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characterize the optimal time profiles of fossil and solar energy supply rates and the ... of the fossil fuel reserves, while the optimal supply of solar energy should.



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


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


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


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


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 .


2.3. 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


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


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 .


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


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


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 ).



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 )).


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 )





w dt 5 ] r

3E 0


c 2rt t



dt 1 P0 2 r Pt e 2rt dt 0



Y. Tsur and A. Zemel

hence `

EwP e




w dt 5 ]] r1r




q tc e 2rt dt 1 P0 .


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 )


and T*

E q (K *, l *e )dt 5 X , c







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


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 ).


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 )),


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: `





Q 5 q (K t ,0)dt.



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.



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


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,



lˆ 5 MS (Kˆ ) 2 MS (K S )



3.3. Increasing energy demand

Tˆ 5 log[(K¯ 2 K0 ) /(K¯ 2 Kˆ )] /d.


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 `


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.


Q 5 q c (K mt , l mt )dt



of fossil stock needed to carry out the conventional energy supply policy with K mt and

l mt 5 l m0 e rt


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.


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 R¯ 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.)


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.


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


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 .


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


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 ˆ KS K,

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Y. Tsur and A. Zemel

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