Hessian matrix based on the parameter estimates is negative semi-definite. This is equivalent to the condition that the matrix of AES is negative semi-definite.

The basic model, with symmetry and constant returns to scale imposed, is estimated using the four factors capital (K), labour (L), energy (E) and materials (M) as set out in the influential Berndt and Wood (1975) study. However, in many international studies of energy demand the lack of data on materials necessitates that authors, such as Fuss (1977b), assume weak separability of K, L and E from M so that the cost function can, in general terms, be written C=C(Q, C1(PK,PL, PE), PM).Weak separability places restrictions on the parameters of (3.18) such that bi/bj=biM/bjM for i, j^M which implies that the marginal rate of substitution between i and j is independent of M (see Fuss et al. 1978). One particularly useful consequence of applying separability, as outlined by Fuss, is the concept of two-stage modelling which allows for the estimation of interfuel substitution and is used in several of the studies considered in Section 3.2.

Assume the existence of a homothetic aggregate (C1) containing coal, oil, gas and electricity which is weakly separable from other non-energy inputs. The non-homothetic cost function is, in general terms, C = C(Q, C\{Pc, Pa, Pv„ PH1), PK, Pl) (3.20)

If the homothetic sub-function Q has a translog form it provides a consistent instrumental variable for the aggregate price of energy PE (which is usually formed as a Divisia index). Economic optimization can therefore proceed in two stages: first, optimize the levels of the fuels in the separable sub-function and then optimize the levels of the aggregates in the higher stage problem.

Varying types of technical progress are considered in the literature but our concern here is limited to Hicks-neutral and non-neutral constant exponential technical change. Consider the homothetic constant returns to scale translog cost function:

In C = b0 + X bt In P, + I £ by (In P,-)(ln Pj) + In Q + y i + X y,r

where t is a linear time trend. Equation (3.21) has Hicks-neutral technical change iff yf 0 and Yi= Q V'« Hicks non-neutral technical change occurs iff 0, in which case the share equation for factor i becomes

The presence of Hicks non-neutral technical change in fuel-share equations is sometimes referred to as 'fuel-efficiency bias'.

The overwhelming majority of the empirical work surveyed in Section 3.2 has utilized versions of the basic static framework outlined above, while recent developments have extended these models to consider dynamic optimization over time explicitly. One such paper is that by Pindyck and Rotemberg (1983) in which the translog functional form is employed to consider the effects of energy price shocks on the four factors K, L, E and M Energy and materials are assumed to be fully flexible but capital and labour are assumed quasi-fixed. Thus the production technology is represented by a restricted cost function, conditional at time I on Kp L, and 0,\

It is also assumed that changes in capital and labour involve adjustment costs represented by convex functions c'(I) and c2(H) where, for simplicity, I is investment and H is net hirings. Firms are assumed to minimize the expected sum of discounted costs:

where ^, is the expectation operator and Rt ris the discount factor applied at t for values at time r. The first-order conditions provide the two-variable factor input demand equations

and the Euler conditions give expressions for the anticipated evolution of the quasi-fixed factors over time. The full solution to these equations is a path for K, L, E and M that depends on the current states of L, K, PE and PM, as well as the expected future prices and output. Equations (3.25a) and (3.25b) and the two Euler conditions form regression equations to estimate the parameters of C, c1 and c2.Pindyck and Rotemberg assume that the restricted cost function (3.23) is a homogeneous and symmetric translog function in PM, PE, K, L, Q with neutral technical ch

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