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Capital-Energy Relationships: An Analysis of Three Panel Data Estimation Methods. “How much drag on the future economic growth of the developed world is likely to be exerted by the narrowing of its resource base, assuming indeed that the resource base is narrowing?” Solow (1978). Nov, 2009.

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Capital-Energy Relationships: An Analysis of Three Panel Data Estimation Methods

“How much drag on the future economic growth of the developed world is likely to be exerted by the narrowing of its resource base, assuming indeed that the resource base is narrowing?”

Solow (1978)

Nov, 2009


Accordingto Saunders (2000), the design of policies that aim to reduce energy consumption in the industrial sector requires knowledge of the Elasticity of Substitution (EoS) between capital and energy. If it is close to zero, then policies must encourage technological diffusion. However, if the elasticity is close to one, taxes could then be used to reduce energy consumption.

previous estimations
Previous estimations

Thefirst estimations were based on flexible forms that are considered an approximation of a general cost function

Berndt and Wood (1975) found that energy price shocks will prompt a shrinkage in the investment of capital for the USA economy. They used a KLME model with aggregate data of 24 years (1947-71).

Griffin and Gregory (1976) found that changes in energy prices will encourageinvestment in capital to reduce energy consumption. The authors used a KLE model for an international panel data of nine manufacturing countries over 14 years from 1955 to 1969


Previous estimations

Apostolakis (1990) analysed several estimations, finding that generally, studies based on time series estimations classified capital and energy as complements whereas the opposite conclusion was reached by panel data studies.

More recently, Frondel and Schmidt (2002) pointed out that in fact none of these arguments were important. They found that, under certaincircumstances, the elasticities obtained by the Translog Cost Function (TCF) ended up being close to the ratio of the factor costs and the total cost, and therefore, the TCF cannot provide reliable estimates


After thirty years of studying EoS issues, researchers have learned that substitutabilityis a relative concept that depends on the country, Pindyck (1979), industry and the kind of capital, Field (1980) and Morrison (1993).

Furthermore, Popp (1997) argues that using a time trend, as a regular practice in the previous estimations, cannot capture technological changes especially for energy efficiency.

Thomsen (2000) suggested a two-step procedure to estimate Cross PriceElasticities (CPEs) based on the shadow price concept.

goal of this exercise
Goal of this exercise
  • To estimate Cross Price Elasticities by using different approaches on a panel data of eight UK industries
  • To estimate a TCF to find evidence of the arguments in the literature (see e.g. Berndt and Wood (1979), Apostolakis (1990) and Frondel and Schmidt (2002)
  • To extend Thomsen’s methodology to a panel data setting and estimate an Error Correction Model (ECM) using investment in R&D for energy efficiency (IR&DEF)
  • To find out which is the more suitable policy to reduce energy consumption in the industrial sector for the UK economy based on empirical evidence.
data set
Data set

The analysis is based on both a cost and production context, and therefore the dataset that it is used contains information of factors and their prices from 1970 to 2006 for 8 UK industries.

Additionally, I consider the following industries: basic metals, chemical, non-metallic minerals, transport equipment, machinery, textiles, food and paper. I also used five factors: "net capital stock" divided into buildings (K1) and machinery and equipment (K2), labour (L) and intermediate materials (M) together with energy (E) and IR&DEF. Regarding ( E), it comprises four fuels, electricity, natural gas, coal and oil.

The information about E, L ,M and R&D and their prices are obtained from the Economic and Social data Service, while the information related to capital is obtained from the National Statistics for the UK.


The TCF assumes symmetry, monotonicity and concavity


The Generalized Leontief assumes also symmetry, monotonicity and concavity

Following Thomsen (2000) the estimation is a two step procedure where inthe first step a System of Input-Output Equations (SIOE) is estimated for the Long Run (LR) parameters. For the short run a SIOE is estimated using a price function for the quasi-fixed factors that is obtained based on the LR parameters. Moreover an error correction equation for the capital motion has to be included in the estimation for the short run.


The Error Correction Model (ECM)

In this estimation N=5 and π is the general technological change in the economy.Theestimation is carried out based on a Cobb-Douglas production function and in the concept of substitution proposed by Frondel (2004)

The estimation methods used in the described models are the Seemingly Unrelated Regression method, the General Method of Moments for dynamic paneldata and cointegration and unit root tests for panel data


Some of the CPE estimated for the TCF

Standard errors are given in parenthesis. "*": significant at 5 percent level. "**": significant at 1% percent level. "***": significant at 0.1 % percent level.


Some ofthe CPE estimated for the TCF (KLE model)


Some of coefficients estimated for the GL in the long run


Some of CPE estimatedfor the GL in the long run


Some of CPE estimated for the GL in the short run (preliminary results)


Unit root test for the series in first differences

IPSW ADF-Fisher PP- Fisher


The Pedroni test for within dimension in the autorregresive coefficient of the residuals

"*": used when the probability that the null cannot be rejected is equal or larger than 1 per cent.


Unit root test for the residuals of the ECM


Elasticities estimated for the ECM in the short run

Elasticities estimated for the ECM in the long run


In conclusion, from the different specifications used, I find clear evidence of complementarity when using the TCF and the ECM, while when using the GL, the results support weak substitutability between capital and energy specially for industries basic metals, chemical and machinery. Moreover, it was found that the TCF and the ECM point out a stronger relationship between energy and investment in machinery than in buildings. Therefore I find evidence that a policy that aims to diminish energy consumption via changes in energy prices will have a more severe contraction in capital demand for those three industries.