WPS4957
Policy Research Working Paper 4957
Substitution and Technological Change
under Carbon Cap and Trade
Lessons from Europe
Timothy J. Considine
Donald F. Larson
The World Bank
Development Research Group
Sustainable Rural and Urban Development Team
June 2009
Policy Research Working Paper 4957
Abstract
The use of carbon-intense fuels by the power sector results indicate that prices for permits and fuels affect the
contributes significantly to the greenhouse gas emissions composition of inputs in a statistically significant way.
of most countries. For this reason, the sector is often Even so, the analysis suggests that the industry's fuel-
key to initial efforts to regulate emissions. But how long switching capabilities are limited in the short run as is the
does it take before new regulatory incentives result in a scope for introducing new technologies. This is because
switch to less carbon intense fuels? This study examines of the dominant role that past irreversible investments
fuel switching in electricity production following play in determining power-generating capacity. Moreover,
the introduction of the European Union's Emissions the results suggest that, because the capacity for fuel
Trading System, a cap-and-trade regulatory framework substitution is limited, the impact of carbon emission
for greenhouse gas emissions. The empirical analysis limits on electricity prices can be significant if fuel
examines the demand for carbon permits, carbon based prices increase together with carbon permit prices.
fuels, and carbon-free energy for 12 European countries The estimates suggest that for every 10 percent rise in
using monthly data on fuel use, prices, and electricity carbon and fuel prices, the marginal cost of electric
generation. A short-run restricted cost function is power generation increases by 8 percent in the short run.
estimated in which carbon permits, high-carbon fuels, The European experience points to the importance of
and low-carbon fuels are variable inputs, conditional starting early down a low-carbon path and of policies
on quasi-fixed carbon-free energy production from that introduce flexibility in how emission reductions are
nuclear, hydro, and renewable energy capacity. The achieved.
This paper--a product of the Sustainable Rural and Urban Development Team, Development Research Group--is part of
a larger effort in the department to understand how climate change policies affect energy markets. Policy Research Working
Papers are also posted on the Web at http://econ.worldbank.org. The author may be contacted at DLarson @worldbank.org.
The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development
issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the
names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those
of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and
its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent.
Produced by the Research Support Team
SUBSTITUTION AND TECHNOLOGICAL CHANGE UNDER CARBON CAP AND
TRADE: LESSONS FROM EUROPE
TIMOTHY J. CONSIDINE AND DONALD F. LARSON
Timothy Considine is an SER Professor of Economics in the Department of Economics and Finance, University of
Wyoming. Donald F. Larson is a Senior Economist in the World Bank's Research Group. The author's gratefully
acknowledge support from the World Bank's Knowledge for Change Trust Fund and the School of Energy
Resources at the University of Wyoming and research assistance from Muhammed Hassan. Comments and
suggestions from brown bag participants from the Economics and Finance Department at the University of
Wyoming are appreciated. Any errors or omissions are the responsibility of the authors.
Substitution and Technological Change under Carbon Cap and Trade:
Lessons from Europe
1. Introduction
The European Union (EU) has pioneered the development of a carbon dioxide (CO2) emissions
trading program, known as the Emissions Trading System (ETS). Operating since early 2005, the
program mandates an overall limit or cap on carbon emissions that originate from large industrial
facilities and electric power generating plants and allows the trading of emission permits under the
cap. By allocating a supply of permits and creating a regulatory demand for CO2, the EU ETS
creates a market for disposing carbon dioxide emissions in the atmosphere. As a consequence,
markets have developed that price CO2 emissions.
Under the legal and regulatory framework established by the EU ETS, producers of carbon
intensive goods and services covered by the program must consider emissions in their production
decisions, weighing the costs of purchasing permits with the benefits of selling excess permits that
are created by using less carbon-intensive inputs or by investing in less carbon-intensive
technologies. The objective of this study is to understand this process and the nature of short-run
relationships among permit use, input substitution and technological change under carbon cap and
trade.
In pursuit of this goal, the study specifies and estimates an econometric model of fuel
substitution in electric power production in Europe. According to Ellerman and Buchner (2007), the
electric power sector accounts for 60 percent of carbon emissions in the EU and constitutes 90
percent of the potential demand and 50 percent of total supply of carbon allowances.
Electric power producers have a variety of options to reduce their carbon emissions. In the
short-run, they can shift their mix of generation, raising their utilization of carbon-free capacity, such
2
as nuclear power, shifting to lower carbon sources, such as natural gas fired generation, and reducing
their use of high carbon sources, such as coal and oil-fired generation. This fuel switching is limited
by installed capacity. Longer term, electricity producers can invest in new capacity, such as advanced
nuclear plants, coal with carbon capture and sequestration systems, or renewable sources, including
wind, solar, biomass, and geothermal capacity.
Dynamically, decisions to invest in new capacity will be influenced in part by the ability of the
existing fleet of generating plants to adapt to carbon emission constraints. As carbon emission
limits become more stringent, the ability of electric power producers to adjust becomes more
difficult and the marginal cost of electricity rises, inducing new investment in carbon-free sources of
electricity. How readily electricity produces respond to price signals remains a key question in
estimating the costs of carbon emission controls.
The model presented below is designed to estimate this short-run adaptability arising from
input substitution and technological change in the electric power sector. The framework uses a
restricted cost function in which electricity producers minimize the variable costs of production
including inputs of coal, natural gas, petroleum, and carbon allowances subject to inputs of carbon-
free energy resources, including nuclear and renewable resources. These last two energy resources
are treated as quasi-fixed inputs because data on prices for carbon-free energy resources are
unavailable. This is the same problem faced by Halvorsen and Smith (1986) in their analysis of
substitution possibilities for internally produced and un-priced ore inputs in metal mining and has
implications for how we interpret our results. We return to this topic later in the paper.
Unlike the Halversen and Smith study, which used a translog (TL) function, the model
presented below is based upon the Generalized Leontief (GL) restricted cost function developed by
Morrison (1988). Caves and Christenson (1980) show that the GL outperforms the TL when
technology has limited substitution, which is likely in electric power generation. Morrison also shows
3
that the GL allows closed form solutions for equilibrium levels of quasi-fixed inputs, which
facilitates computation of substitution elasticities and their standard errors.
The model is estimated using a panel of monthly time series observations from January 2005
through March of 2008 for a cross section of twelve European countries. The relatively large
number of observations and considerable variation in the data allows the estimation of variable
returns to scale and input biases from technological change. For our purposes, an important
advantage is that the model can be used to test whether the carbon cap-and-trade regime induced
carbon saving technological innovation. Moreover, the study provides explicit measures of the
degree of carbon abatement, such as carbon emissions per unit of electricity output, under the EU
ETS and, most importantly, explains how this abatement was achieved.
Specifically, the degree and nature of fuel switching induced by carbon pricing and relative fuel
prices is estimated. This empirical assessment of carbon substitution possibilities sheds light on
whether carbon pricing significantly increases the demand for less carbon intensive fuels, such as
natural gas, at the expense of carbon intensive fuels, such as coal. These substitution possibilities
ultimately determine whether the demand for carbon is price inelastic, which would imply significant
adjustment costs to a low carbon society, or whether carbon demand is elastic, facilitating a less
costly path to achieving significant reductions in carbon emissions.
The next section provides some additional background on the European program. Section 3
presents the economic framework, discussing the theoretical underpinnings for the empirical model.
The parametric specification of the econometric model is then presented in the fourth section along
with a discussion of the estimation techniques. Section 5 provides an overview of the data sample,
including descriptive statistics by country on electricity generation by type, net imports, total
generation, and the composition of so-called combustible fuels, including natural gas, coal, and
petroleum. Trends in the carbon intensity of electricity and in fuel shares in the sample are also
identified and discussed. The sixth section of the paper discusses the econometric results and the
4
implications for assessing substitution and technological innovation under carbon cap and trade.
The final section summarizes the major findings and discusses the policy implications of the results.
2. EU Emissions Trading Scheme
In 2003, the European Council and Parliament approved legislation that eventually launched the EU
ETS in 2005. The ETS is a cap-and-trade program that limits carbon dioxide emissions from more
than 10,000 installations located in the thirty member states of the European Economic Area. 1
Currently, the sectors covered by the program include energy activities (e.g. electric-power
generation greater than 20 megawatts), ferrous metals industries (iron and steel), mineral industries
(cement, glass, ceramics, oil refineries, etc.), and pulp and paper industries. 2 The program is
considered a key element in the European Union's plan to meet its commitment under the Kyoto
Protocol to reduce greenhouse gas emissions by 6 percent compared to 1990 levels by the end of
2012.
Our study period, January 2005 to March 2008, covers two phases of the program. Practical
implementation of the program meant establishing an extensive system of procedures for allocating
allowances, for monitoring how they are used, and for matching allowances with measured
emissions. For this reason, Phase 1 (2005-2007) of the cap-and-trade program was intended in part
as an opportunity to work out operational difficulties in advance of Phase 2 of the program, which
corresponds to the first round of commitments (2008-2012) under the Kyoto Protocol.
As discussed, under the program regulated installations are issued permits, called EU allowances
(EUAs), equivalent to one ton of emitted carbon dioxide. The allocations are made in accordance
with National Allocation Plan (NAP), drawn-up by individual Member States. At the end of each
1The area includes the EU's 27 member states, plus Iceland, Liechtenstein and. Norway
2For more information on the EU ETS see Watanabe and Robinson (2005), Convery and Redmond (2007) and
Europa (2007).
5
year, regulated installations must surrender allowances equivalent to their emissions. Surplus and
short-falls can be matched through sales and purchases. The allowances are tracked in national
registries that were linked to form a system-wide registry during the program's second phase.
Though restrictions apply, the system is open to other tradable units established under the
Kyoto Protocol, including Certified Emission Reductions (CERs) from developing countries. This is
significant, since it potentially links the two types of carbon offsets into a large and liquid market,
making the findings of this study relevant for developing countries. Legislation known as the
"Linking Directive" lays out the relationship between EUAs and the Kyoto-system tradable units. 3
3. The Economic Model
The output of electricity depends upon inputs of labor and maintenance, capital service flows from
generating equipment and structures, and primary fuels. In addition, under the EU ETS producers
of electricity are required to obtain pollution permit allowances to offset their emissions of carbon
dioxide. Hence, the disposal of the carbon dioxide by-products of electricity generation now
becomes a factor of production. These observations imply the following production function for
electricity:
Yt = f (K t , Lt , Et ,Ct ) (1)
where Yt is output of electricity in period t, K t is capital service flows, Lt is salaried and hourly
worker services, maintenance, and non-fuel supplies, Et is an aggregate of energy inputs, and Ct is
carbon emissions.
Assuming capital and labor are fixed in the short run, under duality theory the following long-
run cost function exists:
3For an early assessment of the ETS, see the volume edited by Ellerman, Buchner and Carraro (2007). For more on
carbon markets in general, see Larson et al. (2009).
6
TCt = C (wet , wct K t , Lt ,Yt )+ µ kt K t + µlt Lt (2)
where wet and wct are prices for energy and carbon respectively and µ kt and µlt are the user costs
corresponding with stocks of labor and capital stocks. Prices for carbon emission allowances
represent the societal valuation of the impacts of carbon emissions on common property
atmospheric resources implicit in the target level of allowable emissions and the corresponding
allocation of permits. This specification is similar to the study conducted by Considine and Larson
(2006) of sulfur dioxide pollution allowances.
For the empirical analysis below, K t and Lt are unobservable. To specify an empirical model,
therefore, requires assuming the existence of a weakly separable sub-aggregate of energy and carbon
emissions within the variable cost function. In particular, the weakly separable model implies that
substitution possibilities between fuels and carbon emissions are independent of substitution
possibilities between labor and capital, which is likely a reasonable assumption within a short-run
context. The cost minimization problem, therefore, is to minimize energy and carbon emission
allowance costs subject to output levels.
In the context of the model, there are three types of energy aggregates: i) primary fossil fuels,
including coal, petroleum, and natural gas; ii) nuclear fuel and hydroelectric energy; and iii)
renewable energy resources, including wind, solar, and geothermal energy. As discussed, while
market prices for primary fuels are observable, those for nuclear, hydroelectric, and renewable
energy are not. To accommodate this, the study assumes the existence of a weakly separable sub-
aggregate for primary energy and carbon emissions contingent upon levels of nuclear, hydroelectric,
and renewable energy generation, levels of output and the state of technology. More specifically, this
implies the following short-run restricted energy and emission allowance cost function:
Gt = G (w1t , w2t , w3t N t , Rt ,Yt , Zt ) (3)
7
where Zt is an index of technological change, w1t is the price for carbon emission allowances; w2t is
the is the price for solid and liquids fuels, such as coal and fuel oil with relatively high carbon
content; w3t is price of natural gas with relatively low carbon content; N t is the consumption of
nuclear and hydroelectric energy, which is carbon-free with low operating costs; and Rt is renewable
energy resource use, which are also carbon-free but associated with relatively higher operating costs.
Two sets of substitution possibilities are recovered from this model. The first set provides
estimates of first-order substitution possibilities among primary fuels and carbon allowances when
levels of carbon-free energy are held fixed. The second set of substitution possibilities are recovered
from the convexity conditions, G / N t , G / Rt , and estimate rates of substitution among carbon-
intensive primary fuels, carbon emissions, and carbon-free energy sources.
An engineering perspective on this model can be attained by noting that the consumption of a
primary fuel at a specific plant is equal to the heat rate, which is defined as the amount of fuel
consumed per unit of electricity, multiplied by the level of power generation from that facility. So
from this perspective, the short-run restricted variable cost function specified above in equation (3)
can be viewed as a model that selects the least cost mix of plant capacity operating in any time
period. This model is consistent with least cost scheduling algorithms commonly employed by
electricity companies and system operators.
4. Econometric model
The Generalized Leontief (GL) function developed by Morrison (1988) is best suited for this
particular problem because Caves and Christensen (1980) found it more likely to maintain cost
minimizing curvature conditions under limited input substitution possibilities, which is a reasonable
prior assumption for electric power generation. Another important reason is that the GL provides a
closed-form solution for stocks of quasi-fixed factors, which facilitates computation of long-run
elasticities. For this study, the GL takes the following form:
8
3 3
( )
3 3
ij wit w jt + yi witYt1 2 + zi wit Zt1 2
12
i =1 j =1 i =1 i =1
Gt = Yt 3
it yy t (
+ w Y + 2 Y 1 2 Z 1 2 + Z
yz t t zz t )
i =1
3 3
ni wit N t + ri wit Rt
12 12
i =1 i =1
+Yt1 2 3 (4)
(
+ w Y 1 2 N 1 2 + Y 1 2 R1 2 + Z 1 2 N 1 2 + Z 1 2 R1 2
it yn t t yr t t zn t t zr t t
)
i =1
( )
3 3 12
+ wit nn N t + 2 nr N R 12
t
12
t + rr Rt + wit ci Dct
i =1 i =1 c=1
where the ' s, ' s, and ' s are unknown parameters, the it , kt , and bt are stochastic errors. 4 The
empirical model includes three variable inputs: high carbon fuel, x1t , low-carbon fuel, x2t , and
carbon emissions, x3t . The restricted cost function is symmetric and homogeneous of degree zero in
prices. The asymmetric way that output and technological change enter the cost function facilitates
parametric testing of long-run constant returns to scale.
The input demand functions for high carbon fossil fuels (petroleum and coal), low carbon fuels
(natural gas), and carbon are equal to the derivative of (4) with respect to factor prices. These
expressions are as follows:
12 3 w jt
12
Xit = Dct + Yt ij
i
+ yiYt1 2 + zi Zt1 2 + yyY + 2 yzYt1 2 Zt1 2 + zz Zt
j =1 wit
c
c=1
+ ni (Yt N t ) + ri (Yt Rt ) + ynYt N t1 2 + yrYt Rt1 2 + zn (Yt Zt N t ) + zr (Yt Zt Rt )
12 12 12 12
(5)
+ nn N t + 2 nr (N t Rt ) + rr Rt
12
i
4The countries in order from one to twelve are Austria, Denmark, Finland, France, Germany, Greece, Netherlands,
Portugal, Spain, Poland, Sweden, and United Kingdom.
9
The variable input-output ratios are a function of relative input prices conditional upon electricity
production and the availability of hydroelectric, nuclear, and renewable energy resources. The full
model includes the restricted variable cost function (4) and the three variable input demand
equations (5).
While the model can be estimated with full information maximum likelihood, a more robust
procedure is estimation with Generalized Methods of Moments with country and monthly dummy
variables and lagged values of the right-hand side variables, including those involving input prices,
quasi-fixed factor levels, and output.
5. An Overview of the Data Sample
The above model is estimated with a pooled, monthly data sample across twelve countries in the EU
from January 2002 to March 2008. The International Energy Agency (IEA) reports monthly electric
power generation from nuclear, hydroelectric, geothermal and renewable resources and from the
combustion of fossil fuels. The IEA does not report the types of combustible fuels but the
EuroStat database does report the consumption of petroleum, coal, and natural gas in electric power
generation. The EuroStat database, however, does not report data on generation from geothermal
and renewable electricity generation. Given the rising importance of renewable energy in the
generation portfolio, the more inclusive IEA data on generation is adopted in this study while the
EuroStat data on fossil fuel use is utilized. A comparison of the generation data reported by the two
agencies reveals the average differences are 3.2 percent.
An overview of the generation and net imports data appears in Table 1, which reports the
sample means. The largest producers of electricity are Germany, France, United Kingdom and
Spain. The mid tier includes Sweden and Poland and the other six countries have total indigenous
production between 3,000 and 8,000 gigawatt hours. France and Germany are the largest producers
at 47,624 and 45,666 gigawatt hours repsectively. All twelve countries in the sample produce fossil-
10
fuel-fired electricity. Five countries do not produce nuclear electricity, including Austria, Denmark,
Greece, Portugal, and Poland. Denmark and the Netherlands produce negligible amounts of
hydroelectric power while France and Sweden are the largest producers of hydroelectricity. Germany
and Spain produce rather substantial amounts of renewable electricity (see Table 1).
An overview of coal, petroleum, and natural gas consumption in electric power generation
appears in Table 2. The largest coal consumers include Germany, United Kingdom, Poland, and
Spain. The United Kingdom is the largest consumer of natural gas, Germany is second, and the
Netherlands and Spain are significant consumers as well. Spain consumes significant amounts of
petroleum to generate power along with Greece, the United Kingdom, and Germany.
Trends in the carbon intensity of indigenous electricity production are displayed in Figure 1.
Carbon emissions are computed by multiplying fuel use by its respective carbon emission factor.
The denominator is indigenous electricity production to reflect the shifts between combustible fuels
and carbon-free generation such as nuclear, hydroelectric, and renewable energy resources. For the
aggregate of the twelve countries, carbon intensity decreased from 2004 to 2005 but then increased
very slightly from 2005 to 2007. This aggregate reflects a great deal of variability in carbon intensity
trends among countries. Poland has the highest carbon intensity among the twelve countries, which
actually increased between 2005 and 2007. With the exception of Denmark, Germany, the United
Kingdom, and Greece, five countries reduced the carbon intensity of their electricity production
Portugal, Netherlands, Spain, Austria, and Finland. Electricity generation from renewable energy
increased in each of these countries. Were it not for expanded use of renewable energy, Germany
would have experienced even greater growth in the carbon intensity of their electricity production.
France and Sweden have very low levels of carbon intensity due to their extensive use of nuclear and
hydroelectric resources to generate electric power (see Figure 1).
The shares of natural gas in total fossil fuel consumption by country for the four full calendar
years 2004 to 2007 are displayed in Figure 2. The share of natural gas for the aggregate of the twelve
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countries increased from 33 to 36 percent from 2004 to 2007. Expanding use of gas in Greece, the
Netherlands, the United Kingdom, Portugal, Spain, and Sweden offset declines in Finland and
Denmark and the flat trends in the other countries. The empirical model below may shed some
light on the role of relative fuel and carbon permit prices in these fuel share adjustments.
Nevertheless, the shift to less carbon intensive natural gas, the increasing generation of renewable
energy (see Figure 3), and the declining carbon intensity in several countries suggests that carbon
dioxide emission abatement may have occurred even during the trial period for the EU ETS.
During the first phase of the EU ETS, exchanges emerged to trade spot and futures contracts
derived from Phase I and Phase II. Because the Phase I allowances could not be carried over into
Phase II, future contracts based either on Phase I or Phase II allowances were independently priced.
For a variety of reasons, evidence suggests that allowances were over-issued during Phase I. As a
result, toward the end of the trial period in 2007, emission allowance prices fell to zero (see Figure
4). However, when allocations for the second phase were determined, additional oversight was given
to the European Commission and this appears to have resulted in a binding Phase II cap.
In the short-run, fuel substitution decisions are likely linked to the relative costs of obtaining or
selling marginal allowances and for this reason we use spot prices for carbon permits as reported by
PointCarbon in our model. This is a reasonable proposition and can be tested in the analysis below
by determining whether carbon prices are statistically significant in the input demand functions.
In interpreting our results, we should point out that the pricing incentives for short-run
substitution, which our model measures, may be different that the incentives for new investment in
capacity. Said somewhat differently, fuel-switching and related actions are short-term tactics meant
to minimize costs, while investments are made with an expected flow of profits in mind. Ellerman
(2008), for one, argues that the investment decisions were guided by market valuations of traded
Phase II allowances, as given by prices for the December 2008 futures contract also shown in Figure
4. And while there is no clear reason to expect that such investments were brought forward into
12
Phase I when spot prices were low, investment studies will need to reconcile the conflicting
incentives given by the pricing of Phase I and Phase II allowances.
Prices for natural gas, petroleum, and coal paid by electricity generators in the United Kingdom
are directly observable on a monthly basis from the United Kingdom (2009). For the other
countries, this study must estimate monthly prices based upon regional monthly wholesale prices
published by Platts (2009) for various market hubs in the EU and quarterly prices published by the
International Energy Agency (2009) that measure prices paid by end users including taxes. Quarterly
averages are computed from the monthly data from Platts. Next, we compute the ratio of these
quarterly averages to the quarterly data reported by the IEA. These ratios represent the spreads
between prices in each country and the market hub. Monthly estimates for prices in the remaining
eleven countries result from multiplying these ratios by the monthly data from Platts.
6. Model Estimation Results
The above econometric model of the restricted variable cost function (4) and the three variables
input demand functions given by (5) are estimated as a system of equations. Given that fuel and
carbon permit prices and output could be endogenous, an instrumental variables estimator is
needed. The Generalized Method of Moments (GMM) estimator provides for consistent parameter
estimates and allows correction of the standard errors for heteroscedasticity and autocorrelation in
the error terms. The instruments include lagged values of input prices, generation levels for nuclear,
hydroelectric, and renewable generation, total power generation, and country and monthly dummy
variables. The lagged instruments vary for each equation and correspond with the specific
specification of the right-hand side variables in equations (4) and (5). So, for example, the
instruments for the input demand functions include square roots of lagged price ratios. This
approach is intended to ensure that the instruments are correlated with the explanatory variables but
remain independent of the error terms. Country and monthly dummy variables are included as
instruments in all four equations.
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The test of the over-identifying restrictions for the unrestricted model is 93.2 with a probability
value of 0.51, which suggests that the model given above cannot be rejected. Only one technological
change coefficient was significant at the 5 percent level or less and that was for natural gas, which
indicated gas-saving technological change, most likely reflecting the steady improvements in the
thermal efficiency of combined cycle gas turbine technology. As a result, the hypothesis of non-
neutral technological change is tested by computing a test statistic equal to the difference between
the test statistics of the over-identifying restrictions for the unrestricted model and the restricted
model with neutral technological change imposed via the following parameter restrictions:
zi = 0 i, yz = zz = zn = zr = 0 . The value of this test statistic is 9.16 with a probability value
of 24%, indicating that the null hypothesis of neutral technological change cannot be rejected. This
finding suggests that at least for the early stages of the EU ETS, exogenous technological change has
not induced pervasive changes in the relative factor intensities of power generation. 5
As a result, the following presentation of results will focus on the estimates for the model
assuming neutral technological change. In this case, the instruments involving the trend proxy for
technological change are dropped and the model is re-estimated. The test of the over-identifying
restrictions is 92.3 with 85 degrees of freedom and a probability value of 27.8%. As expected, the
model with neutral technological change cannot be rejected. The parameter estimates appear in
Table 3. Of the 36 country dummy variables, 20 have probability values that suggest less than a 5%
chance of being zero while three have probability values less than 10%. For the 21 coefficients on
the relative price, output, and quasi-fixed factors, 15 have probability values less than 5 percent and
two less than 10 percent. The coefficients involving output, however, have relatively high probability
5The parameter estimates for the neutral technological change model are relatively close to those for the non-neutral
model. There are no sign changes between the two sets of estimates.
14
values, although as we shall see below the output elasticities contain the other parameters and often
are highly significant.
The goodness of fit statistics are reported in Table 4 indicate an excellent fit of the data with R-
squared coefficients ranging from 0.97 to 0.99. The Durbin-Watson statistics indicate first-order
autocorrelation, which is why we allow a first order moving average correction in the GMM
estimation. An explicit structural correction for autocorrelation is not pursued because it could
introduce specification error and would violate the conditions that allow the input demand functions
to be integrated back to the cost function. Given the relatively large sample size used in this study,
the theoretical result that the GMM estimates are asymptotically efficient seems reasonable.
Two sets of elasticities can be computed from the restricted cost function and the input
demand functions. The elasticities of demand holding levels of the quasi-fixed factors are equivalent
to short-run elasticities often defined in the literature. Given that we are estimating a separable cost
sub-function holding capital and labor fixed, this study uses the term partial adjustment for these
elasticities, which are reported in Table 5. 6 The demands for carbon permits and fuels are essentially
perfectly inelastic assuming levels of nuclear, hydroelectric, and renewable resources are fixed. In
some sense, these elasticities are an artifact of this extremely restrictive ceteris paribus measurement
and would explain the violation of the concavity conditions for these partial adjustment elasticities.
Nonetheless, the other elasticities reported in Table 5 show that greater levels of nuclear and
renewable resources reduce the demands for fuels and carbon, as one would expect. Indeed, the
convexity conditions are satisfied for all observations. Likewise, predicted marginal cost is positive
6Using annual data on installed capacity and monthly estimates of replacement costs for new capacity, we generated
monthly time series for capital stocks. Likewise, using quarterly data for employment levels in the electric
generating and transmission sector and quarterly data on wages, we generated price and quantities for labor. We then
estimated a model using (4) and (5) with carbon, labor, and energy as variable inputs and non-carbon energy and
capital as quasi-fixed inputs. The test of the overidentifying restrictions was decisively rejected, suggesting that the
extrapolated data could be introducing measurement errors. This finding verifies our approach to base our analysis
on reported data, which admittedly can only allow measurement of the short-run flexibility of the power grid to
switch generation sources in response to relative prices and output.
15
for all observations. Moreover, the marginal cost function shifts upward with rising carbon and fuel
prices and downward with more nuclear generation.
The full adjustment elasticities allow the levels of quasi-fixed factors to change. These
elasticities result from solving the envelope conditions for the quasi-fixed factors and differentiating
these functions to obtain the elasticities. These derivations appear in Appendix A. The estimated full
adjustment elasticities appear in Table 6. Overall, they reflect very inelastic factor demands. The
demand for carbon permits is very inelastic with an own price elasticity of -0.068 indicating that
ceteris paribus a 10 percent increase in carbon permit prices results in less than a one percent
reduction in carbon use. This inelasticity reflects significant complementarity between carbon
emissions and fuels. While nuclear and renewable energy are substitutes with carbon permits, the
cross price elasticities indicated very limited substitution. For example, a 10 percent reduction in the
price of renewable energy induces slightly less than a 2 percent drop in carbon emissions.
The output elasticities are all positive as expected with the natural gas output elasticity at more
than 3, reflecting the well-known role of gas in leveling peaks and troughs in seasonal demand. The
output elasticity of carbon permits is also significant at more than 2, suggesting that demand side
reductions, if they can be achieved, would substantially reduce the demand for carbon emissions.
The marginal cost elasticities are also all significant. The estimated carbon price elasticity of
marginal cost is 0.211 (see Table 6), indicating that for every 10 percent increase in carbon prices,
the marginal generation cost of electricity increases by 2 percent. If fuel prices increase with carbon
prices, the sum of the carbon and fuel price marginal cost elasticities (see Table 6) suggest that for
every 10 percent increase in carbon prices, the marginal cost of electricity could increase 8 percent. 7
7 A markup pricing model was also estimated in which two additional estimating equations were added to equations
(4) and (5), a demand for electricity and a price markup over marginal cost equation derived by assuming electric
utilities are engage in monopoly pricing. The estimated price elasticities of demand holding output fixed are very
similar to the results presented above. The elasticities in this context, which allows for endogenous output and
16
The last set of elasticities is the Morishima elasticities of substitution, which are a unit-less
measure of substitution. The analysis by Blackorby and Russell (1989) proves that the Morishima
elasticity is a superior measure of substitution for this study because it provides a clear distinction
between substitutions induced by carbon permit prices versus other input price changes. Morishima
elasticities are defined as follows:
M ij =
ln x j
- =
(
ln xi - ln xi x j
.
) (6)
ln wi ln wi ln wi
These elasticities measure the curvature of an isoquant, or the percentage change in a factor input
ratio for a given percentage change in price, holding all other factor prices constant. As the above
equation illustrates, the effect of varying wi on the factor input ratio, xi x j , is composed of two
parts the effect of wi on xi and the effect of wi on x j . Blackorby and Russell [7] show that these
elasticities are inherently asymmetric.
The estimated Morishima elasticities of substitution appear in Table 7. All but four of these 20
substitution elasticities are significant at the five percent level. On the other hand, all of them are less
than one. For instance, the ratio of nuclear resources to carbon emissions rises 0.852 percent for
every percent increase in carbon emission prices. In contrast, the ratio of renewable generation to
carbon emissions increases only 0.345 percent for each percent change in carbon emission prices.
This suggests that nuclear energy serves as an important swing fuel in meeting carbon emission
constraints.
Finally, there is significant complementarity between carbon emissions and high carbon fuels
when the prices for the latter increase but a small an insignificant response of carbon emission
prices, are very complicated and at this juncture of this research would obscure our focus on technological change
and substitution. Nevertheless, this approach may merit future investigation.
17
relative to high carbon fuel consumption as carbon emission permits change. This reflects the very
limited reductions in high carbon fuel consumption in response to carbon permit prices during the
first phase of the EU ETS. Overall, the Morishma elasticities reflect very limited price-induced
substitution between alternative generation fuels in the production of electricity in the short-run.
7. Conclusions
This study provides an analysis of the underlying economic forces inducing adjustments in electricity
production factor intensities during the first phase of the European Union's Emissions Trading
System regulating emissions of greenhouse gas emissions. Our empirical analysis examines the
demand for carbon permits, carbon based fuels, and carbon-free energy for 12 European countries
using monthly data on fuel use, prices, and electricity generation. Our empirical model is unique
because it considers all possible sources of generation within one model. Heretofore, empirical
models of factor substitution in electric power generation were confined to studies of steam power
generation using combustible fuels apart from nuclear of hydroelectric generation because prices for
the latter fuels are not observable. Our approach uses a restricted variable cost function treating
these factors as quasi-fixed to estimate the shadow value of these resources.
Our results suggest several conclusions. Perhaps the most important finding is that very limited
substitution possibilities in combination with low carbon permit prices may explain the limited
success of the EU ETS in achieving carbon emission reductions in the electric power generation
sector. While our empirical results demonstrate that switching to nuclear and renewable energy is
induced by higher carbon permit prices, the extent of this substitution is limited. Other substitution
possibilities are also very limited. These results suggest that the current configuration of electricity
generating assets is inflexible and that to achieve substantial reductions in carbon emissions more
flexibility must be introduced, most likely from significant investments in new generation capacity.
18
A second implication of the model is that the effects of the cap on electricity prices can be
significant, if fuel prices increase together with carbon permit prices as is likely. In this case, our
estimates suggest that for every 10 percent rise in carbon and fuel prices, the marginal cost of
electric power generation increases by 8 percent in the short-run. Consequently, if EUA allocations
are fixed and fuel prices exogenous, the degree to which the costs of a carbon cap are passed on to
consumers in the short run will be determined by how open the system is to alternative carbon
offset, such as CERs, and the relative price of those allowances.
The European experience points to the importance of starting early down a low-carbon path.
Because fixed investments in power generation are long-lived and irreversible, inflexibilities resulting
from past investments will be long-lived as well. Consequently, it is important for countries that do
not currently cap greenhouse gas emissions but hope to promote growth that is less carbon intensive
to find alternative polices that consider the costs of future adjustments.
References
Blackorby, Charles and R.Robert Russell. 1989. Will the real elasticity of substitution please stand
up? American Economic Review 79(4), 882-888.
Caves, Douglas W. and Laurits R. Christensen. 1980. Global properties of flexible functional forms.
American Economic Review 70(3), 422432.
Considine, Timothy J. and Donald F. Larson. 2006. The environment as a factor of production.
Journal of Environmental Economics and Management 52(3), 645-62.
Convery, Frank J. and Luke Redmond. 2007. Market and price developments in the European
Union Emissions Trading Scheme. Review of Environmental Economics and Policy 1(1), 88-
111.
Ellerman, A. Denny. 2008. The EU Emission Trading Scheme: prototype of a global system? The
Harvard Project on International Climate Agreements, Harvard University, Kennedy School of
Government, August, Discussion Paper 08-02.
19
Ellerman, A.Denny and Barbara K. Buchner. 2007. The European Union Emissions Trading
Scheme: origins, allocation, and early results. Review of Environmental Economics and Policy
1(1), 66-87.
Ellerman, A. Denny, Barbara K. Buchner and Carlo Carraro. 2007. Allocation in the European
Emissions Trading Scheme: Rights, Rents and Fairness. Cambridge: Cambridge University
Press
Europa. 2007. Emission Trading Scheme (EU ETS). Available on the Internet at:
http:\ec.europa.eu.
Halvorsen, Robert and Tim R. Smith. 1986. Substitution possibilities for unpriced natural resources:
restricted cost functions for the Canadian metal mining industry. The Review of Economics
and Statistics 68(3), 398-405.
International Energy Agency. 2009. Energy, prices, and taxes (quarterly). Available on the Internet at
http://www.iea.org/Textbase/stats/index.asp .
Larson, Donald F., Philippe Ambrosi, Ariel Dinar, Shaikh Mahfuzur Rahman and Rebecca Entler.
Forthcoming. Carbon markets, institutions, policies and research. The International Review of
Environmental and Resource Economics.
Morrison, Catherine. 1988. Quasifixed inputs in U. S. and Japanese manufacturing: A generalized
Leontief restricted cost function approach. The Review of Economics and Statistics 70(2), 275
287.
Platts Energy Service. 2009. Custom Energy Market Data. Available on the Internet at
http://www.platts.com/Analytic%20Solutions/Custom/.
United Kindgom. 2009. Average prices of fuels purchased by the major UK power producers.
Department of Business Enterprise and Regulatory Reform. Available on the Internet at
http://www.berr.gov.uk/whatwedo/energy/statistics/source/prices/page47818.html
Watanabe, Rie and Guy Robinson. 2005. The European Union Emissions Trading Scheme (EU
ETS). Climate Policy 5(1), 10-14.
20
Table 1: Average annual electric power generation by type and net imports in gigawatt hours, January 2002
to March 2008
Fossil Indigenous
Country Nuclear Hydroelectric Renewables Net Imports Total
Fuel Production
Austria 1,921 0 2,969 96 4,985 372 5,358
Denmark 2,732 0 2 529 3,263 -246 3,017
Finland 3,479 1,837 1,041 14 6,371 739 7,109
France 4,856 35,470 5,164 176 45,666 -4,268 41,398
Germany 30,428 12,734 2,207 2,255 47,624 -694 46,930
Greece 4,061 0 400 105 4,566 251 4,817
Netherlands 7,454 311 8 185 7,958 1,249 9,207
Portugal 2,777 0 827 168 3,773 418 4,190
Spain 13,636 4,846 2,592 1,607 22,681 -182 22,499
Poland 11,480 0 290 20 11,790 -629 11,161
Sweden 1,075 5,645 5,315 82 12,117 112 12,229
United Kingdom 24,784 6,022 646 255 31,706 493 32,199
21
Table 2: Average Annual Fossil fuel consumption in terajoules, January 2002 to March 2008
Country Coal Petroleum Natural Gas
Austria 4,348 529 6,096
Denmark 15,192 1,628 3,382
Finland 6,140 971 6,930
France 18,236 1,390 2,572
Germany 107,066 2,654 44,666
Greece 7 6,991 6,779
Netherlands 21,346 57 24,951
Portugal 11,041 2,625 5,813
Spain 57,891 13,329 25,930
Poland 77,898 8 3,103
Sweden 698 526 252
United Kingdom 102,979 4,510 100,373
22
Figure 1: Carbon intensity of indigenous electricity production by country, 2004 to 2007
23
Figure 2: Shares of natural gas in fossil fuel use in power generation by country, 2004 to 2007
24
Figure 3: Renewable electricity generation, 2002-2007
3,500
3,000
2,500
2,000
Gigawatts
1,500
1,000
500
0
Austria Denmark Finland France Germany Greece Netherlands Portugal Spain Poland Sweden United
Kingdom
2004 2005 2006 2007
25
Figure 4: EU ETS carbon emission allowance prices
30.00
25.00
20.00
Eurodollar / ton
15.00
10.00
5.00
0.00
Dec- Feb- Apr- Jun- Aug- Oct- Dec- Feb- Apr- Jun- Aug- Oct- Dec- Feb- Apr- Jun- Aug- Oct- Dec- Feb- Apr- Jun- Aug-
04 05 05 05 05 05 05 06 06 06 06 06 06 07 07 07 07 07 07 08 08 08 08
EUA Price December 2007 EUA Price December 2008
26
Table 3: Generalized Method of Moments Estimates
Parameter Estimate t-ratio P-value Parameter Estimate t-ratio P-value
1
1
0.006 0.0 [.965] 6
3
-4.158 -3.7 [.000]
2
1
-0.308 -2.7 [.008] 7
3
2.796 1.5 [.141]
1
3 -0.522 -3.1 [.002] 8
3
-1.315 -1.5 [.134]
4
1
1.199 2.3 [.021] 9
3
-0.304 -0.1 [.928]
5
1
0.696 1.8 [.070] 3
10 -31.870 -11.1 [.000]
6
1
-1.346 -9.7 [.000] 3
11 -0.441 -0.2 [.828]
7
1
-1.272 -6.0 [.000] 3
12 41.760 9.5 [.000]
8
1
0.027 0.2 [.814] 11 0.805 13.5 [.000]
9
1
2.092 6.4 [.000] 12 -0.010 -1.9 [.056]
1
10 -0.188 -0.6 [.572] 13 0.005 1.0 [.316]
1
11 -0.168 -0.7 [.510] 22 6.443 13.7 [.000]
1
12 3.129 7.4 [.000] 23 0.061 1.7 [.081]
1
2
-0.251 -0.2 [.823] 33 3.426 9.4 [.000]
2
1
1.857 1.6 [.100] y1 -0.011 -0.9 [.360]
3
2
-5.368 -3.5 [.001] y2 -0.026 -0.7 [.493]
4
2
11.970 2.1 [.033] y3 -0.020 -0.7 [.514]
5
2
18.810 4.2 [.000] yy 0.001 1.2 [.228]
6
2
-10.840 -7.9 [.000] n1 -0.801 -20.0 [.000]
7
2
-14.840 -6.2 [.000] n2 -6.282 -19.0 [.000]
8
2
2.030 1.9 [.055] n3 -3.633 -15.2 [.000]
9
2
25.810 7.0 [.000] r1 -0.457 -7.9 [.000]
2
10 17.240 4.4 [.000] r2 -4.537 -9.2 [.000]
2
11 -0.370 -0.2 [.873] r3 0.389 1.0 [.296]
2
12 11.830 2.1 [.038] yn -0.004 -3.8 [.000]
1
3
2.161 2.0 [.047] yr -0.010 -4.3 [.000]
2
3
-7.926 -8.5 [.000] nn 0.051 2.3 [.019]
3
3
0.530 0.4 [.680] nr 0.072 3.2 [.002]
4
3
3.874 0.9 [.392] rr 0.161 2.7 [.007]
5
3
-22.080 -4.9 [.000]
27
Table 4: Summary Fit Statistics
Mean of Standard Error
Equation Dependent of Regression R-Squared Durbin-
Variable Watson
Variable Cost 331.2 32.90 0.9935 0.6793
Carbon Permits 4.512 0.5289 0.9888 0.6406
High Carbon Fuels 38.38 7.217 0.9706 0.6266
Low Carbon Fuels 21.28 4.968 0.9710 0.5968
Table 5: Partial Adjustment Elasticities of Demand and Marginal Cost (asymptotic t-ratios in parentheses)
Carbon Coal & Oil Natural Gas Nuclear Renewable Total
Price Prices Prices Generation Generation Output
Carbon 0.004 -0.009 0.005 -0.899 -0.122 1.720
(2.3) (1.9) (1.0) (30.4) (10.3) (31.3)
Coal & Oil -0.005 -0.012 0.017 -0.906 -0.184 1.745
(1.9) (1.7) (1.7) (18.8) (9.0) (21.2)
Natural Gas 0.004 0.019 -0.023 -0.936 0.038 1.807
(1.0) (1.7) (1.6) (14.9) (1.4) (15.5)
Marginal Cost 0.258 0.220 0.348 -0.243 -0.029 0.318
(12.7) (31.3) (21.2) (37.2) (10.5) (15.5)
28
Table 6: Full Adjustment Elasticities of Demand and Marginal Cost (asymptotic t-ratios in parentheses)
Natural Renewab
Carbon Coal & Nuclear Total
Input Gas le
Price Oil Price Prices Output
Prices Prices
Carbon -0.068 -0.119 -0.056 0.052 0.191 2.055
(7.7) (8.6) (5.6) (3.6) (12.2) (31.2)
Coal & Oil -0.082 -0.127 -0.036 0.068 0.177 1.952
(7.3) (7.3) (2.8) (3.8) (10.7) (20.8)
Natural Gas -0.066 -0.061 -0.150 -0.022 0.299 3.067
(5.1) (2.6) (5.3) (1.4) (10.2) (14.2)
Nuclear 0.042 0.058 0.054 -0.154 0.000 0.077
(12.1) (13.9) (12.4) (16.5) (2.5) (17.3)
Renewable 0.312 0.539 -0.079 0.028 -0.800 0.413
(4.8) (5.7) (1.1) (2.5) (5.0) (5.2)
Marginal Cost 0.211 0.338 0.284 0.041 0.009 0.239
(29.8) (20.6) (13.8) (11.5) (3.4) (12.0)
Table 7: Full Adjustment Morishima Elasticities of Substitution
(asymptotic t-ratios in parentheses)
Carbon Coal & Natural Nuclear Renewable
Input
Price Oil Price Gas Prices Prices Prices
Carbon -0.014 0.003 0.380 0.110
(3.2) (0.2) (5.2) (9.2)
Coal & Oil 0.009 0.066 0.667 0.185
(1.8) (1.9) (6.2) (9.1)
Natural Gas 0.094 0.114 0.071 0.204
(3.0) (3.1) (0.8) (6.7)
Nuclear 0.852 0.868 0.778 0.801
(4.9) (4.9) (4.8) (5.0)
Renewable 0.345 0.331 0.453 0.182
(14.0) (13.3) (12.3) (10.0)
29
Appendix A
Derivation of the Elasticities
This appendix provides the derivations for the restricted and unrestricted elasticities of
demand and substitution. The derivations all follow from differentiating the following short-run
restricted cost function:
3 3
( )
3 3
ij wit w jt + yi witYt1 2 + zi wit Zt1 2
12
i =1 j =1 i =1 i =1
Gt = Yt 3
it yy t (
+ w Y + 2 Y 1 2 Z 1 2 + Z
yz t t zz t )
i =1
3 3
ni wit N t + ri wit Rt
12 12
i =1 i =1
+Yt1 2 3 (A1)
(
+ w Y 1 2 N 1 2 + Y 1 2 R1 2 + Z 1 2 N 1 2 + Z 1 2 R1 2
it yn t t yr t t zn t t zr t t )
i =1
( )
3 3 12
+ wit nn N t + 2 nr N t1 2 Rt1 2 + rr Rt + wit ci Dct
i =1 i =1 c=1
The variables are defined in the paper above. The three input demand functions are the partial
derivatives of the restricted cost function (A1) with respect to input prices:
Gt 12 3 w jt
12
= Xit = c Dct + Yt ij
i
+ yiYt1 2 + zi Zt1 2 + yyY + 2 yzYt1 2 Zt1 2 + zz Zt
wit c=1
j =1 wit
+ ni (Yt N t ) + ri (Yt Rt ) + ynYt N t1 2 + yrYt Rt1 2 + zn (Yt Zt N t ) + zr (Yt Zt Rt )
12 12 12 12
(A2)
+ nn N t + 2 nr (N t Rt ) + rr Rt
12
i
The marginal cost function is as follows:
30
Gt
( )
3 3
3 3 3
= ij wit w jt + yi witYt + zi wit Zt1 2
12 12
Yt i =1 j =1 2 i =1 i =1
12 12
N R
( )
3
1 3 1 3
+ wit 2 yyYt + 3 yzYt Z 12 12
+ zz Zt + ni wit t + ri wit t (A3)
Yt Yt
t
i =1 2 i =1 2 i =1
1 Zt Rt
12 12
3
1 Zt N t
+ wit yn N t + yr Rt + zn
12 12
+ zr
i =1 2 Yt 2 Yt
The conditional own-price elasticities of demand are defined as follows:
ln Xit 1 Yt 2 w jt
12
= - ij i (A4)
ln wit 2 Xit j =1 wit
While the cross-price elasticities of demand are:
1 Yt w jt
12
ln Xit
= - ij i j (A5)
ln w jt 2 Xit wit
The conditional input demand elasticities with respect to output are:
ln Xit Y 3
12
w jt 3
= ij + yiYt1 2 + zi Zt1 2 + 2 yyY + 3 yzYt1 2 Zt1 2 + zz Zt
lnYt Xit j =1 wit
2
(A6)
1 Nt
12
Rt
12
Zt N t
12
Zt Rt
12
+ ni + ri + zn + zr + yn N t + yr Rt i
12 12
2 Yt
Yt Yt Yt
The conditional input demand elasticity with respect to levels of hydroelectric and nuclear
generation is as follows:
ln Xit N t 1 Yt Rt
12 12
= ni + ynYt + zn Zt + nn + nr
12 12
(A7)
ln N t Xit 2 N t Nt
Likewise, the conditional input demand elasticity with respect to renewable generation is:
31
Rt 1 Yt Nt
12 12
ln Xit
= ri + yrYt + zr Zt + rr + nr
12 12
(A8)
ln Rt Xit 2 Rt Rt
Finally, the conditional input demand elasticity with respect to technological change is:
1 1 1 Yt Rt
12 12 12 12
ln Xit Yt Yt N t
= Yt zi + yz + zz + zn + zr Z (A9)
Z t Xit 2 Zt Zt Zt t
The concavity conditions are determined by calculating the Eigen values of the three-by-three
matrix formed from the partial derivatives of (A1) with respect to input prices.
The elasticity of marginal cost with respect to output is as follows:
ln MCt 1 3 3 3
3 1 2 1 2
= yi witYt + wit 2 yyYt + yzYt Zt
12
lnYt MCt 4 i =1 i =1
2
1 3 1 3
- ni wit N t1 2Yt -1 2 - ri wit Rt1 2Yt -1 2 (A10)
4 i =1 4 i =1
1
3
1
- wit zn Zt1 2 N t1 2Yt -1 2 + zr Zt1 2 Rt1 2Yt -1 2
i =1 4 4
The technical change elasticity of marginal cost is given by:
ln MCt 1 1 3 3
3
zi wit Zt + wit yzYt Zt + zz
-1 2 1 2 -1 2
=
Z t MCt 2 i =1 i =1
2
(A11)
N 1 2 Rt
12
1 3
+ wit zn t
+ zr Y Z
4 i =1 Yt Zt
t t
The partial derivatives of marginal cost with respect to observed variable inputs are as follows:
32
ln MCt wit 3
12
w jt 3
= ij + yiYt1 2 + zi Zt1 2
ln wit MCt j =1 wit
2
12 12
1 N 1 R
(
+ 2 yyYt + 3 yzYt Z 12 12
t )
+ zz Zt + ni t
2 Yt
+ ri t
2 Yt
(A12)
1 ZN
12
1 ZR
12
+ yn N 12
+ yr R
12
+ zn t t + zr t t
2 Yt 2 Yt
t t
The partial derivatives of marginal cost with respect to the quasi-fixed inputs are as follows:
1 1 3 1 N t Zt
12
ln MCt 1 3
ni wit N t Yt + wit yn N t + zn
1 2 -1 2
=
12
ln N t MCt 4 i =1 2 i =1 2 Yt
(A13)
1 1 3 1 Rt Zt
12
ln MCt 1 3
ni wit Rt Yt + wit yn Rt + zn
1 2 -1 2
=
12
ln Rt MCt 4 i =1 2 i =1 2 Yt
The convexity conditions for the quasi-fixed level of hydroelectric and nuclear generation
resources is as follows:
Y 1 2 Zt
12 12
Gt 1 3 Y 1 12 3
= µnt = ni wit t
*
+ Yt wit yn + zn
t
N t 2 i =1 Nt 2 i =1
Nt Nt
(A14)
Rt
12
3
+ wit nn + nr
i =1 Nt
This derivative must be negative. The other convexity condition for renewable energy is:
Y 1 2 Zt
12 12
Gt 1 3 Y 1 12 3
= µrt = ri wit t
*
+ Yt wit yr + zr
t
Rt 2 i =1 Rt 2 i =1
Rt Rt
(A15)
Nt
12
3
+ wit rr + nr
i =1 Rt
33
As Morrison (1988) shows, the convexity conditions can be solved, in this case simultaneously,
for the equilibrium levels of the quasi-fixed inputs. First, consider the solution for equilibrium
levels of nuclear and hydroelectric generation:
1 3 12
( )
3
1 12 3 3
µ - wit nn = N
*
nt t
-1 2
ni witYt + Yt wit ynYt + zn Zt + wit nr Rt
12 12 12
i =1 2 i =1 2 i =1 i =1
2
1 3
( )
3 3
1
2 ni witYt1 2 + Yt1 2 wit ynYt1 2 + zn Zt1 2 + wit nr Rt1 2 (A16)
2
N t = i =1 i =1
3
i =1
µnt - wit nn
*
i =1
Similarly, solving (A11) for the level of renewable generation is as follows:
2
1 3 12
1 12 3
( )
3
2 ri witYt + 2 Yt wit yrYt + zr Zt + wit nr N t
12 12 12
Rt = i =1 i =1
3
i =1
(A17)
µrt - wit rr
*
i =1
Substituting (A10) into (A11) and solving for the equilibrium level of renewable generation
yields:
2
3
GM t RM t + nr wit HM t 2
NN t
Nt =
* i =1
2 = ,
QM RM - NDt
3
t t nr wit
i =1
2
(A18)
3
HM t QM t + nr wit GM t 2
RN t
Rt =
* i =1
2 =
QM RM - RDt
3
t t nr wit
i =1
where
34
3
QM t = µ + nn wit
*
nt
i =1
3
RM t = µ + rr wit
*
rt
i =1
(A19)
3
( )
3
1
GM t = Yt1 2 ni wit + wit ynYt1 2 + zn Zt1 2
2 i =1 i =1
3
( )
3
1
HM t = Yt1 2 ri wit + wit yrYt1 2 + zr Zt1 2
2 i =1 i =1
The elasticities of demand for these quasi-fixed inputs with respect to carbon and fossil
energy prices take the following form:
wit N t w NN t 3
= 2 it* rr GM t + RM t GPit + nr HM t + HPit wit
N t wit
* 2
N t NDt i =1
(A20)
NN t 3
- rrQM t + nn RM t - 2 nr wit i
2
NDt i =1
wit Rt w RN t 3
= 2 it 2
nn HM t + QM t HPit + nr GM t + GPit wit
Rt wit
* *
Rt RDt i =1
(A21)
RN t 3
- rrQM t + nn RM t - 2 nr wit i
2
RDt i =1
where
1
2
(
GPit = Yt1 2 ni + ynYt1 2 + zn Zt1 2 )
(A22)
1
2
(
HPit = Yt1 2 ri + ynYt1 2 + zn Zt1 2 )
The own-price elasticities of demand for the two quasi-fixed inputs are as follows:
µnt N t
*
NN t2 µnt
*
= -2 3
RM t *
N t* µnt
*
NDt Nt
(A23)
µrt Rt
*
RN 2 µ*
= -2 t3 QM t rt
Rt* µrt
*
RDt Rt*
35
While the cross-price elasticities are:
µrt N t*
*
NN t NN t µ*
= 2 GM t - QM t rt
N t* µrt
*
NDt2
NDt Nt
*
(A24)
µnt Rt*
*
RN t RN t µnt
*
= 2 HM t - RM t *
Rt* µnt
*
RDt2
RDt Rt
The output elasticities are respectively:
Yt N t NN t 3
Y
= 2 2
RM t GYt + nr wit HYt t*
N t Yt
*
NDt i =1 Nt
(A25)
Yt Rt RN t 3
Y
= 2 2
QM t HYt + nr wit GYt t*
Rt Yt
*
RDt i =1 Rt
where
1 3 3
1 3
GYt = ni wit + zn wit Zt1 2 + 2 yn wit
4Yt1 2 i =1
i =1 i =1
(A26)
1 3 3
1 3
HYt =
4Yt1 2 i =1
ri wit + zr wit Zt1 2 + yr wit
i =1 2 i =1
Finally, the technological change elasticites are as follows:
12
1 N t* 1 NN t Yt 3
3
= zn RM t + nr zr wit wit
N t Z t 2N t* NDt2 Zt
*
i =1 i =1
12
(A27)
1 R 1 RN t Yt
*
3
3
= zrQM t + nr zn wit wit
t
Rt Z t 2Rt* RDt2 Zt
*
i =1 i =1
The partial adjustment elasticities for carbon and energy inputs allow the quasi-fixed
factors to adjust, at this stage assuming output and prices fixed. The general expressions for these
full adjustment elasticities of demand are as follows:
FA
ln X it ln X it ln X it ln N t* ln X it ln Rt*
= + + i, j (A28)
ln w jt ln w jt ln N t* ln w jt ln Rt* ln w jt
N ,R
36
where
N t* Xit N t* 1 Yt Rt*
12 12
= * * ni + ynYt + zn Zt + nn + nr *
12 12
Xit N t Xit 2 N t
* *
Nt
(A29)
R Xit R 1 Yt N t*
12 12
* *
t
= * ri + yrYt + zr Zt + rr + nr * i
t
12 12
X Rt X 2 Rt
** *
Rt
it it
*
and where Xit are the levels of variable inputs at equilibrium levels of the quasi-fixed factors.
The elasticities of variable input demands with respect to prices for quasi-fixed factors are as
follows:
FA
ln X it ln X it ln N t* ln X it ln Rt*
= +
ln µ nt ln N t* ln µ nt ln Rt* ln µ nt
FA
(A30)
ln X it ln X it ln N t* ln X it ln Rt*
= + i
ln µrt ln N t* ln µrt ln Rt* ln µrt
Similarly, the partial adjustment output and technological elasticities are defined as follows:
FA
ln X it ln X it ln X it ln N t* ln X it ln Rt*
= + +
lnYt lnYt N ,R
ln N t* lnYt ln Rt* lnYt
FA
(A31)
ln X it ln X it ln X it ln N t* ln X it ln Rt*
= + +
ln Zt ln Zt N ,R
ln N t* ln Zt ln Rt* ln Zt
The full adjustment marginal cost elasticities are defined in a similar fashion.
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FA
ln MCt ln MCt ln MCt ln N t* ln MCt ln Rt*
= + + i
ln wit ln wit N * , R*
ln N t* ln wit ln Rt* ln wit
FA
ln MCt ln MCt ln N t* ln MCt ln Rt*
= + i = N,R
ln µit ln N t* ln µit ln Rt* ln µit
FA
(A32)
ln MCt ln MCt ln MCt ln N t* ln MCt ln Rt*
= + +
lnYt lnYt N * , R*
ln N t* lnYt ln Rt* lnYt
FA
ln MCt ln MCt ln MCt ln N t* ln MCt ln Rt*
= + +
Z Z N * , R*
ln N t* ln Z ln Rt* ln Z
All the full adjustment elasticities are evaluated at the grand mean of the observations.
References
Morrison, Catherine. 1988. Quasifixed inputs in U. S. and Japanese manufacturing: A generalized
Leontief restricted cost function approach. The Review of Economics and Statistics 70(2), 275
287.
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