The Valley of Death for New Energy Technologies

Abstract

It is often claimed that a difficulty of raising investment funds prevents promising new energy technologies from attaining commercial viability. We examine this issue using a dynamic intertemporal model of the displacement of fossil fuel energy technologies by non-fossil alternatives. Our model highlights the fact that since capital used to produce energy services from fossil fuels is a sunk cost, it will continue to be used so long as the price of energy covers merely short-run operating costs. Until fossil fuels are abandoned, the price of energy is insufficient to cover even the operating costs of renewable energy production, let alone provide a competitive return on the capital employed. The full long-run costs of renewable energy production are not covered until some time after fossil fuels are abandoned.

Full text

The Valley of Death for New Energy Technologies
by∗
Peter R. Hartley†
and
Kenneth B. Medlock III‡
January 11, 2016
Abstract
It is often claimed that scant financing prevents promising new energy technologies from attaining
commercial viability. We examine this issue using a dynamic intertemporal model of the displace-
ment of fossil fuel energy technologies by non-fossil alternatives. Our model highlights the fact that
since capital used to produce energy services from fossil fuels is a sunk cost, it will continue to be
used so long as the price of energy covers merely short-run operating costs. Until fossil fuels are
abandoned, the price of energy is insufficient to cover even the operating costs of renewable energy
production, let alone provide a competitive return on the capital employed. The full long-run costs
of renewable energy production are not covered until some time after fossil fuels are abandoned.
Keywords: Energy innovation, energy transition, valley of death
∗We thank three referees and seminar participants at the University of Western Australia and the University of
Queensland for valuable comments, and Xinya Zhang and Grace Gao for valuable research assistance.
†George and Cynthia Mitchell Professor, Department of Economics, and Rice Scholar in Energy Studies, James
A. Baker III Institute for Public Policy, Rice University, and BHP Billiton Chair in Resource and Energy Economics,
University of Western Australia
‡James A Baker III and Susan G Baker Fellow in Energy and Resource Economics, and Senior Director, Center
for Energy Studies, James A Baker III Institute for Public Policy and Adjunct Professor and Lecturer, Economics
Department, Rice University
i

1 Introduction
According to data from the US Energy Information Administration (EIA), fossil fuels currently
supply more than 86% of the world’s primary energy, while more than 11% comes from nuclear
power and hydroelectricity. Although large-scale energy production from non-hydroelectric renew-
able sources such as wind, solar, geothermal or biofuels is technologically feasible, in most locations
it is uncompetitive without subsidies.1A frequent justification for subsidizing new energy tech-
nologies is that without such subsidies many promising technologies would perish in a “valley of
death” between discovery and commercial viability.
Markham et al. (2010) illustrated the valley of death notion, repeated as Figure 1. They
claim that while adequate resources for new technology development are available during the basic
research phase, available resources often drop precipitously once the basic research has been com-
pleted. If an idea can make it through the valley of death to prove commercial viability, however,
once again ample resources will be available to take the idea to market.
Resources
Discovery pre-NPD New product development (NPD) Commercialization
Existing
technical
and market
research
resources
Existing resources for
commercialization
Valley of Death
Gap between opportunity
discovery and product
development
Figure 1: Illustration of Valley of Death Concept after Markham et al. (2010)
Markham et al. (2010) claim that the phrase “valley of death” was first used in 1995 to re-
fer to the challenges of transferring agricultural technologies to Third-World countries. Frank et
al. (1996) was perhaps the first paper to use the phrase to describe barriers to commercializing
new technologies, in their case environmental remediation technologies. They claim that govern-
ment funding mainly supports basic research, while the private sector is reluctant to finance mass
production of unproven technologies. Frank et al. contrast environmental remediation with the
1For example, the DSIRE database http://dsireusa.org/ currently lists 28 US Federal policies and 2,613 policies at
the state and local level (including Washington DC) promoting renewable energy and energy efficiency technologies.
1

pharmaceutical industry. They suggest that the latter might not suffer from a valley of death
because government support for medical research extends from basic R&D to human clinical trials,
while the industry is comprised of large, well-capitalized private sector firms involved in all stages
of new product development.
A Report by the US House of Representatives Committee on Science (Sensenbrenner, 1998) also
identified a valley of death for new technology developments as “a widening gap between federally-
funded basic research and industry-funded applied research and development.” The committee
suggested partnerships between universities and firms to span the valley, but argued that the
government should focus its limited resources on supporting basic research.
One of the earliest authors to apply the valley of death notion to energy technologies, Norberg-
Bohm (2000), noted that the phrase is meant to reflect the common experience that many new
technologies “die” before being successfully commercialized. She argued that, “For technologies
such as power plants, which may be standardized but not mass produced, the initial plant is much
more expensive than the 5th or 10th plant” and that early versions may not survive “an extended
period of negative net cumulative cash flow.”
Murphy and Edwards (2003) identified high risk as the main barrier to commercialization of
renewable energy technologies, ascribing the risk to five factors. First, the developer is likely to
know more about the technology than private sector investors, suppliers, or strategic partners,
making the latter hesitant to invest for fear of being exploited. Second, there is uncertainty about
whether the new product is competitive, or how the firm’s rivals will respond. Third, government
support of R&D paradoxically may increase uncertainty about market size, customer benefits and
profitability because these have not been critical to success of the venture to date. Fourth, continued
government support beyond the R&D stage, such as in the form of subsidies or tax credits, may
be uncertain. Finally, many of the firm’s assets – such as trade secrets, patents and key human
resources – are hard to evaluate and cannot be used as collateral for loans.
B¨urer and W¨ustenhagen (2009) surveyed principals and senior managers in 60 venture capital or
private equity fund management firms about the effectiveness of different policies in stimulating new
energy technology investments. Policies grouped into the “push” category included more funding
for R&D, investment subsidies for manufacturing facilities, grants to install equipment, tax breaks
for entrepreneurs and government grants for demonstration plants, the last of which was the most
favored “push” policy. Feed-in tariffs were the most favored “pull” policy, and were preferred
to all other policies, including renewable portfolio standards and tradable green certificates. By
2

contrast, Azar and Sand´en (2011) criticize feed-in tariffs for supporting only specific identified
renewable electricity generating technologies. As a result, “very promising but currently expensive
technologies will not gain the support that is needed to come down in cost.” Azar and Sand´en
also suggest that the fundamental impediment to developing new energy technology is an inability
to capture benefits external to the firm, including some benefits of learning-by-doing, learning-by-
using and network externalities. Failure to capture (some of) the external benefits then limits the
ability of new technology firms to attain the economies of scale that could reduce costs. While this
could be interpreted to mean that renewable technologies are already competitive but inefficiently
excluded from the market, Zindler and Locklin (2010) claim that “Much more work remains to
drive down costs so that renewables can truly compete with . . . their fossil rivals.”
Beard et al. (2009) take the most literal interpretation of a “valley” in the funds available for
new product development, as illustrated in Figure 1. As the authors say, while “several explanations
for the valley of death have been proffered . . . none provides a mechanism that clearly explains the
. . . ‘valley’ characteristic of the phenomenon . . . [that is] a shortfall of funding at an intermediate
stage that is more systematic and profound than the shortfall to either side.” Beard et al. interpret
the vertical axis in Figure 1 as the proportion of projects funded at each stage of the process (or
the probability that any one project at that stage will receive funding). Specifically, they consider
a project requiring three stages of R&D expenditure. At stage i, the probability of success is Pi
and the opportunity cost of funds tied up in the project is Ii. The project pays off V, but only
if it succeeds at stage 3. A risk neutral entrepreneur will undertake the final stage of a project
already at stage 2 only if P3V > I3. Similarly, a project at stage 1 will enter stage 2 only if
V > I3/P3+I2/(P2P3), while a project will be begun only if V > I3/P3+I2/(P2P3) + I1/(P1P2P3).
Since each of the terms on the right side of the inequalities is positive, a successful project at
either stage 1 or stage 2 will automatically be taken to the next stage. From these inequalities,
if an exogenous factor makes stage 2 investment less attractive by raising I2or reducing P2, it
will also make stage 1 investment less attractive. It cannot result in a “valley” of death in stage
2relative to stage 1. However, a “valley” can occur if decisions made in stage 1 proceed in part
independently of what is likely to happen in the later stages. In particular, government subsidies
in stage 1 that are not available for later stages can result in more output of stage 1 projects than
the private sector is willing to finance in stage 2. “So, while pathologies such as risk, uncertainty,
appropriation problems and so forth are present at intermediate stages of an innovation process, a
3

non-economic actor operating at early stages is required for there to be a valley of death.”2
Whereas most of the papers reviewed above assumed (often implicitly) that the valley of death
reflects a misallocation of resources, Beard et al. note that the existence of a valley need not imply
an inefficient outcome. The basic research at stage 1 is likely to have more spillover benefits for
other endeavors than the applied research that occurs at later stages. More of the fruits of stage 1
research thus may not be appropriable and efficiency may require subsidizing stage 1 research but
not the later stages even though a “valley” in technology development is the result.
Weyant (2011) is the only paper we found that suggests research subsidies could be inefficient.
He observes that “technologies that are not likely to be remotely economically competitive (or
maybe not even technically feasible) at commercial scale have often nonetheless been pursued
at great expense, through a pernicious combination of political self-seeking and technological over
optimism.” On the other hand, Weyant also claims that private investors may under-invest in R&D
because of an inability to appropriate some of the benefits, an unwillingness of venture capitalists
to assume technological risks, or regulations that inhibit recovery of R&D costs by large utilities.
Following Schumpeter (1942), Weyant also highlights the phenomenon of “creative destruction”
whereby new technologies make old ones obsolete. Weyant suggests that energy firms with current
technologies that are more profitable than the new ones may have an incentive to delay change,
especially if their industries are oligopolistic or imperfectly regulated.
This paper makes several contributions to the above literature on the valley of death phe-
nomenon as it relates to the energy industries. First, we look at the issue in the context of a
dynamic intertemporal growth model incorporating displacement of fossil energy technologies by
non-fossil alternatives. The intertemporal aspect is important because, as emphasized by Smil
(2012) for example, the displacement of one energy technology by another occurs gradually over
time. Focusing on growth is important because energy technology both has a strong influence on,
and is also strongly influenced by, economic growth.
Second, we allow for progress in the fossil fuel technologies as well as the alternatives. The
unconventional oil and gas revolution is just the latest in a long line of technological improvements
in the fossil fuel producing and consuming industries. These technological changes offset what
would otherwise be a rising cost of energy services produced from fossil fuels, and force non-fossil
alternatives to compete against a moving target.
Third, unlike most of literature on the valley of death phenomenon, we do not assume that
2Beard et al. elaborate on this model in several respects, but the details are not relevant for our discussion.
4

cost reductions result from explicit R&D alone. Instead, following much of the empirical literature
investigating technological progress in renewable energy supply, we assume learning-by-doing also
contributes to the accumulation of knowledge about renewable technology.
Finally, following previous authors, we associate the early stage of the process of displacing old
energy technologies with new ones as consisting largely of R&D expenditure. However, we associate
the “commercialization” phase with the need to build physical capital to supply energy services.
The need to replace the physical capital supplying energy services using fossil fuels by different
capital needed to supply energy services from non-fossil sources is a type of “creative destruction,”
but one that takes time. Smil (2006, Kindle Locations 2678-2679) alludes to this issue when he
states, “capital investment considerations and infrastructural inertia mean that it takes several
decades for any new energy source or conversion to claim a substantial share of the market.”
Our model highlights the fact that, in order to ensure uninterrupted supply of energy services,
investment in new energy technologies is required before fossil fuels are abandoned. Since the cost of
fossil fuel using capital is sunk, however, that capital will continue to be used so long as the energy
price covers its short-run operating costs. Thus, from the time investment in fossil fuel using capital
ceases until the time fossil fuels are abandoned, the operating cost of fossil fuel production sets the
energy price. Furthermore, at the switch point, the price of energy just matches the operating costs
of renewable energy production. Prior to that time, the energy price cannot cover renewable energy
production operating costs, let alone provide a competitive rate of return to the capital employed.
In fact, we show that the full long-run costs of renewable energy (including a competitive rate of
return on capital) are not covered until some time after fossil fuels are abandoned.
It is important to emphasize that although the investment, technological progress and energy
price outcomes in the model are efficient, we make no claim that a competitive equilibrium could
deliver those outcomes. Indeed, a private firm investing in capital to supply renewable energy ser-
vices would have to accept an energy price that is below the long-run cost of supplying renewable
energy for a long time. Conceptually, the return from reducing costs through promoting techno-
logical change ought to pay for some of the costs of investing in renewable energy technology and
productive capital. In practice, however, as emphasized in the literature on the valley of death
phenomenon discussed above, some of the benefits to R&D may fail to yield appropriable returns.
This would discourage early investment in the new technologies. Alternatively, government sub-
sidies for alternative energy technology R&D could produce an overhang of “first stage” research
5

projects as discussed by Beard et al. (2009).3
Some literature on the valley of death phenomenon has observed that it does not appear to
be present in pharmaceuticals or information technology (IT). Perhaps patents and copyrights are
more effective at enforcing property rights in these industries, but our analysis suggests another
reason. Once a new drug has been discovered or invented, or a new piece of software has been
created, these items often can be reproduced at very low marginal cost. In the energy sector, by
contrast, very large capital investments are required after the R&D phase in order to supply energy
services using new energy technologies. Our model points to potential difficulties in financing these
investments to deploy new technologies in the face of competition from fossil fuels.
Our paper is also related to literature on intertemporal optimization models of economic growth
and energy use. We follow Hartley et al. (2014) in the ways that we model economic growth, the
effect of technological change on the cost of fossil fuel energy, and the manner in which learning-
by-doing and investment in R&D affect the cost of renewable energy. However, we we extend that
work by explicitly modeling investment in physical capital needed to supply energy services.4
Chakravorty, Roumasset, and Tse (1997) consider a model with substitution between energy
sources, improvements in extraction, and a declining cost of renewable energy. In contrast to their
paper, we generate an endogenous transition to renewable energy and allow for explicit investment
in physical capital stocks. Unlike Chakravorty et. al., we do not study the implications of energy
use for environmental externalities and we do not conduct policy experiments.
Acemoglu et al. (2012) also examine the transition from fossil fuels to renewable technologies in a
dynamic intertemporal model. Specifically, they study a growth model that takes into consideration
the environmental impact of operating “dirty” technologies. They examine the effects of policies
that tax innovation and production in the dirty sectors. Their paper focuses on long run growth
and sustainability and abstracts from the endogenous evolution of R&D or the need to invest in
physical capital in order to supply energy services.
Our model abstracts from some real world complicating factors. We assume that energy services
supplied by the two sources are perfect substitutes. Imperfect substitutability between energy from
fossil and non-fossil sources would allow non-fossil sources to be more competitive under special
circumstances, such as solar panels in remote locations. The model also abstracts from the fact
3To investigate this further, one would need to examine equilibrium in an explicit decentralized model with various
assumptions about the appropriability of the benefits of R&D and different amounts of subsidization of research.
4It is reassuring for the central results of that paper that the more complicated model in this paper also produces
“an endogenous energy crisis” around the time of the transition between energy sources.
6

that a significant part of current energy supply comes from hydroelectricity and nuclear instead of
fossil fuels. To accommodate these sources, we could add a third type of energy producing capital
that is used with both fossil fuels and renewables. This would add considerable complexity to the
model, but would not change the problems associated with replacing the “fossil fuel only” part of
the current energy supply system.
2 The Model
We model economic activity in continuous time, indexed by t. Although the endogenous variables
are functions of time, we usually simplify notation by omitting time as an explicit argument.
2.1 Goods and services production and consumption
There is a single consumption good in the economy. Letting cdenote per capita consumption, in
common with much of the growth literature we assume that the lifetime utility function of the
representative agent is the constant relative risk averse form:
U=Z∞
0
e−βτ c(τ)1−γ
1−γdτ (1)
where e−βτ is the discount factor and γis the coefficient of relative risk aversion.5
Per capita output yis produced using per capita capital kand per capita energy eas inputs.
We assume for simplicity that there is no substitution between energy and non-energy inputs in
producing y,6and that output depends linearly on k.7The production function can thus be written
y= min{Ak, e}. Since it is costly to produce both kand e, however, it will be optimal at all times
to have y=Ak =e. Finally, we assume that capital depreciates at the constant rate δ, while
investment in new capital is denoted by i:
˙
k=i−δk (2)
5More precisely, since there is no uncertainty in our model, γrelates to intertemporal elasticity of substitution.
6We have examined an extension of the model that allows for investment in end-use energy efficiency. This requires
an additional state, and corresponding co-state, variable and increases the number of regimes in the solution. While
being more realistic, it is more complicated and does not add much to the issues under discussion.
7For the marginal product of capital to remain constant as kaccumulates relative to total hours worked, labor
input would need to expand endogenously through investment in human capital. However, we simplify by excluding
this additional state, and corresponding co-state, variable.
7

2.2 Energy production
We assume that per capita energy input e=F k is required where Fcan be interpreted as the
fuel intensity of the capital stock.8Energy can be provided by two different technologies that also
require capital investments to produce useful output. One, with capital stock denoted kR, mines
depletable fossil fuel resources and converts them into useable energy products Rusing, for example,
refineries and power stations. The other is a backstop or renewable technology that uses a capital
stock denoted kBto “harvest” an energy source that is non-depletable and turn it into useable
energy products B. The two-energy producing capital stocks are accumulated via investment iB
and iRand depreciate at the same rate δas capital kused to produce final output:9
˙
kB=iB−δkB(3)
˙
kR=iR−δkR(4)
Once energy-producing capital is in place it cannot be converted from one type to the other.
However, energy from fossil fuels and renewable sources are perfect substitutes for producing goods.
Total energy input into goods production thus is e=R+B. As with final output, we assume linear
production functions for the energy producing industries. Letting 0 ≤ρR≤1 and and 0 ≤ρB≤1
be the utilization rates10 for kRand kB, energy market equilibrium will require:
F k =ρRGkR+ρBH kB(5)
We assume that Gis constant, but as we explain below we allow Hto increase over time.
We define the units that fossil fuels are measured in so that one unit of kRrequires one unit
of fossil fuel input. Since kRis measured in per capita terms, so also will be the fossil fuel input,
implying that population growth will increase the total amount of fossil fuel resources that are
mined. Letting Qdenote the (exogenous) population and labor supply, the total fossil fuel used
8The usual energy identity relates energy input e, such as gallons of fuel, to utilization uper unit of capital, such
as miles per vehicle, times the capital stock k, such as the number of vehicles, divided by energy efficiency E, such
as miles per gallon. The fuel intensity Fthen is u/E. Although we usually refer to Fas “end-use energy efficiency”
it should be understood that fuel intensity of final production can change for reasons other than improvements in
energy efficiency as usually understood. For example, Medlock (2010) emphasizes that changes in the composition of
production, for example the shift to services, reduce energy intensity as an economy grows.
9Although different types of capital could depreciate at different rates, the data we use to calibrate the model
provides only a single rate of depreciation for capital. Having just one rate also simplifies the analysis somewhat.
10Since capital depreciates exponentially once investment in kRceases, a positive amount of kRwill always remain.
Mining and conversion costs can then be avoided by choosing ρR= 0 and ceasing to use kRto provide energy.
8

will be ρRQkR, and the total quantity of resources mined to date, Swill satisfy:
˙
S=ρRQkR(6)
For simplicity, we assume that Qgrows at the constant rate π, that is, ˙
Q=πQ.
Somewhat analogous to the distinction Venables (2012) draws between costs of extraction on
the intensive margin versus costs of new field development, or expansion on the extensive margin,
we assume there are two components in the cost of producing fossil energy. However, we simplify
by assuming that, for a given value of S, resources can be extracted at a constant marginal (and
average) cost µ. Depletion (an increase in S) raises that cost over time,11 but technological progress
in mining and fossil energy use can offset the cost increases.12 The state of technical knowledge in
energy services production from fossil fuels is encapsulated in a variable N, which is not assumed
to depreciate over time, and where the chosen investment nleads to an accumulation of N:
˙
N=n(7)
Investment ncould be associated with bringing new fields into production as emphasized by Ven-
ables (2012). However, we have in mind longer-run processes, such as new technologies that enable
exploitation of new categories of resources (shale gas and oil, deepwater or pre-salt deposits, oil
sands, methane hydrates or underground gasification of deep coal).13
While the total feasible technically recoverable fossil energy resource ¯
Sis vast we assume that
the maximum recoverable resource S0is far smaller without investment in N. Specifically, if N
were to remain at zero, the marginal mining and conversion costs would be µ(S, 0)ρRkR, where
µ(S, 0) is increasing and convex in Sand unbounded as S↑S0. However, if investment Nwere to
increase without bound, the upper bound on S(where µ(S, N )↑ ∞) would converge to ¯
S. Figure
2 illustrates a simple functional form that incorporates these assumptions:
µ(S, N ) = α0+α1
¯
S−S−α2/(α3+N)=α0+α1(α3+N)
(¯
S−S)(α3+N)−α2
(8)
11Heal (1976) introduced the idea of an increasing marginal cost of extraction to show that the optimal price of
an exhaustible resource begins above marginal cost, and falls toward it over time. This claim is rigorously proved in
Oren and Powell (1985). See also Solow and Wan (1976).
12Depletion can be viewed as “inverse learning-by-doing,” since cumulative past production raises current costs.
Investment in Noffsets this process in the case of fossil fuels, whereas investment in R&D reinforces the cost-reducing
effects of learning-by-doing for renewables.
13Investments that increase the efficiency with which fossil fuel is used to provide useful energy services also extend
fossil fuel resources by allowing the same energy services to be produced using less input of primary fuel.
9

The terms α0, α1, α2and α3in (8) are parameters, and we can associate the afore-mentioned S0
with ¯
S−α2/α3. Investment in Nexpands the temporary capacity limit, and the flat portion of
the marginal cost curve, to the right, extending the competitiveness of fossil fuels.
S
α0
¯
S−α2
α3+N
Figure 2: Unit cost of mining fossil fuels µ(S, N )
By differentiating µ(S, N) one can show that ∂µ/∂S > 0 and ∂µ/∂N < 0, so depletion raises
fossil fuel energy costs while investment in Nlowers them. Also, ∂2µ/∂S2>0, so depletion increases
cost at an increasing rate, while ∂2µ/∂N 2>0, so investment in Ndecreases costs at a decreasing
rate. Investment in Nalso delays the increase in fossil fuel energy costs accompanying increased
exploitation, that is, ∂2µ/∂S∂N < 0. However, since ∂µ/∂N →0 as N→ ∞, it will become
uneconomic at some point to invest further in reducing the costs of fossil fuel energy. The costs
of depletion will then swamp improvements in mining technology and conversion efficiency. Fossil
fuel resources will be abandoned before S=¯
Sas rising costs make renewable energy technologies
more attractive. Once fossil fuel use ceases, the utilization ρRof kRfalls to zero and S, N and µ
will remain constant.
The production of energy from renewable sources incurs operating and maintenance (O&M)
costs in addition to requiring investment in kB. Specifically, for renewable energy capital kB
utilized at the rate ρB, renewable energy production ρBHkBincurs O&M costs mρBkB.
We allow technological progress to increase H, and decrease the amount of kBrequired to yield
given energy output B, until Hattains an upper limit ¯
H.14 Explicitly, we assume that accumulation
14This assumption allows the final regime to be solved analytically, but in reality Hcould not increase forever.
10

of knowledge about renewables requires both the construction of kBand R&D investment j:15
˙
H=




bkψ
Bjα−ψif H≤¯
H,
0 otherwise
(9)
where ψ < α < 1, so there are decreasing returns to investing in renewable energy efficiency. The
parameter ψdetermines the relative contribution from experience versus explicit investment in
R&D. Klaassen et. al. (2005)16 derive robust estimates suggesting that direct R&D is roughly twice
as productive for reducing costs in wind turbine farms as is learning-by-doing. Hence, we assume
that ψ=α/3. The coefficient brelating investment in knowledge to the resulting technological
progress is analogous to the coefficient Ain the production function for final output.
Finally, the above assumptions on goods production and use yield a resource constraint. Per
capita goods output is consumed, used to produce energy, or invested in k, kR, kB, N or H:17
Ak =c+i+iR+n+iB+j+µ(S, N )ρRkR+mρBkB(10)
3 The optimization problem
The objective (1) is maximized subject to the differential constraints (2), (3), (4), (6), (7) and (9)
with initial conditions k(0) = k0, kR(0) = kR0, kB(0) = kB0, H(0) = H0and S(0) = N(0) = 0, the
resource constraint (10), and the energy market equilibrium condition (5). The control variables
are c, ρR, ρB, i, iR, iB, n and j, while the state variables are k, kR, kB, N, S and H. Denote the
corresponding co-state variables by q, qR, qB, ν, σ and η. Let λbe the Lagrange multiplier on the
resource constraint and pethe multiplier on the energy market equilibrium constraint. Use θRL
and θRU for the Lagrange multipliers on the inequality constraints on ρR, and θB L and θBU the
corresponding multipliers on the inequality constraints on ρB. Also use ω, ωR, ωB, ωNand ωHto
denote the multipliers on the non-negativity constraints on investments i, iR, iB, n and j. We can
15Since ρBis absent from (9), learning does not require energy output to be produced. In addition, we assume
that accumulated capacity kB, not just new investments iB, influences growth in H.
16Building on Kouvaritakis et al. (2000), Klaassen et. al. (2005) estimated a two-factor learning curve model that
allowed both capacity expansion (learning-by-doing) and direct public R&D to reduce costs of wind turbine farms in
Denmark, Germany and the UK. They interpret their results as supporting the two-factor learning curve formulation.
17From (10) and (5), c+i+iR+iB+n+f+j=ρRkR[AG/F −µ(S, N)] + ρBkB(AH/F −m). We assume Ais
large enough, and initial fuel intensity Fis low enough, that A/F exceeds µ(0,0)/G and m/H0.
11

then define the current value Hamiltonian and hence Lagrangian by
H=c1−γ
1−γ+q(i−δk) + qR(iR−δkR) + qB(iB−δkB) + νn +σρRQkR
+ηbkψ
Bjα−ψ+λAk −c−i−iR−iB−n−j−µ(S, N )ρRkR−mρBkB
+peρRGkR+ρBHkB−F k+θRL ρR+θRU (1 −ρR) + θBL ρB
+θBU (1 −ρB) + ωi +ωRiR+ωBiB+ωNn+ωHj
(11)
The first order conditions for a maximum of (11) with respect to the control variables are:
∂H
∂c =c−γ−λ= 0 (12)
∂H
∂ρR
=σQkR−λkRµ(S, N ) + peGkR+θRL −θRU = 0
θRLρR= 0, θRL ≥0, ρR≥0, θRU (1 −ρR) = 0, θRU ≥0, ρR≤1
(13)
∂H
∂ρB
=−λmkB+peHkB+θBL −θBU = 0
θBLρB= 0, θBL ≥0, ρB≥0, θB U (1 −ρB)=0, θBU ≥0, ρB≤1
(14)
∂H
∂i =q−λ+ω= 0; ωi = 0, ω ≥0, i ≥0 (15)
∂H
∂iR
=qR−λ+ωR= 0; ωRiR= 0, ωR≥0, iR≥0 (16)
∂H
∂iB
=qB−λ+ωB= 0; ωBiB= 0, ωB≥0, iB≥0 (17)
∂H
∂n =ν−λ+ωN= 0, ωNn= 0, ωN≥0, n ≥0 (18)
∂H
∂j =η(α−ψ)bkψ
Bjα−ψ−1−λ+ωH= 0, ωHj= 0, ωH≥0, j ≥0 (19)
The differential equations for the co-state variables are:
˙q=βq −∂H
∂k = (β+δ)q−λA +peF(20)
˙qR=βqR−∂H
∂kR
= (β+δ)qR−σρRQ+ρRλµ(S, N )−ρRpeG(21)
˙qB=βqB−∂H
∂kB
= (β+δ)qB−ηψbkψ−1
Bjα−ψ+ρBλm −ρBpeH(22)
12

˙ν=βν −∂H
∂N =βν +λρRkR
∂µ
∂N (23)
˙σ=βσ −∂H
∂S =βσ +λρRkR
∂µ
∂S (24)
˙η=βη −∂H
∂H =βη −ρBpekB(25)
We also recover the resource constraint (10), the energy market equilibrium condition (5), and the
differential equations for the state variables, (2), (3), (4), (6), (7) and (9).
Let Vdenote the maximized value of the objective function (1) subject to the constraints. Since
His the current value Hamiltonian, from the Hamilton-Jacobi-Bellman equation −Vt=e−β t max H.
Furthermore, the (current value) co-state variables satisfy q=eβt∂V /∂k ≥0, qR=eβt∂V /∂kR≥0,
qB=eβt∂V /∂kB≥0, σ=eβ t∂V /∂S ≤0, ν=eβ t∂V /∂N ≥0 and η=eβ t∂V /∂H ≥0. Also note
that the utility function ensures that c > 0 and hence from (12), λ > 0. The multiplier peon the
energy market equilibrium constraint can be interpreted as the shadow price of energy.
4 Energy price and kBinvestment
The main result of our paper is that investment in renewable energy producing capital kBis needed
for a long period of time before the price of energy peis sufficient to cover the long run cost of
renewable energy production. In this section, we show that this result is generic in the sense that
it holds for a very wide range of parameter values.
We first derive expressions that can be interpreted as the short run and long run costs of fossil
fuel and renewable energy production. Analogous to the peak load problem in electricity supply,
the short run costs correspond to the operating costs, while the long run costs exceed the short
costs by an amount sufficient to cover the costs of capital kRand kB.
We develop the argument through a number of lemmas and more significant propositions.
Lemma 1. For kR>0,ρR= 0 if peG < λµ −σQ,ρR= 1 if peG > λµ −σQ and if 0< ρR<1,
peG=λµ −σQ. Similarly, for kB>0,ρB= 0 if peH < λm,ρB= 1 if peH > λm and if
0< ρB<1,peH=λm
Proof. These results follow from (13) and (14).
Comment. Since each unit of kRdelivers Gunits of energy services each valued at pe,peGis
marginal revenue from fossil energy supply. Explicit marginal production costs of fossil energy
13

measured in output units are λµ, while −σQ > 0 is an implicit depletion cost. For renewables,
each unit of kBdelivers Hunits of energy services and marginal operating costs are just the explicit
marginal costs λm. Hence, the lemma essentially says that for an energy source to be used marginal
revenue has to at least cover marginal operating costs.
Lemma 2. The current shadow value qRof kRequals the discounted value of the implicit rents
to scarce capacity when ρR= 1, with a discount factor given by the time discount rate βplus the
depreciation rate δof kR:
qR=e(β+δ)tZ∞
t
e−(β+δ)τθRU
kR
dτ
Proof. From (13), when ρR= 1, θRU /kR=peG−(λµ −σQ)>0 and the result follows from
Lemma 1, (21) and the assumptions that βand δare constants.
Comment. We can interpret θRU /kR, or the excess of marginal revenue over short run marginal
cost, as an implicit return to scarce capacity when ρR= 1. In the case of fossil fuels, observe that
the terms not involving qRon the right hand side of (21), namely ρR(λµ −peG) equal zero unless
peG > λµ −σQ. The analogous result does not hold for (22) since the term ηψbkψ−1
Bjα−ψdoes not
depend on peor ρB. This term represents an additional implicit return to investment in kBthat
arises because of the learning benefits of additional kB. Analogous to the fossil fuel case, we can
interpret θBU /kB=peH−λm as an implicit return to scarce capacity when ρB= 1.
Lemma 3. If both i > 0and iR>0over some interval of time
pe=λ(A+µ)−σQ
F+G(26)
over that same interval
Proof. When both i > 0 and iR>0, (16) and (15) imply qR=λ=q. The result then follows from
˙qR= ˙q, (20), (21) and Lemma 1.
Comment. From (12), λalso equals the marginal utility of consumption, which is the opportunity
cost of investing in kor kR.
In preparation for discussing an analogous result to Lemma 3 for the case of iB>0, it is
convenient to define a function Y≡[η(α−ψ)b/λ]s, where s≡1/(1 + ψ−α)>1. The solution for
jfrom (19) can then be written j=Y ksψ
B. Also, the term ηψbkψ−1
Bjα−ψin (22) can be written as
λψk(α−1)s
BY/(α−ψ).
14

Lemma 4. Suppose both i > 0and iB>0over some interval of time. If, in addition, peH > λm
then
pe=λ
F+HA+m−ψ
α−ψk(α−1)s
BY(27)
while if peH≤λm then
pe=λ
FA−ψ
α−ψk(α−1)s
BY(28)
Proof. When both i > 0 and iB>0, (17) and (20) imply qB=λ=q. The result then follows from
˙qB= ˙q, (21), (20) and Lemma 1.
In accordance with these results, we define the energy price:
pR
SR =λµ −σQ
G(29)
to be the short run cost, and the energy price pR
LR given by (26) to be the long run cost, of fossil
energy production. We similarly define a short run cost of renewable energy analogous to (29) as:
pB
SR =λm
H(30)
We also define pB
LR given by (27) to be the “full” long run cost of renewable energy supply. The
energy price pB
ρ0given by (28) can be interpreted as the price necessary to incentivize investment
in kBduring an “R&D only phase” when kBis not used to supply energy services. Investment in
it is still worthwhile because of the benefits from learning and future cost reductions. It is useful
to note the following result:
Lemma 5. When pB
SR > pB
ρ0, we must also have pB
SR > pB
LR > pB
ρ0and when pB
SR < pB
ρ0, we must
also have pB
ρ0> pB
LR > pB
SR .
Proof. This result follows from the definitions of the various costs.
Comment. These inequalities imply that the long run cost of renewable energy supply is always
between the other two critical costs unless all three are equal.
We now note the following key result:
Proposition 1. kRalone is used to supply energy services if and only if pR
SR ≤pe< pB
SR while kB
alone is used to supply energy services if and only if pB
SR ≤pe< pR
SR . Furthermore, if both kRand
15

kBare used to supply energy services and if pR
SR < pB
SR then pR
SR < pe=pB
SR and kBmay be less
than fully utilized, while if pB
SR < pR
SR then pB
SR < pe=pR
SR and kRmay be less than fully utilized.
Proof. Follows from Lemma 1 and the definitions of the short run costs.
The main assumption about parameter values that we make in this section is that they are
chosen to ensure that pR
SR < pe=pR
LR < pB
SR at t= 0. Then only fossil fuels are used to supply
energy services at t= 0. Furthermore, we will have i, iR>0 and investment will be occurring in
both kand kR.18 While renewable energy would not be used to supply energy services at t= 0, we
can nevertheless show:
Proposition 2. Both kBand jmust be strictly positive at t= 0 and jmust continue positive until
H=¯
H
Proof. Since η > 0, for H < ¯
H, and recalling that 1 > α > ψ > 0, the additional return
λψk(α−1)s
BY/(α−ψ) to investing in kBarising from learning-by-doing becomes unbounded as
kB→0. In particular, kBmust be positive at t= 0. Similarly, since η(α−ψ)bkψ
Bjα−ψ−1→ ∞ as
j→0, (19) would violate the condition ωH≥0 unless jis strictly positive.
Since the initial value kB0of kBis specified exogenously, we could begin with excess kB. Then
we would have qB< λ =qRwhile iB= 0 and kBdepreciates. Eventually, it would be worthwhile
to start investing in kBonce again. However, it is more natural to assume that investment in kB
up to t= 0 has been optimal and hence iB>0 at t= 0. In that case, the price of energy at t= 0
also has to equal pB
ρ0.
Energy production from fossil fuels alone will continue with iR>0, ρR= 1 and ρB= 0 so
long as pB
SR > pe=pR
LR > pR
SR . However, investment nin fossil fuel technology cannot indefinitely
keep µfrom rising as a result of depletion (increasing S). Furthermore, increases in Hfrom the
accumulation of kBand R&D investment jwill reduce the short run cost of renewable energy
production pB
SR . Eventually, the economy must reach a time TQwhen pR
SR < pR
LR =pe=pB
SR . For
t>TQ,kBwill also be used to supply energy services:
Proposition 3. As tincreases above TQ, for some interval of time we will have pR
SR < pB
SR =pe
with both kBand kRused to supply energy services and ρR= 1,0< ρB<1
18In our view, this realistically approximates the current situation. Non-fossil sources other than hydroelectricity
and nuclear power have a small share of primary energy production and, except in special circumstances, would be
uncompetitive without subsidies or mandates.
16

Proof. With pR
SR < peat TQwe must continue to have pR
SR < peand ρR= 1 as tincreases from
TQ. Energy market equilibrium (5) as tincreases from TQwith ρR= 1 will require
ρB=F k −GkR
HkB
(31)
with ρB= 0 at TQ. In particular, since the state variables k, kR, kBand Hare continuous and G
is a constant, ρBmust be continuous at TQ. Hence, ρB↓0 as t↓TQand we must have 0 < ρB<1
and pe=pB
SR as tincreases above TQ.
We then have the following result that may at first sight appear surprising:
Proposition 4. Investments iRin kRand iBin kBboth cease at TQ
Proof. For t<TQ, we have pR
LR =pe=pB
ρ0with positive investment in both kRand kB. From
Proposition 3, kRwill fully utilized to supply energy services as tincreases from TQwith pR
SR and
pR
LR continuing to rise as depletion effects overwhelm investments nin fossil fuel technology. On
the other hand, continuing growth in Hfrom positive R&D investment jmeans that pB
SR =pe
must continue to decline as tincreases from TQ. But then we must have pe< pR
LR as tincreases
above TQand hence investment iRin kRmust cease at TQ. Furthermore, since pB
ρ0=pR
LR for all
t≤TQwe must have ˙pB
ρ0= ˙pR
LR as t↑TQand hence pB
ρ0also will be increasing at TQ. Thus, pB
SR
must fall below pB
ρ0as tincreases from TQ. Then Lemma 5 implies pB
SR also falls below pB
LR as t
increases from TQand investment iBin kBalso ceases at TQ.
Comment. The key explanation for this outcome is Lemma 5, which shows that the ordering of pB
SR
relative to pB
LR flips at TQ. While investment in kRand kBcease at TQ, end use capital continues
to grow:
Proposition 5. Investment iin kremains positive as tincreases from TQ
Proof. Differentiating the energy equilibrium condition (5) with respect to time and noting that
since iR=0=iB,˙
kR=−δkR,˙
kB=−δkBwe find F i = ˙ρBH kB+ρB˙
HkB>0
As depletion continues to increase pR
SR while investment in jcontinues to increase Hand thus
reduce pB
SR =pe, we must reach a time Tρwhen pR
SR =pe=pB
SR . Further increases in pR
SR and
reductions in pB
SR as tincreases above Tρwill, from Proposition 1, then lead to incomplete use of
kRwhile ρB= 1. This is the reverse of the situation for TQ<t<Tρ. Proposition 1 also implies
that we will have pB
SR < pe=pR
SR as tincreases above Tρ. Further, we can also show:
17

Proposition 6. Investments iRin kRand iBin kBboth remain zero at Tρ
Proof. At Tρ,pB
SR =pe=pSR and therefore peis less than pR
LR and pB
LR, which in turn is also now
less than pB
ρ0.
Since the short run cost of fossil fuel energy supply continues to increase beyond Tρwhile the
short and long run costs of renewable energy supply continue to fall, we must eventually reach
a time TBwhen peequals pB
LR. Investment iBin renewable energy supply capital kBthen will
resume. Renewable energy costs, now pB
LR, will once again set the energy price pe.
With accumulation of kBadding to investment j,Hshould continue to increase and thus
continue to reduce pB
LR =pe. It follows that beyond TB,pR
SR must be less tan peand hence ρR= 0
and fossil fuels will not be used to supply energy services.
Nevertheless, there is no necessary reason for the time TBthat investment in kBresumes to
correspond exactly to the time TRthat ρRfirst becomes zero and thus fossil fuels are abandoned.
Once fossil fuels are abandoned, kBsupplies all the energy services and investments iand kBhave
to maintain energy market equilibrium. However, a rapidly increasing Hby itself reduces the
amount of kBneeded to provide the required energy services input to final production. In the
numerical example examined later, the purely renewable regime has an initial period following the
abandonment of fossil fuels at TRwhere iBremains zero.19
Between Tρ, when ρBfirst equals one and ρRfirst drops below 1, and TR, there is another
critical juncture TNwhen nfirst becomes zero. We first discuss the evolution of the co-state
variables corresponding to Sand N:
Proposition 7. σ(t) = ν(t) = qR(t) = 0 for all t≥TRand σ(t)<0< ν(t)for all t<TR
Proof. Since fossil fuels are not used beyond TR, changes in S, N and kRhave no effect on V, and
hence σ=ν=qR= 0, for t≥TR. Also, since TRis the first time fossil fuels cease to be used,
ρR>0 for t<TR. Then since ∂µ/∂N < 0, (23) implies that ν > 0 and ˙ν < 0 for t<TR, while
∂µ/∂S > 0 and (24) imply σ < 0 and ˙σ > 0 for t<TR.
Comment. In contrast to νand σ, ˙qRcan be zero for t<TRand ρR>0. Specifically for
Tρ≤t≤TR,pe=pR
SR and (21) and qR(TR) = 0 will imply qR= 0 for all Tρ< t ≤TR, with qRleft
continuous at Tρ.
19The appendix shows how peis determined as it rises from pB
SR at TRto pB
LR at TB.
18

Although ν=qR= 0 < λ at TR, (18) implies ν=λwhile n > 0, and (16) implies qR=λ > 0
while iR>0. Define TN< TRas the first time that n= 0. Essentially, as the time TRat which
fossil fuels will be abandoned approaches, it no longer becomes worthwhile investing to offset the
effects of depletion on fossil fuel costs.
We illustrate the above discussion in Figure 3. The upper part of Figure 3 shows the different
investment regimes, while the lower part shows the different regimes of energy production and use.
The economy passes through six regimes before entering the final regime at THwhere H=¯
Hand
an analytical solution is possible.
0
Investment in fossil fuels technology (n > 0)
Investment in renewables capital (iB > 0)
TQ
Investment in fossil fuel capital (iR > 0)
TN
Regime 1 Analytical Solution
Investment in end-use capital (i > 0)
Investment in renewables efficiency (j > 0)
Fossil fuel capital fully used (ρR = 1)
Renewable energy capital fully used (ρB = 1)
Regime 2
TR
Regime 4 Regime 5
Renewable energy capital not used (ρB = 0)
TH
Fossil fuel capital not used (ρR = 0)
TB
Regime 6
Investment in renewables capital (iB > 0)
iB = 0
1 > ρR > 0
0 < ρB < 1
Tρ
Regime 3
Figure 3: A schematic representation of the different regimes
In an appendix available online, we derive and discuss the system of simultaneous differential
equations that determine the evolution of the state and co-state variables in each regime. In
contrast to the discussion in this section, these equations are developed working backwards through
time from the analytical solution in the final regime. That is also how the equations are solved
numerically. The appendix also discusses how the initial values are chosen to determine a unique
solution for a given set of parameter values and contains the MatLab code that was used to solve
the model.
While it is important to show that the model can be solved for a given set of parameter values,
19

we emphasize that the key feature of the model holds quite generally and does not require a narrowly
selected range of parameter values. To summarize that result, we have shown in this section that
in an optimal transition from fossil fuel energy sources to renewables, investment in renewable
technologies will start long before the price of energy is sufficient to cover all the costs of renewable
energy, including the required return to capital.
More specifically, for t≤TQ, the energy price pewill equal both the long run cost of fossil
fuel energy supply pR
LR and the price pB
ρ0for “R&D only” renewable investment. The latter will,
however, be below both the short run and long run costs of renewable energy supply and kB
will not be used to supply energy services. For TQ< t ≤Tρ,pewill equal the short run cost of
renewable energy supply pB
SR , which in turn will remain below the long run cost needed to stimulate
further investment in kB. Even after Tρ, where exiting renewable capacity is fully utilized to supply
energy services for the first time, pewill remain below the long costs of renewable energy supply.
Investment in new renewable capacity accompanied by full utilization of kBcapacity to supply
energy services will not commence until TB, at which point pe=pB
LR.
There is a “valley of death” in the sense that investment into R&D and development of new
technologies will commence long before these technologies are deployed commercially to supply
energy services. Furthermore, the renewable technologies will also be used to supply energy services
before the energy price is sufficient to cover the long run costs of renewable energy supply. There
will also be a gap in investment in kBbetween the period where “R&D only” investment is justified
and when the new technology takes over as the new dominant source of energy services.
5 Calibration
We need to solve the model numerically for “reasonable” parameter values in order to judge the
quantitative significance of the main result. As far as possible, we choose values for the parameters
and variables at t= 0 that are consistent with observations from the world economy.
In line with standard assumptions made to calibrate growth models, we assume β= 0.05. From
previous analyses of macroeconomic and financial data, we would expect the coefficient of relative
risk aversion γto lie between 1 and 10, but there is no strong consensus on what the value should
be. The per capita growth rate of the economy depends inversely on γ. However, larger values of
γalso extend the time it takes the economy to enter the terminal regime, which makes the model
more difficult to solve. We set γ= 5 as a compromise between these considerations.
20

By definition, we start the economy with S=N= 0 and initial values for Q=Q0, H =
H0, k(0) = k0, kR(0) = kR0and kB(0) = kB0. For convenience, we take the current population
Q0= 1 and effectively measure future population as multiples of the current level. We will assume
that the population growth rate is 1%.20
To calibrate values for the initial production, capital stocks and energy quantities we used data
from the EIA,21 the Survey of Energy Resources 2010 produced by the World Energy Council
(WEC),22 and The GTAP 8 Data Base produced by the Center for Global Trade Analysis in the
Department of Agricultural Economics, Purdue University.23 The last mentioned data source is
useful for our purposes because it provides consistent international macroeconomic data that take
account of energy flows. We also use the GTAP depreciation rate δon capital of 4%.
National accounts include government spending in GDP, which does not appear in the model.
Since a substantial amount of government spending, such as that on health care or education, is
directly substitutable with private consumption spending, we classified government spending as
part of consumption.
Converting the GTAP data base estimates of the total capital stock to units of GDP, we obtain24
k+kR= 2.6551. Similarly, we identify investment in physical capital in the GTAP data as i+iR
in the model. Rescaling units so output equals 1, we conclude that i+iR= 0.2299.
The GTAP data gives firm purchases of the capital resource endowment (and other factors of
production) by sector. We identify the “energy sector” to include production of the primary fuels
(coal, oil and natural gas), plus the energy commodity transformation sectors of refining, electricity
generation and natural gas distribution, plus the transportation services sector. After doing so, we
find that 12.35%, or 0.3278 of the capital would be kRwhile k= 2.3273. Defining units so that
output equals 1 implies that A= 1/k = 0.4297. Similarly, choosing units of energy service inputs
into final production so R=GkR= 1 implies G= 1/kR= 3.0506. The energy market equilibrium
condition then requires F k =GkR= 1, so we must also have F=A= 0.4297.
20This is consistent with a simple extrapolation of recent world growth rates reported by the Food And Agriculture
Organization of the United Nations, http://faostat.fao.org/site/550/default.aspx
21International data is available at http://www.eia.doe.gov/emeu/international/contents.html
22The data are available at http://www.worldenergy.org/publications/2010/survey-of-energy-resources-2010/ and rep-
resent estimates as of the end of 2008.
23We extracted data pertaining to 2007 from the database. Further information on the database can be found at
https://www.gtap.agecon.purdue.edu/databases/v8/default.asp.
24Nuclear, hydroelectric, wind and solar energy, currently supply some electricity, and some transportation is
powered by electricity and biofuels, but as an approximation we are assuming that fossil fuels supply all energy
services at t= 0, that is, ρB(0) = 0. Using the remaining calibrated values, we also find that the equations linking
the capital stocks at t= 0 imply that kB(0) and iB(0) are positive but negligible, so they are ignored in calibrating
the remaining capital stocks and investments.
21

Next we calibrate the initial marginal cost of producing energy. From the resource constraint,
the difference between total output and i+iR, namely 0.7701, would equal c+µkR+n, which is
all classified as consumption expenditure in the GTAP data. We then associate µkR+nwith the
share of consumption falling on the output of the energy sector as defined above and cwith the
remainder. The result is c= 0.7140 and µkR+n= 0.0561.
The International Energy Agency (IEA) publishes statistics on energy sector investment in
R&D.25 The IEA also reports real GDP for the economies in their panel, and taking the average
ratio from 2005–2010 for all countries in the sample, the R&D part of nwas just 0.00644% of GDP.
However, the sample consists of mostly higher income countries from the OECD. If we assume that
energy sector R&D as a proportion of GDP in the remaining countries is on average one fourth that
of the included countries, we can use the GTAP data to find that a weighted average of 0.0051%
of GDP would be invested in the R&D part of n.
The variable nin the model should also include expenditures to expand fossil fuel reserves
through exploration and investment in new mines. The EIA reports26 exploration and development
expenditures by oil and natural gas firms over the period 2003–2009, and these averaged more than
seventy-two times the total fossil fuel R&D expenditure recorded by the IEA. The Australian
Bureau of Statistics reports27 exploration expenditures by Australian coal mining firms, which
according to EIA data produce around 20% of the total coal output from those countries included
in the IEA R&D expenditure data set. Multiplying the recent annual Australian coal exploration
expenditure by five, we arrive at a figure that is only about 3% of the oil and gas exploration
and development expenditure reported by the EIA. Adding these exploration expenditures to the
previously calculated fossil fuel R&D spending, we arrive at a total value for nof 0.31% of GDP.
Expressing this in units defined so output equals 1, we have n= 0.0031.
Subtracting nfrom the previously calculated µkR+n= 0.0561, we obtain28 µkR= 0.0530, and
25The data is available at http://www.iea.org/stats/rd.asp. Expenditure categories (as a percentage of the total
from 2005–2010) are: energy efficiency (18%), fossil fuels (14%), renewable energy sources (15%), nuclear (33%),
hydrogen and fuel cells (6%), other power and storage technologies (5%), and other cross-cutting technologies and
basic research (8%). Much of the explicit R&D expenditure would be part of the variable jin the model, but a
substantial amount of this expenditure also would not exist without extensive government support. Thus, we take
j(0) to be negligible as implied by the model solution for the remaining calibrated values. Items in the other power
and storage technologies category relate mainly to electrical system improvements that are not specific to renewable
energy sources and thus were grouped with R&D into fossil fuels as part of n.
26The data is available at http://www.eia.gov/finance/performanceprofiles/ as Table 15.
27See publication 8412.0 – “Mineral and Petroleum Exploration, Australia” – available at http://www.abs.gov.au
by catalog number.
28For comparison, the EIA Annual Energy Review gives energy expenditures as amounting to around 5% of gross
output in the U.S.
22

using the previously obtained kRwe then obtain µ= 0.1614.29 After we set the initial values of S
and Nto zero and Rto 1 (by defining energy units), the initial value for µalso would imply
0.1614 = α0+α1
¯
S−α2/α3
(32)
Next, we evaluate ¯
S, which must be measured in the same units as the fossil fuel used to produce
energy services. The EIA web site gives world wide production of oil in 2007 of 178.596 quads (where
one quad equals 1015 BTU), of natural gas 107.391 quads and of coal 133.367 quads. Summing
these gives a total of 419.354 quads for fossil fuel use at t= 0, which we will denote f0.30 To obtain
an estimate of total fossil fuel resources ¯
Sin the same units, we used data from the WEC and the
US Geological Survey (USGS). The millions of tonnes of coal, millions of barrels of oil, extra heavy
oil, natural bitumen and oil shale and trillions of cubic feet of conventional and unconventional
natural gas were converted to quads using conversion factors available at the EIA. The result
is 70.282 quintillion BTU of coal, 72.122 quintillion BTU of conventional and unconventional oil
and 13.821 quintillion BTU of conventional and unconventional natural gas. These resources are
nevertheless relatively small compared to estimates of the volume of methane hydrates that may
be available.31 According to the National Energy Technology Laboratory (NETL),32 the USGS
has estimated potential resources of about 200,000 trillion cubic feet in the United States alone.
According to Timothy Collett of the USGS,33 current estimates of the worldwide resource in place
are about 700,000 trillion cubic feet of methane. The latter figure would be equivalent to 718.900
quintillion BTU. Adding this to the previous total of oil, natural gas and coal resources yields a
value for ¯
S= 875.125 quintillion BTU or around 2086.8425 times f0.
Using ¯
S, (32) will give us one equation in the four parameters αi, i = 0,...,3. The quan-
tity ¯
S−α2/α3represents the limit of total fossil fuels that could be extracted in the absence of
any investment n. Thus, we associate ¯
S−α2/α3with current producing reserves of fossil fuel.34
We deduce these from current production and decline rates.35 A report from Cambridge Energy
29Observe that, since we have defined units so R= 1, this value of µand the previously calculated value for A, F
and Gimply fossil fuels yield positive net output at t= 0, that is, AG/F −µ=G−µ= 2.8892 >0.
30In the model solution, we defined f0to have the same numerical value as kR, but the units of the two variables
are different.
31Although experiments have been conducted to test methods of exploiting methane hydrates, a commercially
viable process is yet to be demonstrated. Partly as a result, resource estimates vary widely.
32http://www.netl.doe.gov/technologies/oil-gas/FutureSupply/MethaneHydrates/about-hydrates/estimates.htm
33http://www.netl.doe.gov/kmd/cds/disk10/collett.pdf
34Current official reserves are not the relevant measure since many of these are not available for production without
further investment, denoted nin the model.
35Note that the rate of exploitation of fossil fuels is determined endogenously in the model. We are using data on
23

Research Associates (CERA, 2009),36 for example, gives weighted average decline rates for oil pro-
duction from existing fields of around 4.5% per year, but observes that, “the average decline rate
for fields that were actually in the decline phase was 7.5%, but this number falls to 6.1% when the
numbers are production weighted.” As an approximation, we shall use 6% as a decline rate for
oil fields. Using United States natural gas production and reserve figures as a guide, we find that
natural gas decline rates are closer to 8% per year. The United States data on coal mine decline
rates approximate 6% per year. In accordance with these figures, we assume the ratio of fossil
fuel production to producing reserves equals the share weighted average of these figures, namely
(178.60∗0.06 + 107.391 ∗0.08+133.367∗0.06)/419.354 = 0.0651. Thus, the initial value of producing
reserves ¯
S−α2/α3would equal f0/0.0651 = 15.3559f0. Using the previously calculated value for
¯
S, this leads to α2/α3= 2071.4867f0.
We obtain another equation by examining investment in expanding fossil fuel production at
t= 0. As noted above, we calibrated the initial value of n= 0.0031. We assume that this level
of investment at t= 0 is sufficient to increase producing reserves by a percentage amount equal
to the average annual growth over 2004-10 of around 2.43%.37 In other words, we assume that
the investment n= 0.0031 increases producing reserves to 15.729f0, that is, α2/(α3+ 0.0031) =
2071.1135f0. Then using α2/α3= 2071.4867f0, we find α3≈17.0411. The previously calculated
value for α2/α3then implies α2≈35300f0. We thus have determined the µfunction up to one
degree of freedom. Once we specify either α0or α1, (32) along with the previously determined ¯
S
and α2/α3will give us the remaining αiparameter. We chose α1in an attempt38 to ensure that
λ(0) = c(0)−γat t= 0.
To calibrate technological progress in renewable efficiency, we rely on empirical estimates based
on experience with subsidized installations of wind turbines and solar panels to set ψand α. In a
study of wind turbines, Coulomb and Neuhoff (2006) found values corresponding to the parameter
ψin our model of 0.158 and 0.197. Gr¨ubler and Messner (1998) found a value for ψ=.36 using data
on solar panels, while van Bentham et. al. (2008) report several studies finding approximate values
of ψ= 0.322 for solar panels. We will take ψ= 0.25. Klaassen et. al. (2005) estimated a model
of technological change in renewables that allowed for both learning-by-doing and direct R&D.
decline rates solely to arrive at an estimate of current producing reserves.
36“The Future of Global Oil Supply: Understanding the Building Blocks,” Special Report by Peter Jackson, Senior
Director, IHS Cambridge Energy Research Associates, Cambridge, MA.
37These calculations are again based on data from the EIA.
38The model is highly non-linear making it difficult and time consuming to solve. Each time α1is changed, the
model needs to be solved many times over to find solution paths with values at t= 0 that approximate initial values
of the state variables. We therefore could obtain only an approximate match for c(0).
24

Although they assume the capital cost is multiplicative in total R&D and cumulative capacity,
while we assume the change in knowledge is multiplicative in new R&D and cumulative capacity,
we take their parameter estimates as a guide. They find direct R&D is roughly twice as productive
for reducing costs as is learning-by-doing. Consequently, we assume that α= 0.75.
Finally, we need to establish values for the initial H(0) and final ¯
Hvalues of the productivity
of kBin producing energy services, and the operating and maintenance costs mfor renewable
energy production. Although a substantial amount of current primary energy consumption is
direct rather than indirect through the consumption of electricity,39 we focus on the relative costs
of producing electricity using fossil and non-fossil sources. One justification may be that electric
vehicles represent the main way of using non-fossil fuels in the transportation sector. Electricity
may also be the main alternative to fossil fuels for indoor space heating, heating water and other
residential and commercial uses.
We focus on natural gas and coal generated electricity for the fossil fuel cost and nuclear,40
wind and pumped storage for non-fossil generation. Not all locations have suitable geography for
pumped storage, however, so a certain fraction of capacity needed to provide ancillary services will
have to take more expensive forms such as batteries, flywheels, or compressed air.
In the Annual Energy Outlook, 2010 the EIA gives indicative costs for different types of gener-
ation capacity and heat rates for the natural gas and coal plants.41 We used these cost estimates
along with fuel prices for 2009,42 indicative load factors,43 and assumed operating lives of 40 years
39Data from the EIA for the US shows that around 28% of primary energy is consumed in transportation, 20% in
industry and another 10% in residential and commercial activities. The remainder is used to generate electricity and
allocated as primary energy to users based on their consumption of electricity.
40Although nuclear fuels also are mined, the energy content of known uranium, thorium and other fissionable
material is huge. The available fuel supply can also be extended using breeder reactors, while if electricity production
from nuclear fusion is ever perfected, the fuel supply would, for all practical purposes, be inexhaustible. We can also
use nuclear power costs as a proxy for the costs of unconventional geothermal energy based on “hot rocks”.
41The heat rate for nuclear plants comes from the average realized heat rate in the US in 2010 as reported in Table
5.3 of the EIA publication Electric Power Annual.
42The natural gas and coal fuel prices were obtained from Table 3.5 of the Electric Power Annual. The
uranium price is the 2009-2011 average monthly price of U3O8per pound obtained from the IndexMundi
web site available at http://www.indexmundi.com/commodities/?commodity=uranium&months=60 divided by the
average energy content of U3O8, namely 180 MMBTU per pound obtained from the TradeTech web site
http://www.uranium.info/unit conversion table.php.
43The load factors for the coal and nuclear plants were obtained by dividing net generation from Table 1.1 of Electric
Power Annual by net summer capacity from Table 1.1.A of the same publication and then averaging the result for
2007-2010. Performing the same calculation for natural gas fired plants produces an average capacity factor of 0.2396.
However, this would cover combined cycle and conventional steam plants fired by natural gas, which are operated as
base or intermediate load, and combustion turbines, which are operated as peaking plants at a very low load factor.
A technology brief from the IEA, available at http://www.iea-etsap.org/web/E-TechDS/PDF/E02-gas fired power-GS-
AD-gct.pdf, claims that typical international values for load factors of combined cycle plants range from 0.2–0.6, while
corresponding values for combustion turbines range from 0.1–0.2. We have assumed that combined cycle plants are
operated at the top of the IEA range (0.6) and combustion turbines at the low end of the IEA range (0.1). We also
25

for the coal, nuclear and pumped storage plants, 30 years for the single and combined cycle tur-
bines, and 25 years for the wind generators44 to calculate costs of electricity generation for the
different types of plants. We separated these costs into those that would be part of investment in
the GTAP data (capital costs) and those that would be part of “consumption” (operating and main-
tenance expenditures). The equivalent annual capital cost (EAC) of capital per MW of capacity
was calculated based on an assumed annual real required rate of return of 7.5%.
The annual capital costs for the system are calculated as a capacity-weighted average of the
EAC for each type of plant.45 Similarly, the annual O&M costs for the system are calculated as
a capacity-weighted share of the fixed O&M costs plus an output-weighted sum of the combined
variable O&M and fuel costs.46 The ratio of the annual capital costs for the non-fossil system to
the annual capital costs for the fossil-based system were then calculated to be 9.9807, while the
ratio of the annual O&M costs for the two generating systems was 0.4948.47
We already calculated above that the energy output to capital ratio for fossil fuels as G= 3.0506.
We therefore assumed that the corresponding initial energy output to capital ratio for non-fossil
fuels was H= 3.0506/9.9807 ≈0.3056. Similarly, since we calculated above that the initial value
for µ= 0.1614, we take the value of m= 0.1614 ∗0.4948 ≈0.0799.
Finally, we need to specify a final limiting value for Hand the coefficient bin the renewable
assumed that the load factor for pumped storage would equal the load factor (0.1) of natural gas combustion turbines
in the fossil fuel world and that the cost of other forms of storage would be double the cost of pumped storage.
Using EIA data on annual electricity generation and installed generation capacity by country and type, available at
http://www.eia.gov/countries/data.cfm, we found that the average load factor for wind generators for the world as a
whole in 2009 was 0.20.
44The Environmental Protection Agency National Electric Energy Data System (NEEDS) database, available at
http://www.epa.gov/airmarkt/progsregs/epa-ipm/past-modeling.html#needs, gives an average age of US coal-fired gen-
erators of 38 years, but many plants will be far from the end of their useful life. The conventional oil or gas-fired
steam plants had an average age of 44 years. The average age of the nuclear plants was 24 years, but many more of
these plants would still be a long way from retirement. The average age of the combustion turbines in the NEEDS
database was 27 years, while the combined cycle plants, which embody a relatively new technology, averaged just
13 years. The average age of the pumped storage plants in the NEEDS database was 30 years. Finally, the wind
generators in the NEEDS database were also constructed recently, so we do not have a good indication of how long
they may last. However, several sources on the internet gave a design life of 20 years for wind turbines, while the
maximum estimated lifespan we found was 30 years.
45We weighted the EAC/MW for each technology by the share of that technology in the overall generating capacity.
Allowing for the share of combined cycle plants to be somewhat higher in the future to reflect the higher efficiency
of those plants, we assumed that combustion turbines would constitute 10% of total capacity (and be used only for
10% of total hours) and then divided the remaining required capacity equally between coal and natural gas combined
cycle plants. For the non-fossil world, we increased the share of backup storage to 20% of capacity to account for
the intermittency of wind power. The intermittency of wind generation also makes it unlikely that network stability
could be maintained with such sources constituting more than 30% of system wide capacity. We therefore set the
wind capacity at 30% and nuclear at 50% in the non-fossil world.
46The latter were converted to an annual basis by multiplying by the number of hours in a year that each type of
plant would be operated, namely the load factor, times 8760.
47Details of these calculations are available from the authors.
26

energy technological change function. We arbitrarily assumed that Hcould be increased by a factor
of 4 to ¯
H≈1.2224. Then the final ratio m/ ¯
H≈0.0653 would approximate the current ratio for
fossil fuels µ/G ≈0.0529. Once the values of ¯
Hand mhave been set, the long run per capita
growth rate can also be calculated as 4.14%. We could not find a suitable data source to calibrate
b, so we arbitrarily set it to 0.006 to obtain an approximate 100 years until the renewable sources
attain their final long-run efficiency level.
6 Numerical results
Figure 4 graphs the solution paths for the state variables while Figure 5 graphs the paths for
the control variables.48 The critical times for transition between the various regimes are TQ=
73.5248, TN= 79.0248, TR= 80.0424, TB= 85.7931 and TH= 99. Thus, fossil fuels are abandoned
after about 80 years, but investment in kRceases after about 73.5 years. Investment in fossil
fuel technology Ncontinues for another 5.5 years, ceasing after slightly more than 79 years. The
solution for Simplies that slightly over 77% of the initial stock of fossil fuel resources are exploited.
Investment in fossil fuel technology Ndoes not play a large role until t≈50.
[Figures 4 and 5 around here]
A striking feature of the optimal paths of the investment variables in Figure 5 is their somewhat
volatile character. Some of the fluctuations in i, iRand iBappear to be to facilitate investments n
and j. There is a large spike in investment nas investment in kRceases, with a smaller jump in
noccurring right before ndrops off quickly to zero. Similarly, investment iBin kBand jin Hdo
not take off until investment in kRceases at TQ. The need to replace fossil fuel energy production
at TRmeans that there has to be considerable investment in kBprior to TR. In addition, the rapid
increase in Hafter TRmeans that less kBis needed to supply a given level of energy services. The
result is excess kBat TRand, as illustrated in the third graph in Figure 5, investment iBin kBis
therefore zero for about five years (from TRuntil TB) and kBdeclines over this period. Similarly,
kdeclines along with kRbetween TQand TRas both iand iRare zero over this interval. While kR
48The solution discussed here has k(0) = 2.3285, H (0) = 0.3070, S(0) = 0.0000004 and N(0) = −0.0924 compared
to target values of 2.3273, 0.3056, 0 and 0. The value of S(0) is closest to its target because Sresponds most sensitively
to changes in the terminal values used to determine the solution to the model’s differential equations. The initial
value of Nis least sensitive, making it hardest to match. The model’s differential equations are highly non-linear,
with small changes in the terminal values often making the differential equations impossible to solve. Hence, we could
not use an automatic solution procedure to find the best terminal values to hit the required target values at t= 0.
27

remains positive for t>TR, it is not used to supply energy services because renewable sources are
then cheaper.
The economy also is severely disrupted by the switch from using fossil fuels to renewables to
supply energy services. In particular, per capita consumption actually declines for nearly 7 years
(between TQand TR). Part of the explanation is that the real price of energy rises substantially
between TQand TRand the need to spend on energy takes resources away from consumption and
investment. Around the same time, the growth in energy R&D in the form of increases in both
nand jalso reduces the resources available for consumption. However, the decline in kover this
interval also reduces output available for consumption and investment.
[Figure 6 around here]
Figure 6 graphs the solution paths for the main49 co-state variables. The shadow price νof
investing in Nis equal to λwhile n > 0 (that is, between t= 0 and TN) after which point it declines
quickly to zero and remains there. The shadow price σof the fossil fuel resource mined to date S
is negative until fossil fuels are abandoned at TR, at which point σbecomes zero and remains at
zero thereafter. The negative shadow price reflects that assumption that increased mining raises
future costs. The real (or utility) value of the shadow price σ/λ declines continuously (increases
in absolute value) until TN, after which the increase to zero is swift. Finally, the shadow price η
of Hincreases to a peak at TRand then declines to zero at THwhen additional gains in Hare no
longer possible. This implies that the incentive to invest in Hfollows the same pattern. The ratio
η/λ looks much more similar to the graph of investment jin H. However, it does start to increase
earlier than j(after 60 years) and the “bump” immediately prior to TRis longer lasting and more
noticeable in η/λ than in j.
7 Concluding remarks
We have presented an intertemporal optimizing growth model where energy inputs essential to final
production can be supplied by either fossil fuels or renewable sources. Initially, the fossil fuels are
much cheaper and supply all the energy. However, the fossil fuels are eventually displaced (at TR)
as depletion raises their cost, while R&D and learning-by-doing allows the cost of renewable sources
to decline over time. Although technical progress in fossil energy production can offset the cost
49A graph of the small interval over which qBdiffers from λhas been omitted. Also, qR=λuntil TQ, at which
point it declines to zero and remains at zero thereafter.
28

increases from depletion and delay the displacement of fossil fuels by renewables, it cannot forestall
the transition indefinitely.
Physical capital is needed to turn fossil fuel resources into useful energy services or to “harvest”
the non-fossil energy sources (such as sunlight, wind, or geothermal resources) and make them
available as energy that can be used in final production. We assume that capital used to provide
energy services from fossil fuels kRcannot be re-purposed as kBto be used for renewable energy.
An important consequence is that the economy has to invest in kBprior to TRso that renewables
are ready to take over from fossil fuels at TR.
For t≤TR, the cost of fossil fuels sets the price of energy, which thus is rising for some time
from t < TRuntil TR. Also, the short-run cost of renewable energy, which exceeds the price of
energy at t= 0, is falling over time as a result of R&D investment and learning-by-doing. At
TR, the short-run cost of producing energy from the two sources is equal. Thus, prior to TR, the
short-run cost of renewable energy exceeds the price of energy.
For most of the time prior to TR, there is virtually no investment in kB. The gains from learning-
by-doing imply that some investment in kBand renewable R&D is optimal in order to hasten the
transition and lower the costs of energy supply from renewables at the transition. Furthermore,
for a few years immediately before the transition, a burst of investment in renewable capacity is
needed to ensure continuity in the supply of energy services. At the same time, however, kBis not
used to supply energy services until the transition occurs. Prior to the transition, the energy price
is not sufficient to cover even the operating cost of kB.
In the numerical example, renewable sources take over from fossil fuels after about 80 years,
when slightly over 77% of the initial stock of fossil fuel resources have been exploited. Investment
in kBbegins in earnest after about 74 years. Energy prices are not sufficient to cover the full costs
of producing renewable energy (including a competitive return on capital) until after almost 86
years. While these specific transition dates depend on the chosen parameter values, the qualitative
characteristics of the solution were shown to hold much more generally.
We emphasize that the investment paths calculated in the model are part of the efficient out-
come. Whether all the benefits of such investments would be fully appropriable to private en-
trepreneurs and allow a competitive equilibrium to support the Pareto optimum is an entirely
different matter. Our model may in this sense complement much of the literature explaining the
difficulty of establishing new energy technologies as akin to transitioning a “valley of death.” That
literature has focused on the inability of entrepreneurs to appropriate all of the benefits of early re-
29

search into alternative energy technologies. Our model has added the additional point that, unlike
the pharmaceutical and IT industries, the energy industry requires substantial capital investment
beyond the R&D phase in order to commercialize the new technologies. These investments will
be difficult to finance while the capital invested in existing technologies is a sunk cost and sets a
low energy price as the old technology is phased out. Nevertheless, in the model presented, the
subsidies to renewable energy in the initial phase ought not extend to subsidizing energy production
on a commercial scale.
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years
0 10 20 30 40 50 60 70 80 90 100
multiples of y(0)
0
100
200
300
400
500
600 k
years
0 10 20 30 40 50 60 70 80 90 100
multiples of y(0)
0
50
100
150
200
250
kB
years
0 10 20 30 40 50 60 70 80 90 100
multiples of y(0)
0
5
10
15
20
25
30
35
40
45
kR
years
0 10 20 30 40 50 60 70 80 90 100
multiples of R(0)/y(0)
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3 H
years
0 10 20 30 40 50 60 70 80 90 100
multiples of f0
0
200
400
600
800
1000
1200
1400
1600
1800 S
years
10 20 30 40 50 60 70 80 90 100
multiples of y(0)
0
10
20
30
40
50
60 N
Figure 4: Calculated paths of the state variables
33

years
0 10 20 30 40 50 60 70 80 90 100
multiples of y(0)
0
50
100
150 c
years
0 10 20 30 40 50 60 70 80 90 100
multiples of y(0)
0
5
10
15
20
25
30
35
40
45
50 i
years
0 10 20 30 40 50 60 70 80 90 100
multiples of y(0)
-10
0
10
20
30
40
50
60
iB
years
0 10 20 30 40 50 60 70 80 90 100
multiples of y(0)
0
1
2
3
4
5
6
7
iR
years
0 10 20 30 40 50 60 70 80 90 100
multiples of y(0)
0
1
2
3
4
5
6
7j
years
0 10 20 30 40 50 60 70 80 90 100
multiples of y(0)
0
5
10
15
20
25
30
35
40
45 n
Figure 5: Calculated paths of the control variables
34

years
0 10 20 30 40 50 60 70 80 90 100
utils/y(0)
0
1
2
3
4
5
6
7
8λ
years
0 10 20 30 40 50 60 70 80 90 100
utils*y(0)/R(0)
×10-7
0
0.2
0.4
0.6
0.8
1
1.2 η
years
0 10 20 30 40 50 60 70 80 90 100
utils/f0
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0σ
years
0 10 20 30 40 50 60 70 80 90 100
utils/y(0)
0
1
2
3
4
5
6
7
8ν
years
0 10 20 30 40 50 60 70 80 90 100
y(0)2/R(0)
0
50
100
150
200
250 η/λ
years
0 10 20 30 40 50 60 70 80 90 100
y(0)/f 0
-0.16
-0.14
-0.12
-0.1
-0.08
-0.06
-0.04
-0.02
0σ/λ
Figure 6: Calculated paths of the co-state variables
35