Pergamon
PII:
Solar Energy Vol. 68, No. 5, pp. 379–392, 2000
 2000 Elsevier Science Ltd
S 0 0 3 8 – 0 9 2 X ( 0 0 ) 0 0 0 1 8 – 9 All rights reserved. Printed in Great Britain
0038-092X / 00 / $ - see front matter
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LONG-TERM PERSPECTIVE ON THE DEVELOPMENT OF SOLAR ENERGY
YACOV TSUR* , ** and AMOS ZEMEL *** , **** , †
*Department of Agricultural Economics and Management, The Hebrew University of Jerusalem,
P.O. Box 12, Rehovot, 76100, Israel
**Department of Applied Economics, University of Minnesota, Minneapolis, MN, USA
***Department of Energy and Environmental Physics, The Jacob Blaustein Institute for Desert Research,
Ben Gurion University of the Negev, Sede Boker Campus, 84990, Israel
****Department of Industrial Engineering and Management, Ben Gurion University, Beer Sheva, Israel
Received 31 August 1999; revised version accepted 5 January 2000
Communicated by ARI RABL
Abstract—We use dynamic optimization methods to analyze the development of solar technologies in light of
the increasing scarcity and environmental pollution associated with fossil fuel combustion. Learning from solar
R&D efforts accumulates in the form of knowledge to gradually reduce the cost of solar energy, while the
scarcity and pollution externalities associated with fossil fuel combustion come into effect through shadow
prices that must be included in the effective cost of fossil energy. Accounting for these processes, we
characterize the optimal time profiles of fossil and solar energy supply rates and the optimal investment in
solar R&D. We find that the optimal rate of fossil energy supply should decrease over time and vanish
continuously upon depletion of the fossil fuel reserves, while the optimal supply of solar energy should
gradually increase and eventually take over the entire energy demand. The optimal solar R&D investment
should initially be set at the highest feasible rate, calling for early engagement in solar R&D programs, long
before large scale solar energy production becomes competitive.  2000 Elsevier Science Ltd. All rights
reserved.
fuel, have reached maturity leaving little room for
a significant cost reduction. Indeed, current research on these technologies is mainly concerned
with pollution abatement (e.g. clean coal technologies or the use of hydrogen) rather than with
improving fossil fuel conversion efficiencies
which are nearing their theoretical limits. Similarly, the important progress in energy conservation
serves mainly to mitigate the rapid increase in
energy demand, but does not contribute directly to
reduce the production costs of fossil energy. In
contrast, solar technologies still have large potential for improvement, pending appropriate
R&D. Moreover, the true price of fossil energy
must include scarcity and pollution components to
allow a valid evaluation of social costs and
benefits of alternative energy options.
Nuclear energy is also often mentioned as a
viable alternative. However, the nuclear reactor
industry has seen serious setback due to public
perception of the risks involved and the controversial waste disposal practices. For this reason,
future progress of the nuclear option depends on
considerations that belong mainly to the political
arena and lie outside the scope of the present
work.
The present economic value of future events or
ongoing processes (such as the depletion of the
1. INTRODUCTION
In this work we offer long-term perspectives of
fossil-solar energy tradeoffs by formulating
energy policies within a dynamic optimization
framework, seeking to maximize social welfare
defined over an extended planning horizon rather
than myopic goals. This presentation draws heavily on Tsur and Zemel (1998b) that lay out the
model on which we base our policy analysis. The
model is based on the observation that the development of solar technologies is driven by two
major concerns: fossil energy is limited by finite
reserves of non-renewable deposits, and the combustion of fossil fuels entails the emission of
various pollutants and greenhouse gases into the
atmosphere with undesirable environmental consequences. In contrast, solar energy is practically
unlimited (so can serve as a backstop resource)
and is clean.
At present, large-scale fossil energy production
is cheaper than the available solar alternatives
(Chakravorty et al., 1997). However, conventional energy generation technologies, based on fossil
†
Author to whom correspondence should be addressed. Tel.:
1972-7-659-6925; fax: 1972-7-659-6921; e-mail:
[email protected]
379
380
Y. Tsur and A. Zemel
fuel stock or the buildup of atmospheric pollution)
is a direct outcome of the dynamic optimization
methodology. In particular, overall effects of fuel
scarcity and atmospheric pollution manifest themselves via dynamic ‘shadow prices’ that must be
included in the effective costs of fossil energy. In
a dynamic optimization framework these shadow
prices are derived as an intrinsic part of the
optimal solution and need not be included via
ad-hoc assumptions. Moreover, the dependence of
the shadow prices on key parameters (such as the
initial size of the fuel stock) can be derived.
The oil crises of the 1970s have led to a surge
in R&D efforts dedicated to the development of
the solar alternative. These efforts, however, were
strongly correlated with the (fluctuating) market
price of energy, and suffered a serious setback as
this price later plunged. The missing ingredient, it
appears, was a long-term perspective that treats
R&D policy within the wider context of fossil and
solar energy tradeoffs rather than reactions to
temporary price fluctuations. The present work
attempts to consider the development of solar
energy from this perspective.
The same events also gave rise to a rich
literature on the optimal exploitation of natural
resources and the desirable rate of R&D efforts to
promote competitive backstop technologies (see
Tsur and Zemel (1998b) for a literature survey).
A common assumption in this literature is that the
backstop technology arrival (or improvement) is a
discrete event whose occurrence (which may be
governed by uncertainty) is affected by the R&D
policy. A typical pattern for R&D efforts under
this framework is to follow a single-humped path
(an increase in R&D efforts followed by a decrease) with a possible delay in initiating the
R&D program. Here we depart from this characteristic aspect by substituting the discrete-event
nature of the backstop technology arrival date
with a technological progress that evolves continuously in time as R&D efforts accumulate in
the form of knowledge to reduce the cost of solar
energy. The result is an early engagement in
R&D, as the optimal policy calls for maximal
R&D efforts in the early stages. We also assume,
like Hung and Quyen (1993), that the marginal
cost of energy production is rate-dependent. This
entails a gradual shift from fossil to solar sources
and the rate of fossil energy supply decreases
continuously in time and approaches zero as the
stock of fossil fuel is nearing depletion.
Another strand of related literature is the growing body of research on energy management in
light of atmospheric pollution and global warming
processes (see, e.g., Nordhaus (1979, 1991, 1992,
1993), Edmonds and Reilly (1983, 1985), Cline
(1992), Weyant (1993), Hoel and Kverndokk
(1996), Tsur and Zemel (1996, 1998a), Chakravorty et al. (1997)). Here, however, our interest
is focused on endogenizing the development of
the backstop technology. Our description of the
energy market and the treatment of the process of
atmospheric pollution and its potential damage
are, therefore, simplistic. As such, the model
presented here cannot pretend to provide a quantitatively realistic description of future energy
trends, nor is it our aim to accurately predict the
depletion date of fossil fuel or the rate of penetration of solar energy. Rather than that, our purpose
here is to present a methodology of considering
these issues in a systematic and consistent manner. In terms of this methodology, the incorporation of the scarcity and pollution damage components into the cost of fossil energy is based on
solid economic theory and not restricted to intuitive arguments. For this reason, this work might
contribute to bridge the gap between those who
advocate, on ‘economic’ grounds, to delay the
introduction of alternative technologies until they
turn competitive, and the vision of solar energy
pioneers who stressed the urgency to develop
these technologies well in advance. To apply
these theoretical concepts in actual practice, the
functional relations of the model must be
specified and expanded so as to reflect the richness and intricacies of the global energy market,
the model parameters should be reliably estimated
and numerical methods need to be applied to
derive optimal policies in different circumstances.
This task is outside the scope of the present work.
These qualifications notwithstanding, some
simple and robust policy rules can be derived by
our methodology. Most notable among these are
the smooth (rather than abrupt) transition from
fossil to solar energy supply and the recommendation for early engagement in solar R&D programs,
long before large scale solar energy production
becomes competitive. The complete characterization of the optimal energy policy is highly technical and its derivation can be found in Tsur and
Zemel (1998b). Here we present the main results
and discuss their policy implications. The typical
pattern of the optimal processes is illustrated via
numerical examples.
2. A DYNAMIC FORMULATION OF THE
ENERGY TRADEOFFS
We consider a simplified model of the energy
sector, focusing attention on the components that
underlie the dynamic tradeoffs we wish to investi-
Long-term perspective on the development of solar energy
gate. In particular, we aggregate the various
sources of supply into two competing classes,
namely fossil (representing conventional sources)
and solar (representing alternative sources), disregarding important differences among technologies
within each class. This approach allows a rigorous
analysis of the optimal solar R&D policy and its
characterization in terms of some key parameters
for which simple expressions can be derived.
2.1. Demand
Instantaneous energy demand D( p) is a decreasing function of the price of energy p. The
inverse demand function, D 21 (q), represents the
price that can be obtained for the last energy unit
at any level of power supply q. Since this price
serves as a measure of the value to consumers of
this last energy unit, the area G(q) 5 e0q D 21 (z)dz
below the curve of this function gives the total
rate of user benefit flow from the supply of q. The
inverse demand is measured in $ / MJ, (or more
often, in $ / kWh). In the dynamic framework
considered here, it is also convenient to think of
this function as the rate of benefit flow derived
from the last power unit produced (i.e. $ per unit
time per W). Thus, the total benefit flow G(q) is
measured in $ per unit time.
Typically, energy demand fluctuates daily and
seasonally, and follows a long-term trend. In the
context of long run planning, short run (daily,
seasonally) fluctuations can be smoothed out
(Tsur and Zemel (1992) considered policies that
mitigate short-term demand fluctuations) and only
long-term trends are of interest. These long-term
trends will be discussed in a following section.
Here we consider stationary demand.
2.2. Supply of fossil energy
The marginal cost of energy — the cost of
generating the last energy unit — is measured in
$ / MJ, (or $ / kWh). Again, we can think of this
cost as the rate of expenses flow (in $ per unit
time) for the last power unit (W) produced. It is
with this interpretation in mind that we speak of
this cost as the (instantaneous) marginal cost of
power. For fossil energy, the marginal cost is
composed of direct and external components. The
direct (or engineering) costs are those borne by
the energy producers themselves and consist of
the extraction, processing and delivery cost of
fossil fuel, as well as cost of operation and
maintenance and disposable equipment. The external costs, which affect society as a whole,
account for environmental pollution and scarcity
of fossil fuels.
The marginal engineering cost for a particular
381
power plant may decrease over some range of
power supply due to economies of scale in power
generation. Yet, at the aggregate level, when
plants differ in efficiency, the marginal cost of
supply increases. This is so because the last
energy unit should be generated by the cheapest
plant that is still operating below capacity and
larger quantities of energy require the operation of
the more expensive plants. With C(q c) ($ per unit
time) representing the instantaneous engineering
cost of producing the conventional fossil power
q c , it is assumed that the marginal cost Mc (q c) 5
dC(q c) / dq c increases with q c .
To the engineering cost of fossil energy one
must add the external costs due to pollution.
Measuring the pollution in terms of the cumulative fossil energy produced, the pollution-induced
damage flow is given by wPt , where Pt represents
atmospheric concentration of pollutants at time t
and w converts pollution units into pecuniary cost.
A methodology to estimate the pollution costs is
presented in Rabl et al. (1996). The pollution
state Pt evolves in time according to
dPt / dt 5 q tc 2 r Pt
(1)
where r .0 is a cleansing parameter representing
the natural rate of pollution decay.
The other component of external costs — due
to the increased scarcity of fossil resources —
will show up below in the formulation of the
dynamic solution through the shadow price (the
costate variable) associated with the fuel stock.
For convenience, the fuel stock is measured in
terms of the energy that can be actually obtained
from it (after accounting for conversion inefficiencies), hence its level changes over time according
to
dXt / dt 5 2 q ct .
(2)
2.3. Supply of solar energy
Solar energy generation technologies improve
as R&D activities are translated into knowledge
via learning processes. This implies that the
marginal cost Ms of solar energy is a decreasing
function of the state of knowledge Kt available at
time t. The latter, in turn, accumulates with the
learning associated with the R&D investments
that had taken place up to time t. Assuming that,
at a given knowledge level, solar energy generation technology admits constant returns to scale,
the instantaneous cost ($ per unit time) of producing the solar power q s at time t is specified as
Ms (Kt )q s . It should be acknowledged, at this
point, that restricting the marginal cost to depend
on the knowledge state alone disregards important
382
Y. Tsur and A. Zemel
cost determinants. One expects some dependence
of the marginal cost on the instantaneous power q s
and, at least for the earlier stages of market
penetration, on the cumulative solar energy produced up to the time t. The latter dependence is
usually described via the ‘learning curves’. We
note, however, that accounting for these factors
does not affect the general conclusions derived
below, although the analysis turns more cumbersome. Moreover, our interpretation of the R&D
efforts is somewhat more general than the usual
meaning assigned to this concept. For example,
the installation of a demonstration solar power
station and its ongoing operation are viewed, in
this context, as part of the R&D efforts. The
contribution of this activity to the learning curve
is, therefore, included in the knowledge state K.
Similarly, the R&D activity is not restricted to the
resolution of technical questions that affect the
production cost of solar energy but includes
research aiming at understanding and removing
social, cultural and political barriers that hamper
the market penetration of solar technologies.
The balance between the rate of R&D investment, (R t , measured in $ per unit time) and the
rate at which existing knowledge is lost or
becomes obsolete due to aging or new discoveries, determines the rate of knowledge accumulation
dKt / dt 5 R t 2 d Kt
(3)
where K is measured in monetary units ($) and d
is a knowledge depreciation parameter.
2.4. Social benefit
From the point of view of society as a whole,
the energy bill is just a transfer from consumers to
producers of no effect on the overall welfare
hence it can be ignored. (It is assumed, of course,
that the price to consumers is determined so as to
ensure production at the socially optimal rates
derived below.) The gross consumer’s surplus
from the supply of power q is specified above as
G(q) — the area below the demand curve and to
the left of q (see Fig. 1). The cost of supplying
q c 1 q s is C(q c) 1 Ms (Kt )q s . The net consumer
and producer surplus generated by q 5 q c 1 q s is
G(q c 1 q s ) 2 [C(q c) 1 Ms (Kt )q s ]. Adding the
costs of R&D and of environmental pollution, the
net rate of social benefit flow at time t is
G(q ct 1 q st ) 2 C(q ct ) 2 Ms (Kt )q st 2 R t 2 wPt .
(4)
2.5. Energy policy
An energy policy consists of three control
(flow) and three state processes: the flow pro-
Fig. 1. Power supply and demand at time t, given Kt and lt .
The area ABCD represents the sum of consumer and producer
surpluses.
cesses are q tc (supply of conventional power), q ts
(supply of solar power) and R t (R&D in solar
technologies). The state variables are Xt (available
reserves of fossil fuel), Kt (solar knowledge), and
Pt (atmospheric pollution). Initiated at the states
hX0 , K0 , P0 j, a policy G 5 hq tc , q ts , R t , t $ 0j
determines the evolution of the state variables via
Eqs. (1)–(3) and gives rise to the instantaneous
net benefit, Eq. (4). The optimal energy policy is
the solution to
`
V(X0 , K0 , P0 ) 5 MaxG
EfG(q 1 q ) 2 C(q )
c
t
s
t
c
t
0
2 Ms (Kt )q st 2 R t 2 wPtge 2rt dt
(5)
¯
subject to Eqs. (1)–(3), q ct , q st $ 0, 0 # R t # R,
Xt $ 0, and X0 , K0 , P0 given. In Eq. (5), r is the
time rate of discount and R¯ is an exogenous upper
bound on the affordable R&D effort. Together
with Eq. (3), this bound implies the upper bound
K¯ 5 R¯ /d on the knowledge state. Note that at this
stage the pollution externality is expressed in
terms of the pollution state Pt , but not yet as a
cost component imposed on fossil power production. Similarly, the scarcity externality is
expressed only indirectly, via the constraint Xt $ 0
imposed on the fuel stock.
Long-term perspective on the development of solar energy
3. THE OPTIMAL ENERGY POLICY
The complete characterization of the optimal
energy policy requires the specification of the
three control variables (q ct , q st and R t ) and of the
state variables (Xt , Pt and Kt ) derived thereof.
Indeed, the dynamic optimization problem, Eq.
(5), is formulated in a way which is most readily
handled by optimal control methods (see, e.g.
Leonard and Long, 1992). The mathematical
derivation of the optimal policy is highly technical and will not concern us here. (The details can
be found in Tsur and Zemel, 1998b.) In this
section we present the main results in terms of
simple and explicit rules, and illustrate their
application via numerical examples in the next
section. For the problem at hand, the solution is
divided into two steps. First, the optimal supply
rates of fossil and solar energy are determined in
much the same way as one would do in a static
problem, where the dynamics enters through the
scarcity rent of fossil fuel that is added to the
marginal cost of conventional energy. The second
step involves the determination of the optimal
solar knowledge and fossil fuel scarcity processes.
3.1. Optimal supply rates of fossil and solar
energy
The energy supply rates are determined such
that (a) the overall supply meets demand, and (b)
the effective marginal cost of fossil energy (which
includes the scarcity rent and the atmospheric
pollution cost) equals that of solar energy (which
depends on knowledge). The effective marginal
cost of fossil energy consists of the direct production cost Mc (q c) 5 dC(q c) / dq c and indirect
costs due to the external effects. For the latter, the
optimal control derivation yields the form l0 e rt 1
w /(r 1 r ), where the first term is the fossil fuel
scarcity rent (or shadow price), with l0 a nonnegative constant depending on the initial fuel
stock as described below, and the second term is
the marginal cost due to atmospheric pollution.
Indeed, w /(r 1 r ) is a levy imposed on fossil
energy due to its contribution to atmospheric
pollution; as such it is a manifestation of what is
generally referred to as ‘carbon tax’. The exponential rise obtained for the shadow price
lt 5 l0 e rt until the depletion date accounts for the
fact that as the fuel reserves are being used, the in
situ value of the resource (i.e. the value of stock
remaining in the ground) must increase. Thus, the
effective marginal cost of fossil energy
increases with time when l0 is positive (for a
fixed supply rate q c). With a strictly convex cost
function C(q c), the marginal cost Mc (q c) increases
with q c .
As solar energy entails no external effects, its
marginal cost of supply is simply Ms (Kt ) — a
decreasing function.
Conditional on Kt and lt , we use q c (Kt , lt ) and
s
q (Kt , lt ) to represent the optimal supply rates of
conventional and solar energy, respectively, and
let q(Kt , lt ) 5 q c (Kt , lt ) 1 q s (Kt , lt ) denote total
power supply. The optimal mix of conventional
and solar power at each point of time is determined such that each additional unit of energy
is supplied from the cheapest available source. So
long as the fossil fuel stock is not depleted,
conventional power is supplied up to the level
where its effective marginal cost just equals the
marginal cost of solar power, i.e. the power q c at
which M(q c , lt ) intersects Ms (Kt ) (see Fig. 1).
Thus, the supply of fossil power is the level
q c (Kt , lt ) that satisfies
Ms (Kt ) 5 Mc (q c (Kt , lt )) 1 lt 1 w /(r 1 r ).
(6)
(7)
Power demand beyond q c (Kt , lt ) is provided by
solar plants. The overall power supply q c (Kt , lt )
1 q s (Kt , lt ) is determined by the point in which
demand (D 21 ) intersects the minimal unit cost of
power production, i.e. the minimum between
Ms (Kt ) and Mc (q c) 1 lt 1 w /(r 1 r ) (see Fig. 1).
Assuming that, at their intersections with the
demand curve, the unit cost of solar power
generation is smaller than that of fossil power, the
(energy) market clearing condition, i.e. demand
equals supply, reads
q c (Kt , lt ) 1 q s (Kt , lt ) 5 D(Ms (Kt )).
(8)
Together, Eqs. (7) and (8) determine the optimal
supply mix when Kt and lt are given.
The role of the shadow prices ought to be
explained. Indeed, the supply rules (7) and (8)
and the explicit form of the shadow prices are
derived in Tsur and Zemel (1998b) using the
maximum principle, which comes out of the
necessary conditions for optimal control problems. However, the simple form assumed for the
damage term in (5) allows interpreting this role in
a straightforward manner. Using Eq. (3) to eliminate Pt and integrating the resulting dP/ dt term by
parts, one obtains
`
EwP e
t
M(q c , lt ) 5 Mc (q c) 1 l0 e rt 1 w /(r 1 r )
383
0
`
2rt
w
dt 5 ]
r
3E
0
`
c 2rt
t
qe
E
dt 1 P0 2 r Pt e 2rt dt
0
4
384
Y. Tsur and A. Zemel
hence
`
EwP e
`
2rt
t
w
dt 5 ]]
r1r
0
3E
4
q tc e 2rt dt 1 P0 .
0
It follows that the damage term 2 wP, appearing on the right-hand side of Eq. (5), is equivalent
to adding a cost term 2 q c w /(r 1 r ) to the
objective’s integrand and a constant term 2 P0 w /
(r 1 r ) to the value function V(X0 , K0 , P0 ). The
former term entails a contribution of w /(r 1 r ) to
the effective marginal cost of fossil power while
the latter term implies that ≠V/ ≠P0 5 2 w /(r 1 r ),
establishing the interpretation of this shadow price
as the value to society of changing the pollution
level by one unit. A similar interpretation can be
assigned to any shadow price with respect to its
corresponding state variable (Leonard and Long,
1992). In particular, the process lt measures the
change in the value V(Xt , Kt , Pt ) due to an
increase in the fossil fuel stock Xt by one unit.
Thus, the lt component of the effective marginal
cost accounts for the loss of value associated with
the decreasing stock as more fossil power is
produced.
Considering Fig. 1 again, we see that the
optimal supply rule (7) and (8) is equivalent to
maximizing the area ABCD, which is the net
consumer and producer surplus (see the previous
section) when the marginal production cost of fossil power is taken as the effective cost M(q c , lt )
of Eq. (6) rather than the marginal engineering
cost Mc (q c) alone. This result is typical of static
economic optimization. Here, however, the dynamics enter via the incorporation of the shadow
prices into the marginal cost.
A difficulty with implementing the supply rule
(7) and (8) arises when the fossil power supply it
implies is positive but the stock of fossil fuel is
already depleted. Fortunately, this situation cannot
occur under the optimal policy. This is so because
the optimal knowledge and shadow price processes are so chosen that at the time of depletion
(and thereafter) it is not optimal to use fossil
energy. Thus, as the fossil reserve is nearing
depletion, i.e. as time t approaches the optimal
depletion date T * (the * superscript indicates
optimal values), the supply of fossil energy apc
proaches zero, i.e., q (K *t , l *t ) → 0. This result
follows from the conditions
Ms (K *T * ) 5 Mc (0) 1 l 0* e rT * 1 w /(r 1 r )
(9a)
and
T*
E q (K *, l *e )dt 5 X ,
c
rt
t
0
0
0
(9b)
which hold at the depletion date T * (see Tsur and
Zemel (1998b) for details).
Condition (9a) implies that, as time approaches
the depletion date, the optimal rate of fossil
energy supply approaches zero (c.f. Eq. (7)).
Thus, fossil energy supply does not undergo a
discontinuous drop at the depletion time and solar
energy takes an increasing share prior to depletion. Note again the importance of the shadow
prices in this context. Without these terms, some
fossil power would still be desirable at the
depletion date, but the empty stock would not be
able to supply it. The instability associated with
discontinuous supply must entail considerable
damage to society. However, the shadow prices
do not carry out the task alone. The knowledge
process K must obtain its appropriate value to
ensure the smooth transition. Condition (9b) is a
restatement of the depletion event at T *. Together, conditions (9a) and (9b) serve to determine the optimal parameters T * and l 0* , as
explained below.
The particular form of (9a) has been derived for
the specific model considered here. However, the
conclusion derived thereof, of a smooth transition
from fossil to solar energy, follows from general
properties of the optimality conditions, and its
validity extends for more general circumstances.
It is formulated as
Policy Rule 1: the optimal fossil energy production rate vanishes continuously at the depletion date.
The energy supply rates at each time depend on
the R&D policy, represented by the knowledge
process Kt , and on the remaining reserves of fossil
energy, represented by the scarcity rent process
lt . We turn now to characterize the optimal
trajectory of these processes.
3.2. Optimal R& D policy and fossil reserve
scarcity rent
Spence and Starrett (1975) defined a most rapid
approach path (MRAP) as the policy that drives
the underlying state process Kt to some steady
state Kˆ as rapidly as possible. Let
K mt 5 (1 2 e 2d t )R¯ /d 1 K0 e 2d t
(10)
be the knowledge path that departs from K0 when
R&D investment is set at its maximal rate R¯ (see
Eq. (3)). Then, the MRAP policy initiated at
K0 , Kˆ is defined by Min(K mt , Kˆ ).
In the present case, we find that the optimal
R&D policy is to steer the optimal knowledge
process, K *t , as rapidly as possible to some
prespecified target process (rather than a fixed
Long-term perspective on the development of solar energy
steady state) and to proceed along the target
process once it is reached. We call such a policy
nonstandard
most
rapid
approach
path
(NSMRAP). Characterizing a NSMRAP, then,
requires specifying the target process. To that end,
we introduce the function
L(K, l) 5 2 M 9s (K)q s (K, l) 2 (r 1 d ).
(11)
It turns out that the target process corresponding to the optimal R&D policy is the root of
L(K, l), i.e. the solution K( l) of L(K( l), l) 5 0
evaluated at the optimal l-process. The intuition behind this property stems from the fact that L(K, l)
can be viewed as the derivative with respect to K
of a utility to be maximized by the R&D policy
(Tsur and Zemel, 1998b). Thus we seek the root
of L(K, l) over the K-domain in which L(K, l)
decreases in K. It is assumed that the root K( l) is
unique in this domain for any non-negative l and
that it lies above the initial level K0 and below the
maximal knowledge level K¯ 5 R¯ /d. A relaxation
of this assumption would imply corner solutions
or an ambiguity concerning the ‘correct’ root, but
otherwise adds no further insight to the analysis.
Thus, the optimal R&D policy is a NSMRAP
with respect to the K( l t* ) process, (denoted the
root process), driven by the optimal lt process i.e.
m
K t* 5 MinhK t , K( l t* )j;
R *t 5 R¯ if K *t , K( l *t );
sponding to q c 5 0) for any given K, the root Kˆ is
an upper bound on the root process, corresponding to sufficiently high scarcity rents (c.f.
Eq. (11)). Let
K S 5 M 21
s (Mc (0) 1 w /(r 1 r )),
(14)
be the minimal K-level that renders conventional
energy too expensive to be used even with an
infinite stock of fossil fuel (hence with zero
scarcity rent l). At this knowledge level the unit
cost of solar energy (Ms (K S )) equals the cost of
generating the first unit of fossil power (Mc (0) 1
w /(r 1 p)).
When Kˆ $ K S the root process reduces to the
ˆ In view of the NSMRAP feature of
singleton K.
the optimal R&D process, the optimal R&D
policy reduces to the standard MRAP K *t 5
MinhK mt , Kˆ j in this case.
The optimal fossil fuel scarcity process, l *t ,
depends on the initial reserves in the following
0
way. Let Q be the total amount of fossil energy
reserves consumed under the maximal R&D
investment policy K *t 5 K tm with an unbounded
initial fossil reserves and a vanishing scarcity
rent:
`
0
E
c
m
Q 5 q (K t ,0)dt.
(15)
0
R t* 5 K9( l t* )rl 0* e rt 1 d K( l t* ) if K t* 5 K( l *t )
(12)
(c.f. Eq. (3)).
This result gives rise to
Policy Rule 2: the optimal R&D policy must
begin immediately at the highest possible rate.
Again, Eq. (12) is derived for the specific
model of this work. However, the recommendation for substantial early engagement in solar
R&D holds under more general formulations. The
root process K( l *t ) bears a simple economic
interpretation. Increasing the knowledge level by
dK reduces the cost of solar power production by
2 M 9s (K)q s dK but incurs the cost of (r 1 d )dK
due to interest payment on the investment and the
increased depreciation. The root K( lt ) represents
the optimal balance between these conflicting
effects at time t.
It remains to determine the optimal scarcity
rent process l *t . Let Kˆ be the root of
2 M 9s (Kˆ )D(Ms (Kˆ )) 2 (r 1 d ) 5 0.
385
(13)
Since D(Ms (K)) is the upper limit of q s (corre-
It turns out that if Q 0 # X0 (i.e. if the initial
fossil reserve X0 is large enough to support the
fossil energy supply policy hq c (K mt , 0), t $ 0j),
then l t* 5 0. Smaller initial reserves imply depletion and a positive scarcity rent.
In sum, if Kˆ $ K S then:
(i) the optimal R&D policy is the standard
MRAP K t* 5 MinhK tm , Kˆ j;
(ii) if Q 0 # X0 then l t* vanishes identically for
all t;
(iii) if Q 0 . X0 then l t* 5 l *0 e rt . 0, the fossil
fuel reserves will be depleted at a finite date
T *, and l 0* and T * are found by solving Eqs.
(9a) and (9b).
The dependence of the scarcity rent on the
initial fossil stock X0 is manifest via the conditions of (ii) and (iii) and Eq. (9b). When the
stock is large enough, scarcity is not an issue and
the shadow price vanishes. Otherwise, the latter
equation implies that the shadow price depends
strongly on the stock (c.f. the numerical example).
The case Kˆ , K S is somewhat more involved.
Since the initial knowledge level K0 lies below
K( l) for any l $ 0, the NSMRAP property
implies that the optimal process K t* evolves
initially along K tm . If K tm overtakes the root
386
Y. Tsur and A. Zemel
ˆ then
process K( l t* ) before the latter reaches K,
K t* switches to K( l t* ) and continues with it until
ˆ Otherwise, the optimal process
they arrive at K.
evolves along K mt all the way to Kˆ as a simple
MRAP.
Whether or not the processes K mt and K( l *t )
cross before they reach Kˆ depends on the initial
scarcity rent l *0 . For example, when l 0* 5 0,
K( lt ) is fixed at K(0) and will surely be overtaken
by K mt ; at the other extreme, for large enough l *0 ,
K( l *t ) 5 Kˆ already at t50. It turns out that
whether K tm crosses K( l *t ) prior to reaching Kˆ
depends on whether l *0 exceeds
ˆ
l 0m 5 lˆ e 2rT,
(16)
where
lˆ 5 MS (Kˆ ) 2 MS (K S )
(17a)
and
3.3. Increasing energy demand
Tˆ 5 log[(K¯ 2 K0 ) /(K¯ 2 Kˆ )] /d.
(17b)
Tˆ is the date at which K mt reaches Kˆ (see Eq.
(10)). Our assumption that Kˆ , K S ensures that
lˆ . 0. However, l *0 is not known a priori and the
above criterion cannot be readily applied. For an
equivalent criterion, we consider the quantity
`
m
when (i) the fuel stock is not depleted and
conventional production is feasible (as represented
by the root K( l) of L(K, l)), and (ii) after depletion (as represented by Kˆ ). If the fossil reserves
ˆ investing in R&D at the
are not depleted by T,
maximum possible rate (i.e. the standard MRAP)
entails knowledge depreciation in excess of what
is justified by the solar energy cost reduction,
hence cannot be optimal. The investment rate,
therefore, is decreased at an earlier date. We note
that the slowdown in R&D investment occurs at
the final, singular part of the knowledge process,
and the initial investment is always at the maximum rate, in accord with Policy Rule 2.
Given K *t and l t* 5 l *0 e rt , the optimal mix of
power supply is given by Eqs. (7) and (8). The
characterization of the optimal energy policy is
now complete.
E
Q 5 q c (K mt , l mt )dt
(18)
0
of fossil stock needed to carry out the conventional energy supply policy with K mt and
l mt 5 l m0 e rt
(19)
as the knowledge and scarcity rent processes, and
compare it with the initial stock X0 . We obtain the
following classification.
If Kˆ , K S and Q m $ X0 , then (i) the optimal
R&D policy is the simple MRAP K t* 5 Min(K tm ,
Kˆ ); (ii) the initial scarcity rent l 0* ( $ l 0m ) and the
depletion date T * are obtained from Eqs. (9a) and
(9b).
If Kˆ , K S and Q m , X0 , then, at some date
ˆ the optimal process K t* switches from K tm
t , T,
to the root process K( l t* ). The parameters l *0
( , l 0m ), T * and t are obtained by solving
simultaneously Eqs. (9a) and (9b) and t 5
log[(K¯ 2 K0 ) /(K¯ 2 K( l t* ))] /d, which defines t as
the time the process K mt overtakes the root process
K( l t* ).
The latter case gives rise to a NSMRAP policy.
In this case the balance between solar energy cost
reduction and knowledge depreciation is different
The above characterization is derived assuming
a stationary energy demand. However, in spite of
significant conservation efforts, global energy
demand increases with time due to the rising
standard of living and the accelerated population
growth. This effect has been incorporated into the
model by allowing the demand D( p, t) to depend
explicitly on time (Tsur and Zemel, 1998b). This
change induces a corresponding time dependence
on various parameters of the problem, but the
characteristic property of the optimal knowledge
process in the stationary model — of a MRAP to
the appropriate root process — is extended to this
case. The optimal knowledge policies under
stationary and non-stationary demands differ only
inasmuch as the respective root processes are
different. This result appeals to intuition, because
an increase in demand cannot reduce the benefits
derived from the R&D efforts. Thus, the notion
that substantial investments in solar energy research should not await the next energy crisis is
robustly supported by this analysis.
4. NUMERICAL ILLUSTRATION
In this section we illustrate the procedure to
determine the optimal energy policy by means of
numerical examples. While presenting these examples, we show how the key parameters are
derived and used to characterize the optimal
policy, as described in Section 3. It is emphasized
that the purpose of presenting these examples is
purely expository, hence the demand, supply and
learning are specified in terms of the simplest
Long-term perspective on the development of solar energy
possible functional forms and the parameters are
so chosen as to display the transition from standard to non-standard MRAP in a clear fashion.
Accordingly, we have not attempted to relate
these parameters to any realistic cost estimates,
and the corresponding results must not be interpreted as predictions (or recommendations) concerning the actual fossil fuel depletion date or the
rate of penetration of solar technologies. Indeed,
the two important policy implications of the
present work (namely Policy Rules 1 and 2)
follow from the general optimization methodology and are not related to the oversimplified
specifications of the examples below.
The following specifications are adopted: the
inverse demand for energy (in $ / MJ) is D 21 (q) 5
0.12 2 0.035q, where q is measured in 10 20 J per
year. The marginal cost of fossil energy is
Mc (q c) 5 0.009 1 0.02q c $ / MJ. The unit cost of
solar energy (as a function of knowledge) is
]]
Ms (K) 5 0.002 1 0.048 /œK /K0 $ / MJ, where the
initial knowledge state is normalized at K0 510 13
$. The maximal rate of R&D investment is set at
R̄ 5 0.05K0 $ per year and the rate of knowledge
depreciation is d 50.2% per year. The maximal
attainable knowledge state is thus K¯ 5 R¯ /d 5
25K0 . (By the current standards, these numbers
are enormous. We shall return to this point below.)
387
The discount rate is set at 5% per year. The pollution cost parameter is taken as w51.8310 24 $ / MJ
per year and the natural cleansing rate for the
pollution process is r 51% per year, yielding
w /( r 1 r) 5 0.003 $ / MJ for the fixed shadow
price of pollution.
With these specifications we find that Kˆ /K0 5
5.49 (Eq. (13)) and K S /K0 5 23.04 (Eq. (14)).
S
Thus Kˆ , K , precluding the possibility that the
fossil fuel stock is not depleted at equilibrium.
The optimal knowledge process, K t* , approaches
ˆ Whether the steady state is
the steady state K.
approached as a standard MRAP (along K mt of Eq.
(10)) or as a NSMRAP (Eq. (12)) depends on
whether the initial fuel stock X0 falls short or
exceeds the benchmark quantity Q m of Eq. (18).
To determine the latter, we find from Eq. (17) that
lˆ 5 0.0105 $ / MJ and Tˆ 5 103.5 years. Thus,
straightforward integration yields the value Q m 5
84.44 3 10 20 J.
In order to illustrate the two types of solutions,
we obtain the optimal policy for X0 570310 20 J
(Figs. 2–4) and for X0 5100310 20 J (Figs. 5–7).
Figs. 2 and 5 compare the root process and the
optimal process in each case. Figs. 3 and 6 depict
the corresponding shadow price processes and
Figs. 4 and 7 display the fossil and solar energy
supply rates. Both cases involve the determination
Fig. 2. The normalized optimal knowledge (solid line) and root (dotted line) processes when the initial stock (X0 570310 20 J)
m
ˆ
falls short of the benchmark quantity Q . The optimal knowledge process is a standard MRAP to the steady state K.
388
Y. Tsur and A. Zemel
Fig. 3. The optimal fossil reserves shadow price process when the initial stock (X0 570310 20 J) falls short of the benchmark
m
quantity Q . The process increases until the depletion date T *, then decreases back to the steady state level lˆ .
Fig. 4. The optimal fossil energy (solid line) and solar energy (dotted line) supply rates when the initial stock (X0 570310 20 J)
falls short of the benchmark quantity Q m . The fossil energy supply rate vanishes at the depletion date T *, while the solar energy
ˆ
supply rate increases until T.
Long-term perspective on the development of solar energy
389
Fig. 5. The normalized optimal knowledge (solid line) and root (dotted line) processes when the initial stock (X0 5100310 20 J)
is above the benchmark quantity Q m . In this NSMRAP solution, the optimal knowledge process follows K tm until the date t, then
ˆ
it switches to the root process until the depletion date T *, at which time both processes settle at the steady state value K.
Fig. 6. The optimal shadow price (scarcity rent) process under a NSMRAP policy — when the initial stock (X0 5100310 20 J) is
m
above the benchmark quantity Q . The process rises until the depletion date T *, then settles at the steady state value lˆ .
390
Y. Tsur and A. Zemel
Fig. 7. The optimal fossil energy (solid line) and solar energy (dotted line) supply rates under a NSMRAP policy — when the
initial stock (X0 5100310 20 J) is above the benchmark quantity Q m . Note the kinks at the crossing date t. Both rates enter their
corresponding steady states at the depletion date T *.
of the initial scarcity price l 0* and the depletion
date T * via the simultaneous solution of the
continuity Eqs. (9a) and (9b). The NSMRAP
corresponding to the higher initial stock also
requires the simultaneous determination of the
switching date t.
The initial stock X0 570310 20 J falls short of
m
Q and the high scarcity price leads to a high root
process which is not crossed by the MRAP K mt
ˆ hence a
prior to arrival at the steady state K,
switch to the root process never occurs (Fig. 2).
One sees that depletion of the fossil reserves
occurs at T * 5 81.2 years — before the optimal
ˆ but after the
process enters its steady state at T,
date at which the root process settles at the steady
state. The latter observation clearly follows from
the fact that the exponential branch of the fossil
stock shadow price lt (Fig. 3) overshoots its
steady state level lˆ prior to the date T * and then
decreases back towards the steady state level until
ˆ Although the shadow price loses its
the date T.
active role following depletion (because fossil
energy cannot be supplied at any cost), lt retains
its meaning as the value gained by society if an
additional unit of fossil fuel suddenly becomes
available. The decrease in lt indicates a decline in
this value as incoming knowledge further reduces
the cost of the solar alternative.
Fig. 4 depicts the corresponding fossil and solar
power supplies: both energy sources are used
simultaneously during a significant part of the
planning period. However, the fossil power supply vanishes at the depletion date T * in accord
with Policy Rule 1. At the same time, solar power
supply continues to increase slowly due to the
ongoing knowledge accumulation until the
ˆ The increase in the total
equilibrium date T.
power supply (relative to the initial supply) is
noticeable, and is explained in terms of the cost
reduction of solar energy due to the R&D activities.
When the initial stock X0 5100310 20 J exceeds Q m , the fossil energy shadow price is
smaller, which in turn implies a lower root
process that is crossed by the standard MRAP K mt
at t 582.4 years. From that date on, the optimal
knowledge process switches to the root process
and cruises along with it until it reaches the
steady state Kˆ at the depletion date T * 5 129.8
years. The result is the NSMRAP depicted in Fig.
5. The shadow price in this case increases exponentially until it reaches its steady state lˆ at T *
Long-term perspective on the development of solar energy
(Fig. 6). In view of the larger initial stock, the
initial shadow price l 0* 5 1.6 3 10 25 $ / MJ is
about a factor of 13 smaller than the value l 0* 5
25
20
21.3 3 10 $ / MJcorrespondingtoX0 570310 J,
demonstrating the sensitivity of the shadow price
to changes in the initial stock. Finally, Fig. 7
depicts the corresponding trajectories of fossil and
solar power supplies. The kinks at the switching
date t and at the depletion date T * are noticeable.
A comment on the upper bound set on the
R&D expenditures is in order. The values of K0
and R¯ specified in these examples are orders of
magnitudes higher than present investments on
the development of alternative energy technologies. Evidently, had we used more ‘realistic’
values for these parameters, the maximal attainable knowledge state K¯ would fall short of the
root process and the optimal policy would be a
standard MRAP to this limiting state. Both Policy
Rules would remain valid. However, current R&D
investment rates are not necessarily the correct
scale to measure optimal R&D policies. In fact,
estimates of the carbon tax levels required to
mitigate global warming threats amount to $200
billion per year in the United States alone (Nordhaus, 1993; Poterba, 1993; Chakravorty et al.,
1997). The question who will collect and control
these enormous revenues cannot be addressed
here, but it is clear that they cannot be removed
out of the energy sector without serious consequences. Thus, an important fraction of the tax
collected will have to be devoted to the development of alternative energy sources so that the tax
can meet its goals. The appropriate investment
rate should therefore be measured by this scale.
5. CONCLUDING COMMENTS
The diminishing reserves of fossil fuel and the
pollution caused by its combustion have long
been used as arguments for the promotion of the
development of solar technologies. The present
work employs the methods of dynamic optimization to present these arguments in a systematic
manner within a comprehensive intertemporal
framework. The shadow prices obtained by this
analysis offer a precise meaning to the notion of
externalities that must be included in the social
cost of fossil energy. Indeed, environmental and
economic processes dictate that alternative energy
sources will eventually capture an increasing
share of energy supply. It is up to us, however, to
decide whether this occurs abruptly and painfully
or whether substantial and persistent R&D programs prepare the solar industry well in advance
391
towards its future role and ensure a gradual and
smooth transition. Although our model is grossly
oversimplified, the analysis presented here lends
support to the latter view.
Two properties of the optimal energy policy are
particularly relevant in this context. Firstly, the
postulated gradual cost reduction of solar energy
as knowledge accumulates and the rate-dependent
marginal cost of fossil energy production entail a
smooth transition from conventional to solar
technologies, which are simultaneously employed
during a significant period within the planning
horizon. Indeed, the persistent reductions in the
cost of photovoltaic cells accompanied by increasing cell efficiency, as well as substantial, though
gradual, advances in solar thermal technologies,
lend credence to this description.
Secondly, the model advocates substantial early
engagement in solar R&D programs that should
precede, rather than follow, future increases in the
price of fossil fuels. Extending the model to
non-stationary demand suggests that these policy
rules hold in more general and realistic situations.
NOMENCLATURE
cumulative cost of fossil energy, $ year 21
demand function, J year 21
inverse demand function, $ J 21
consumer surplus, $ year 21
(monetary value of) knowledge state, $
MRAP process (Eq. (10)), $
steady state and critical knowledge levels (Eqs.
(13) and (14)), $
K( l)
root process (defined by the root of L(K, l) 5 0), $
L(K, l)
utility function defining the root process (Eq. (11)),
% year 21
Mc , Ms
marginal costs of fossil and solar energy, $ J 21
P
cumulative pollution level, J
q c0, q s m
fossil and solar power supplies, J year 21
Q ,Q
reference fuel stock levels (Eqs.
(15) and (18)), J
r
social discount rate, % year 21
R
investment rate in solar R&D, $ year 21
T
depletion date, year
Tˆ
date when the MRAP process reaches Kˆ (Eq.
(17b)), year
V
social value of the energy policy (Eq. (5)), $
w
unit cost of pollution damage, $ J 21 year 21
X
fossil fuel stock, J
Greek symbols
d
knowledge depreciation rate, % year 21
l
shadow price of fossil fuel stock, $ J 21
lˆ
steady state shadow price (Eq. (17a)), $ J 21
r
natural decay rate of pollution, % year 21
t
switching date from the MRAP to the root process,
year
*]
indicates optimal quantities
indicates an upper bound
C
D
D 21
G
K
K mt
S
K̂, K
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