**Optimizing the Level of Renewable Electric R&D Expenditures Using Real
Options Analysis**[^1]

Graham A. Davis[^2]

Colorado School of Mines, Golden, CO 80401

Brandon Owens[^3]

National Renewable Energy Laboratory

Golden, CO 80401

December 18, 2001

# Abstract

One of the primary objectives of the United States\' federal non-hydro
renewable electric R&D program is to promote the development of
technologies that have the potential to provide consumers with stable
and secure energy supplies. In order to quantify the benefits provided
by continued federal renewable electric R&D, this paper uses real option
pricing techniques to estimate the value of renewable electric
technologies in the face of uncertain fossil fuel-prices. Within the
real options analysis framework, the current value of likely future
supply from renewable electric technologies, net of federal R&D
expenditures, is estimated to be \$30.6 billion. Of this value, 86% can
be attributed to past federal R&D efforts, and 14% can be attributed to
future federal R&D efforts, assuming that federal R&D funding remains at
its current level. In addition, real options analysis shows that the
value of renewable electric technologies increases as current and future
R&D funding levels increase. This indicates that the current level of
federal renewable electric R&D funding is sub-optimally low.

## Introduction

The purpose of this paper is to quantify the value of the United
States\' federal non-hydro renewable electric R&D program in order to
determine if these research efforts represent a sound national
investment. Should we find that these activities hold positive economic
value, we will recommend the optimal level of annual investment in such
activities. With this information, the U.S. Department of Energy will be
able to clearly assess whether or not its renewable electric R&D
activities are economically warranted; and, if warranted, to show why it
is a good idea to continue to invest millions of dollars annually in
these activities. Furthermore, through the use of an Internet-based
valuation model, the Department of Energy (DOE) will be equipped with
the ability to determine the appropriate level of annual renewable
electric R&D funding as new market information unfolds.

In order to quantify the value of renewable electric, we first examine
these technologies from the traditional discounted cash flow
perspective, a valuation perspective that does not consider the
program\'s insurance value. We then examine renewable electric
technologies using a real options analysis framework. Real options
analysis draws upon insights from financial markets in order to value
all technology benefits including insurance value.

## Background

In 1973, several Arab nations, angered at U.S. support of Israel in the
1973 Arab-Israeli War, instituted an oil embargo against the U.S. and
Holland. The Arab oil embargo came at a time of declining domestic crude
petroleum production, rising demand, and increasing imports. The embargo
was also accompanied by decreased production from the Organization of
Petroleum Exporting Countries (OPEC). Because non-OPEC nations had
minimal excess production capacity, the embargo created petroleum
shortages and price increases. World crude petroleum prices more than
doubled in a six-month period (Figure 1) and annual U.S. consumer
petroleum expenditures nearly doubled, rising from \$56 billion to \$97
billion dollars (EIA 2000a). The federal government\'s renewable
electric (RE) R&D program was initiated in 1974 to promote the
development of technologies that have the potential to provide consumers
with stable and secure renewable electric supplies in a high fossil
fuel-price environment. Reducing U.S. vulnerability to energy supply
disruptions is still one of the primary missions of the RE R&D program,
which is managed by the Department of Energy\'s (DOE) Office of Energy
Efficiency and Renewable Energy (EERE).

![](media/image1.png){width="5.999305555555556in"
height="4.100694444444445in"}Between fiscal years (FY) 1974 and FY 2000,
a total of nearly \$13.5 billion (1992) dollars have been spent on the
federal RE R&D program (Figure 2). During this time, approximately 90%
of program dollars have been distributed among six key RE technologies:
concentrating solar power (CSP), photovoltaics (PV), geothermal energy,
electric storage systems, wind, and bioenergy (Figure 3).

![](media/image2.png){width="5.847222222222222in"
height="3.9965277777777777in"}

![](media/image3.png){width="5.999305555555556in"
height="4.100694444444445in"}In the 27 years since its conception, the
federal RE R&D program has achieved numerous technical successes. For
example, in 1980 wind energy costs were about 45 cents per kilowatthour
(kWh) (McVeigh et al. 2000). Today, large, utility-scale wind machines
are being installed that can produce electricity for 4 to 6 cents/kWh,
depending on the wind resource and the financial and ownership
structure. This cost is actually lower than the projections made by
technology specialists in 1980. Photovoltaic (PV) energy is another
technological success story. Thin-film PV modules have recently achieved
conversion efficiencies of greater than 12% (EERE 2000). Conversion
efficiency improvements have helped reduce the cost of electricity from
PV systems from more than \$1/kWh in 1980 to just more than 20 cents/kWh
today (McVeigh et al. 2000). In fact, the costs of *all* RE technologies
under development have decreased dramatically since the federal R&D
program\'s inception, and recent studies project continuing declines in
costs in the coming decades due to continued technical research (PCAST
1997).

RE technologies have also experienced a measure of market success. For
example, between 1990 and 1998, nonutility geothermal energy consumption
increased by more than 40% in the United States (EIA 2000c). Wind
technologies have also experienced impressive market growth. Between
1990 and 1998, U.S. wind energy generation increased tenfold, from
300,000 kWh/yr. to nearly 3 million kWh/yr. (EIA 2000c). Still, in the
context of total U.S. energy consumption, RE technologies have yet to
emerge as dominant players. In 1999, nonhydro RE technologies accounted
for only 4% of total U.S. energy consumption (EIA 2000a) (Figure 4).

An examination of energy market conditions reveals that the lack of
widespread adoption of RE technologies has more to do with changes
outside the R&D program than with the performance of RE technologies
(McVeigh et al. 2000). An increasingly competitive world petroleum
market has led to a decline and stabilization in the price of petroleum
to such an extent that by 1998 the real price of petroleum was the
lowest since prior to the 1973 oil embargo (Figure 5). Changes in the
supply of natural gas during the past 27 years have led to natural gas
price declines that are

![](media/image4.png){width="5.999305555555556in"
height="4.100694444444445in"}![](media/image5.png){width="5.999305555555556in"
height="4.100694444444445in"}

almost as impressive as the decreases in petroleum prices. In the
mid-1970s, when the federal RE R&D program was initiated, natural gas
was thought to be a diminishing resource.[^4] However, between 1975 and
2000 the picture was decidedly different. The regulations of the 1970s
were replaced by a restructured and highly competitive gas market, and
natural gas is now being counted on to fuel much of America\'s economic
expansion well into the 21^st^ century (for example, see NPC (1999)).
Advances in technology led to new discoveries, and new production tools
led to more gas being extracted from both existing and newly discovered
fields. The net result of all of this is that, until quite recently, the
cost of generation from conventional fossil fuel sources declined in
tandem with, although at a less rapid rate than, the cost of electricity
generation from renewable electric technologies.

By the 1990s, the limited amount of U.S. RE consumption led some
analysts to question the value of federal RE R&D investments (for
example, see Taylor 1999; Management Information Services 1998; Bradley
1997; Cohen and Noll 1991; and Ball and Tabors 1990). Skeptics began
asking why the United States is continuing to invest millions of dollars
annually in high-cost renewable electric technologies in an era of
low-cost fossil fuels. This question reflects a basic lack of
understanding of the mission of the federal RE R&D program. One of the
primary program missions is to serve as an insurance policy that ensures
domestic energy security.[^5]^,^[^6] Nevertheless, these skeptical
examinations of the federal RE R&D program have raised some important
questions. Namely, what is the value of the program, and given this
value, if any, what is the appropriate level of national investment in
these activities?

The purpose of this paper is therefore to quantify the value of the RE
technologies under various R&D scenarios in order to determine if
ongoing federal RE R&D activities are likely to yield a substantial
return to the nation. If the federal RE R&D program is expected to yield
a favorable return, we will then recommend an optimal R&D investment
level. We start by examining the value of RE technologies from the
traditional discounted cash flow perspective, a perspective that does
not consider the program\'s insurance value. We then value RE
technologies using a real options analysis framework. Real options
analysis draws upon insights from financial markets in order to value
all technology benefits including insurance value. After we develop the
real options model and present the results, we use the model to
determine the optimal level of annual federal RE R&D expenditures. We
conclude the paper by discussing our results and suggesting areas for
future research.

## The Discounted Cash Flow Model

The most commonly used investment valuation framework is discounted cash
flow (DCF) analysis. Within this framework, future benefits in terms of
cash flows, are estimated, usually on an annual basis, and then these
cash flows are discounted at a risk-adjusted rate so that they are
expressed in present-value dollars. Initial investment costs are then
subtracted from the present value of future cash flows to yield the
investment\'s Net Present Value (NPV). Within this framework, the
investment decision rule is simple: if NPV \> 0, the investment is
economic and decision makers are advised to proceed; if NPV \< 0, the
investment is uneconomic and should be abandoned.

Thus, in order to determine the NPV of RE technologies within the DCF
framework, we need first to estimate future technology-generated cash
flows. This is a challenging task because it requires the development of
an energy market model and assumptions about the rate of RE technology
adoption. Although difficult, we believe it is important to undertake
this exercise here because of the valuable insights that can be gained
through the economic modeling process. However, as we proceed, it is
important to remember that the usefulness of this exercise lies more in
the insights provided by the model than in the specific numerical values
obtained.

In order to estimate future cash flows generated by RE technologies, we
must first create a simplified model of the U.S. electricity market.[^7]
Positive cash flows, in the form of consumer cost savings, will arise
when RE technologies are installed and provide electricity to consumers
at a cost lower than that of traditional nonrenewable electric (NRE)
technologies. A combination of RE R&D success and fossil fuel-price
increases will create an environment in which RE technologies are
adopted in the marketplace and become the lowest cost suppliers of
electricity.

The consumer cost savings generated by RE technologies in any given year
can be calculated as the difference between supplying incremental demand
at the expected price of NRE electricity and that of supplying that
market segment with the best available RE electricity generation
technology (Figure 6).[^8] In this analysis, we abstract from the daily
and seasonal trends in wholesale power prices and use the
levelized-cost-of-electricity (LCOE) as a proxy for the expected retail
electricity price, assuming that consumers pay only the marginal cost of
generation from new electricity sources.[^9] LCOE is the average cost of
production per kWh over the technology investment life.[^10] In Figure
6, we denote RE electricity with the subscript *R*, and NRE electricity
with the subscript *F* (for fossil fuel). In Figure 6, *S~F~* and *S~R~*
are the electricity supply functions for NRE and RE technologies, *D~R~*
is the annual incremental U.S. electricity demand that could be
fulfilled by RE technologies, and *D~T~* is the total annual incremental
U.S. electricity demand. To simplify the analysis, we assume infinitely
elastic supply and infinitely inelastic RE demand. In this case, the
shaded area in Figure 6 shows the surplus in the current year from RE
technologies that are able to provide energy quantity *q* at cost *P~R~*
instead of at cost *P~F~*. The surplus is simply consumers\' cost
savings from replacing the higher cost NRE electricity with the lower
cost RE electricity.

![](media/image6.wmf){width="5.433333333333334in" height="4.2625in"}

Taking the current time period as zero, the NPV of the future cash flows
created by RE technologies, if they were to be installed today and
annually hereafter at rate *q*,[^11] is the difference between the
expected cost of meeting current and future incremental electricity
demand using the most cost-competitive RE technology and the expected
cost of meeting current and future incremental demand using the most
cost-competitive NRE technology:

$E\left\lbrack V(P_{F_{0}},P_{R_{0}}) \right\rbrack = \int_{0}^{\infty}{P_{F_{0}}eq_{t}}dt - \int_{0}^{\infty}{P_{R_{0}}eq_{t}}dt$
(1)

In this equation, $\hat{\alpha}$ and $\hat{g}$ are risk-adjusted
discount rates, *α* is the expected rate of change of the cost of NRE
electricity, *γ* is the expected rate of change of the cost of RE
electricity after switchover, and *q~t~* is the incremental electricity
demand at time *t* met by RE technologies.[^12]^,^[^13] The future costs
of NRE electricity, *P~F~*, and RE electricity, *P~R~*, are
uncertain.[^14]

For simplicity, we assume incremental electricity demand to be constant
at rate *q* in all future periods, and rising linearly at rate *q*
through time. Then, given deployment time zero, the initial demand for
RE will be *q*, and *q~t~* = *q* + *qt* for all subsequent periods *t*.
If we assume that $\alpha < \hat{\alpha}$ and $g < \hat{g}$ to avoid
infinite present values or division by zero, then Equation 1 simplifies
to

$E\left\lbrack V(P_{F_{0}},P_{R_{0}}) \right\rbrack = \frac{P_{F_{0}}}{(\hat{\alpha} - \alpha)}q\left\lbrack 1 + \frac{1}{(\hat{\alpha} - \alpha)} \right\rbrack - \frac{P_{R_{0}}}{(\hat{\gamma} - \gamma)}q\left\lbrack 1 + \frac{1}{(\hat{\gamma} - \gamma)} \right\rbrack$.
(2)

This value may be positive or negative, depending on the current costs
of RE and NRE electricity, their expected rates of change over time, and
their relative discount rates. A negative value indicates that
immediately deploying RE technologies to meet incremental electricity
demand will lower consumers' expected aggregate present wealth.

There are certain real-world complications that we now introduce into
this valuation model. It is often hypothesized that the transition from
a NRE-only electricity market to a mixed NRE-RE electricity environment
will involve market conditioning and other infrastructure expenditures
that take time to complete. These "switching" expenditures are expected
to be necessary in order to eliminate technical, institutional, and
market barriers (OTA 1995), and to overcome technological lock-in
(Arthur 1989) enjoyed by NRE technologies. In this model, we define
total switching costs remaining at time *t* as *K~t~*.[^15] Initial
switching costs are *K~0~*, which we will denote as *K*. Annual
switching expenditures are made at up to maximum rate *I~max~* (\$/yr.).
We take the rate of switching expenditures, *I*, to be a linear function
of the cost of NRE electricity: *I*= *iP~F~*.[^16] We assume these to be
irreversible expenditures: if RE technologies do not take off, ∑*I*
cannot be recovered, and so *i* ≥ 0. We take *i* to be the control
variable, and the expected time to deployment at time *t* for *i* \> 0
is
$T_{t}(k_{t}) = \frac{\text{ln}(1 + \frac{{\alpha k}_{t}}{i})}{\alpha}$
years, where *k~t~* = *K~t~/P~Ft~*. We assume that there is a maximum
rate of switching expenditure, such that 0 ≤ *i* ≤ *i~max~* \< ∞. This
means that RE technologies cannot be instantaneously deployed, with the
minimum expected time to initial deployment
$T_{t_{\text{min}}}(k_{t}) = \frac{\text{ln}(1 + \frac{{\alpha k}_{t}}{i_{\text{max}}})}{\alpha}$
years.[^17] No RE supply is possible until the entire switching cost *K*
is spent, after which *q* units of RE become available for immediate
installation.

We also assume that prior to the completion of switchover, the cost of
RE electricity changes at the rate *υ* due to continued R&D. DOE
currently manages a portfolio of RE R&D projects, therefore, *υ*
represents the rate of change associated with the entire RE R&D
portfolio. The portfolio rate of change is expected to be greater than
the rate associated with an individual RE technology because there are
considerable advantages to optimally managing interrelated R&D
projects.[^18] For simplicity, we also assume that once the switching
period is complete, and the electricity market is prepared to adopt RE
technologies, federal RE R&D program funding will be curtailed.[^19]
Thus, after switchover, the rate of change, $\gamma$, is a function of
market, rather than R&D, activities.

We also must charge against the program annual R&D expenditures, *M*
(\$/yr.) during the switching period.[^20] We assume that *M* is the
linear function *M* = *mP~F~*, where *m* is a given positive
constant.[^21] We will later allow for the optimization of *M*.

Taking these switching cost and switching time complexities into
account, the NPV of the RE technologies, if switchover were to begin
today at the maximum rate of expenditure, *I~max~*, is

$$\begin{array}{r}
E\left\lbrack F(P_{F_{0}},P_{R_{0}},K) \right\rbrack = \frac{P_{F_{0}}}{(\hat{\alpha} - \alpha)}e^{(\alpha - \hat{\alpha})T_{0\text{min}}(k_{0})}q\left\lbrack 1 + \frac{1}{(\hat{\alpha} - \alpha)} \right\rbrack \\
\text{                   } - \frac{P_{R_{0}}}{(\hat{\gamma} - \gamma)}e^{(\upsilon - \hat{\gamma})T_{0\text{min}}(k_{0})}q\left\lbrack 1 + \frac{1}{(\hat{\gamma} - \gamma)} \right\rbrack - \int_{0}^{T_{0\text{min}}(k_{0})}{P_{F_{0}}(i_{\text{max}} + m)e^{- \hat{\alpha}t}}dt \\
\text{                         } = \frac{P_{F_{0}}}{(\hat{\alpha} - \alpha)}e^{(\alpha - \hat{\alpha})T_{0\text{min}}(k_{0})}q\left\lbrack 1 + \frac{1}{(\hat{\alpha} - \alpha)} \right\rbrack \\
\text{                    } - \frac{P_{R_{0}}}{(\hat{\gamma} - \gamma)}e^{(\upsilon - \hat{\gamma})T_{0\text{min}}(k_{0})}q\left\lbrack 1 + \frac{1}{(\hat{\gamma} - \gamma)} \right\rbrack\text{   } - \frac{P_{F_{0}}(i_{\text{max}} + m)}{(\hat{\alpha} - \alpha)}\left\lbrack 1 - e^{(\alpha - \hat{\alpha})T_{0\text{min}}(k_{0})} \right\rbrack
\end{array}$$

\(3\)

Under this framework, the only choices are to initiate switching period
now or never. The optimal investment rule is to immediately proceed with
deploying RE technologies if *F* \> 0. If *F* \< 0, it is optimal to
instead abandon the federal RE R&D program because RE technologies, net
of R&D expenditures, are uneconomic.

Since the NPV function above is linear homogeneous in costs and *K*, it
is convenient to express these valuation functions using the cost of NRE
electricity as the numeraire. That is,

$\begin{array}{r}
G(p_{0},k_{0}) = \frac{1}{(\hat{\alpha} - \alpha)}e^{(\alpha - \hat{\alpha})T_{0_{\text{min}}}(k_{0})}q\left\lbrack 1 + \frac{1}{(\hat{\alpha} - \alpha)} \right\rbrack \\
\text{                    } - \frac{p}{(\hat{\gamma} - \gamma)}e^{(\upsilon - \hat{\gamma})T_{0_{\text{min}}}(k_{0})}q\left\lbrack 1 + \frac{1}{(\hat{\gamma} - \gamma)} \right\rbrack\text{   } - \frac{i_{\text{max}} + m}{(\hat{\alpha} - \alpha)}\left\lbrack 1 - e^{(\alpha - \hat{\alpha})T_{0_{\text{min}}}(k_{0})} \right\rbrack
\end{array}$ (4)

where *p* = *P~R~*/*P~F~*, and *F*(•) = *P~F~G*(•). The value of *F* can
then, for a given level of *k*, be plotted as a linear function of the
ratio of electricity costs, *p*, at the time of investment.

**Discounted Cash Flow Parameter Values**

As we indicated above, we believe it is important to estimate the DCF
value of RE technologies because of the insights that can be gained
through the economic modeling process. As such, the usefulness of this
exercise lies in the insights provided by the models, more than in the
specific numerical values. This is particularly important to remember in
the context of estimating model parameters. Many of the parameters
required for this analysis are complex enough to warrant their own
independent research effort. As such, the goal here is to develop
simplified parameter estimates that can be used to glean insights from
the valuation model. At the end of this paper, interested readers are
referred to an Internet-based application to run the models presented
here.

One of the most contentious issues in calculating any net present value
of a risky cash flow stream is the selection of appropriate discount
rates $\hat{\alpha}$ and $\hat{g}$. Here, the *relative* level of the
two discount rates also affects the value of RE technologies. For
example, if NRE generation is less risky (in a systematic sense) than RE
generation, the discount rate for NRE technologies will be lower. This
increases the present value of these costs---risk in this case is
valuable to the consumer, since costs provides a hedge over deviations
in wealth, with costs being high when income is high, and low when
income is low---and provides an incentive to switch to the riskier RE
technologies (Cadogan et al. (1993) raise such risk-reduction issues).
Awerbuch and Deehan (1995) find that fossil fuel-prices may indeed be
negatively correlated with the market, making them less risky than
renewables, whose cost uncertainty we assume to be technical, and
therefore unsystematic. However, they find that the Beta of fossil
fuel-generated electricity is approximately zero. Bessembinder and
Lemmon (1999) speculate that wholesale electric power prices have no
systematic risk, which is in agreement with Awerbuch and Deehan, and we
have found that near-term natural gas futures prices during the past six
years show no significant trend and no correlation with the market.[^22]
From this we assume that the appropriate discount rate for fossil energy
is the risk-free rate. Since the uncertainties in the cost of supplying
renewable electric are technical, and therefore unlikely to be
systematic, we also assume a risk-free discount rate when discounting
these costs.

The values of the parameters in the DCF model, Equation 4, are presented
in Table 1. All values are expressed in nominal terms. Because wind
technologies are currently the most cost-competitive RE technologies in
the bulk electric-power market, the parameter values in Table 1 were
derived using wind energy as the representative RE technology. We
recognize the existence of niche markets in which other RE technologies
currently hold a competitive advantage. PV technologies, for example,
often have an advantage in providing electricity in off-grid, or
distributed energy, applications. We do not examine these niche markets
here. We expect that the cost-competitiveness of all of the technologies
within the federal RE R&D portfolio will continue to improve over time,
perhaps even to the extent that wind energy may not be the most
cost-competitive RE technology during the analysis period. We therefore
allow for the possibility that another RE technology may become the
representative technology within the forecast horizon, and this
possibility is reflected in the parameter values of *υ* and $\gamma$.

+-----------------+-------+---------------+----------------------------+
| **Table 1: DCF  |       |               |                            |
| Model Parameter |       |               |                            |
| Values**        |       |               |                            |
+=================+=======+===============+============================+
| **Parameter**   | **    | **Value**     | **Notes**                  |
|                 | Model |               |                            |
|                 | Sym   |               |                            |
|                 | bol** |               |                            |
+-----------------+-------+---------------+----------------------------+
| Current         | *P    | 4.5 cents/kWh | The approximate current    |
| renewable       | ~R0~* |               | cost of class 4 wind       |
| electricity     |       |               | energy in 2000, according  |
| cost            |       |               | to the Electric Power      |
|                 |       |               | Research Institute\'s      |
|                 |       |               | renewable electric         |
|                 |       |               | Technology                 |
|                 |       |               | Characterizations (RETC)   |
|                 |       |               | (EPRI 1997). The actual    |
|                 |       |               | cost of generation from    |
|                 |       |               | both renewable and         |
|                 |       |               | nonrenewable systems will  |
|                 |       |               | vary depending upon        |
|                 |       |               | resources and financial    |
|                 |       |               | and ownership structure.   |
+-----------------+-------+---------------+----------------------------+
| Renewable       | *υ,   | -1%/yr.       | *υ* is an estimate of the  |
| electricity     | γ*    | (assuming an  | marginal productivity of   |
| annual rate of  |       | annual \$300  | RE R&D efforts. This       |
| cost reduction  |       | million R&D   | estimate is consistent     |
| (2000-2020)     |       | budget)       | with the technical goals   |
|                 |       | before        | outlined in the RETC, and  |
|                 |       | i             | the report of the          |
|                 |       | mplementation | President\'s Committee on  |
|                 |       | (*υ*), 0%/yr. | Science and Technology     |
|                 |       | after         | (PCAST 1997), both of      |
|                 |       | i             | which indicate that        |
|                 |       | mplementation | current R&D funding levels |
|                 |       | (*γ*)         | must be roughly doubled to |
|                 |       | (assumes      | achieve program            |
|                 |       | supply        | objectives. The estimate   |
|                 |       | constraints   | of *γ* assumes that market |
|                 |       | offset        | learning effects and       |
|                 |       | technological | international R&D efforts  |
|                 |       | i             | cause the cost of RE to    |
|                 |       | mprovements). | remain flat, with          |
|                 |       |               | technological gains just   |
|                 |       |               | offsetting inflation in    |
|                 |       |               | costs.                     |
+-----------------+-------+---------------+----------------------------+
| Renewable       |       | 7%            | The risk-free discount     |
| discount rate   |       |               | rate.                      |
+-----------------+-------+---------------+----------------------------+
| Current         | *P    | 3.5 cents/kWh | The estimated marginal     |
| fossil-fuel     | ~F0~* |               | cost of electricity        |
| electricity     |       |               | generation from a natural  |
| cost            |       |               | gas advanced               |
|                 |       |               | combined-cycle generator   |
|                 |       |               | in 2005 (EIA 1999, p. 67)  |
+-----------------+-------+---------------+----------------------------+
| Fossil-fuel     | *α*   | 0.3%/yr.      | This is the rate of        |
| electricity     |       |               | marginal generation cost   |
| annual rate of  |       |               | increase for natural gas   |
| cost increase   |       |               | advanced combined-cycle    |
| (2000-2020)     |       |               | generation. This rate      |
|                 |       |               | assumes mid-range natural  |
|                 |       |               | gas prices (EIA 1999, p.   |
|                 |       |               | 67).                       |
+-----------------+-------+---------------+----------------------------+
| Fossil-fuel     |       | 7%            | The risk-free discount     |
| discount rate   |       |               | rate.                      |
+-----------------+-------+---------------+----------------------------+
| Annual          | *q*   | 37.3 billion  | This value assumes that    |
| incremental     |       | kWh/yr.       | 50% of annual incremental  |
| electricity     |       |               | U.S. demand will be met by |
| demand met by   |       |               | RE technologies. 37.3      |
| renewables      |       |               | billion kWh/yr. is         |
|                 |       |               | approximately equal to 50% |
|                 |       |               | of expected annual         |
|                 |       |               | incremental demand between |
|                 |       |               | 2000 and 2020. This value  |
|                 |       |               | is reasonable given that   |
|                 |       |               | total U.S. wind resource   |
|                 |       |               | potential (for class 4 and |
|                 |       |               | greater) is estimated to   |
|                 |       |               | be more than 1.5 trillion  |
|                 |       |               | kWh/yr. (Elliot and        |
|                 |       |               | Schwartz 1993).            |
+-----------------+-------+---------------+----------------------------+
| Total switching | # *K* | \$10 million  | A rough estimate of the    |
| cost            |       |               | cost of legislation and    |
|                 |       |               | standards that may be      |
|                 |       |               | required to facilitate the |
|                 |       |               | switchover to renewable    |
|                 |       |               | electric technologies.     |
|                 |       |               | This value is small        |
|                 |       |               | relative to the level of   |
|                 |       |               | RE implementation since it |
|                 |       |               | is assumed that markets    |
|                 |       |               | will operate efficiently,  |
|                 |       |               | thus minimizing switching  |
|                 |       |               | expenditures.              |
+-----------------+-------+---------------+----------------------------+
| Annual maximum  | # *I~ | 1/3 total     | A switchover time          |
| switching       | max~* | switching     | (*T~0min~*) of             |
| expenditure     |       | costs         | approximately 3 years is   |
|                 |       |               | used.                      |
+-----------------+-------+---------------+----------------------------+
| Annual R&D      | *     | \$0.3 billion | The approximate federal FY |
| funding         | M~0~* |               | 2000 RE R&D funding level. |
+-----------------+-------+---------------+----------------------------+

## 

## Discounted Cash Flow Model Results

Using Equation 4 and the parameter values presented in Table 1, the
value of RE technologies is estimated to be negative \$35.3 billion
given the current RE/NRE cost ratio of approximately 1.29 (4.5/3.5). For
these parameter values, the DCF model indicates that the RE technologies
are uneconomic and therefore the RE R&D program should be abandoned if
the current cost of electricity for the most competitive RE technology
is more than 13% higher than the current NRE cost of electricity of 3.5
cents/kWh. This general result seems to buttress the arguments of RE
opponents who claim that the national RE R&D program should be
abandoned.

![](media/image7.png){width="5.996527777777778in"
height="4.100694444444445in"}The NPV of RE technologies, as a function
of the current ratio of the costs of RE and NRE electricity, is
presented in Figure 7. This figure provides insights into the perceived
value of RE technologies given changes in the cost of RE and the cost of
NRE. Decreases in the cost of RE electricity due to the technical
success of the RE R&D program, or increases in the cost of NRE
electricity due to fossil fuel-price increases, appear as rightward
movements along the horizontal axis. Decreases in the cost of NRE
electricity due to fossil price decreases appear as a leftward movement
along the horizontal axis. This shows how reductions in the price of
fossil fuels decrease the value of RE technologies and may create the
perception, on an NPV basis, that continued R&D funding is unwarranted.
As we noted above, the NPV becomes positive for cost ratios of 1.13 or
lower.

**Discounted Cash Flow Shortcomings**

Using the parameters from Table 1, the DCF model indicates that the RE
R&D program should be abandoned. This may be disconcerting to RE
technology advocates who take exception with the valuation of RE
technologies from a narrowly defined market-based perspective. This is a
legitimate concern, particularly since there is strong evidence that the
potential social, environmental, and auxiliary economic benefits of RE
technologies are greater than the market-based benefits. The central
purpose here, however, is to determine the value of RE technologies from
a narrowly defined economic perspective without introducing benefits
that are external to the electricity market. We prefer to preach to the
entire congregation of energy analysts, as it were, rather than to the
choir of RE sympathizers. Note that, if indeed RE technologies provide a
favorable return on a market basis, then the need to introduce
environmental and auxiliary economics benefits is diminished to the
extent that these benefits will only serve to make an already favorable
investment even more attractive.

Recall, however, that one of the primary missions of the federal RE R&D
program is to \"reduce U.S. vulnerability to energy supply disruptions\"
(EERE 2000). The DCF model assumes that RE market conditioning begins
immediately, even though RE electricity costs are greater than NRE
electricity costs. During the three-year switching period, RE
electricity costs are assumed to decline by 1% per year due to ongoing
federal R&D efforts, while NRE electricity costs are assumed to climb by
0.3% per year. After installation is complete and federal R&D efforts
are curtailed, RE electricity costs are assumed to remain constant.
Given these parameter assumptions, RE technologies will eventually pay
off in terms of lower electricity supply costs, but those benefits do
not begin for 74 years. With the benefits of RE so far off, the present
value of installing RE technologies now is negative. In fact, given
these assumptions, the optimal policy is to delay deploying RE and
continue R&D funding until RE technologies become more competitive,
which will take about 25 years according to the DCF model. In fact,
given these parameter assumptions, the NPV of RE technologies is
maximized if deployment is delayed for 22.7 years while ongoing R&D
efforts reduce technology costs. The expected NPV of the RE technologies
under this "commit in 22.7 years" deployment plan then rises to \$4.8
billion, net of the present value of ongoing R&D expenditures.

But even by assuming optimal deployment timing, we still do not address
the full value of RE in mitigating the impact of fossil fuel-price
increases. What are the chances that a fossil fuel supply disruption
will occur during this timeframe and significantly increase the cost of
NRE electricity? What is the value of having RE systems available as
\"backstop\" technologies in the event of severe fossil fuel-price
increases? Unfortunately, these issues are not addressed within the DCF
model. The DCF framework uses only mean values and disregards the
potential for decision makers to create value by reacting to
uncertainty. This is one of the most significant shortcomings of DCF
analysis. This shortcoming means that the DCF framework is unable to
quantify the insurance value of RE technologies. Clearly, a more
insightful analytic framework is required to properly value these
technologies.

The DCF framework has several other significant limitations when
assessing technologies in which there is considerable technical and
financial uncertainty and where R&D is ongoing. First, since there is
still scope for improving the economics of technologies under
development, R&D continues -- and these ongoing R&D efforts must be
evaluated. The sequential nature of R&D has led economists to reject the
usual evaluation criteria, such as the NPV technique we used above, as a
method of valuing ongoing R&D programs (e.g., Roberts and Weitzman
1981). It is, rather, a dynamic programming problem. Second, given the
uncertainty of the outcomes of the R&D process, and the fact that the
option to install the technologies limits the downsides from the RE
technologies, there is considerable difficulty in assessing the risk --
and therefore the discount rate -- for the future benefit flows, were
the technologies under development to become economic. That is, we have
difficulty evaluating the eventual payoffs to the R&D effort using DCF.
Third, the switchover to new RE technologies takes time, and the amount
of time needed for the switchover is uncertain. This creates uncertainty
about the deployment time, or "time-to-build" as it is known in the real
options literature. DCF analysis can allow for non-instantaneous
investments, but has difficulty evaluating the value of an uncertain
time-to-build (Ott and Thompson 1996). In addition, since there is a
time to build, the riskiness of the project will change as the remaining
amount of investment, and hence the project leverage, decreases (Berk et
al. 1998). DCF analysis, which uses a constant discount rate for all
period cash flows, does not take this changing risk into account.
Finally, during technology switchover, the ratio of energy costs can
vary, suddenly making the project uneconomic. The option to temporarily
halt or even permanently abandon switchover midstream adds value. DCF
techniques cannot value this flexibility.

It is important to note here that the shortcomings described above can
be addressed by constructing a modified DCF model. For example, DCF
techniques combined with analysis tools such as decision trees, in which
probability distributions are assigned to uncertain variables and
utility functions are used to specify risk preferences over uncertain
outcomes, can be used to model the optionality of investments. We
believe, however, that the need to continually modify to the DCF
framework in order to correct its deficiencies is an indicator of the
need to adopt a new theory. In its basic form, the DCF framework simply
does not capture many of the key aspects of many investment problems
under uncertainty. We use the historical analogy of the astronomical
practice of adding epicycles to the geocentric model of planetary
motions. Eventually so many revisions are added that the model becomes
too convoluted to be tractable. From our perspective, transition from
the DCF to the real options framework is akin to the progression from
the geocentric to the heliocentric theory.

## The Real Options Approach

The real options approach, which uses the concepts embedded in financial
options to value non-financial investment opportunities under
uncertainty, is ideally suited to value RE technologies and determine
the benefits of continued federal R&D. In fact, economists already have
successfully applied real options valuation techniques to R&D
investments and have found that the real options framework substantially
clarifies the theory and practice of R&D decision making. [^23] The
seeds of real options theory were sown in 1973 when Myron Scholes,
Robert Merton, and the late Fischer Black made a Nobel prize-winning
breakthrough in how to price financial options. The Black-Scholes
formula transformed financial options trading and helped create a global
derivatives business. Stewart C. Meyers of the Massachusetts Institute
of Technology coined the term \"real options\" in 1984 to describe the
valuation of non-financial assets using options theory. Academics have
long recognized that real options can bring the discipline of financial
markets to bear on strategic investment decisions. However, due to its
complex nature, the real options approach has only recently made its way
out of academia and into the hands of decision makers.

In order to understand how one might use the real options approach to
evaluate the portfolio of available RE technologies as an option whose
value is affected by the rate of R&D expenditures, consider an option
(or a right) to irreversibly invest in a particular asset (the
underlying asset) at a specific price (the exercise price or investment
cost) at or prior to a predetermined date in the future (the expiration
date). This is the framework under which financial call options are
created and valued, and under which we can evaluate RE technologies.
Given that, among the technologies within the federal RE R&D portfolio,
a most cost-competitive RE technology exists at any moment, we have the
option at each instant to meet incremental energy demand with
RE-generated electricity. The expected payoff to meeting electricity
demand is the present value of any perpetual cost savings from
installing the most cost-competitive RE technology. These savings are
equivalent to the value of the underlying asset in a financial option.
Receiving these cost savings requires that the RE technology be
deployed, which involves an irreversible investment in infrastructure
and other switching costs. This is the exercise price. Irreversibility
of the switching costs means that once the deployment is made, the costs
cannot be recovered.[^24] With RE, the option is to invest in and
install RE technology and infrastructure, receiving the benefits of the
difference in cost between RE generation and NRE generation. The option
is virtually perpetual, since the RE technologies that have been
developed have a long shelf life.[^25] The final component of the option
analogy is the uncertainty in future value of the underlying asset. For
RE technologies, the benefits depend on the difference between RE and
NRE in cost of supply. This difference can be affected by ongoing
federal RE R&D expenditures, which can be seen as a holding cost while
waiting to exercise the option, and by movements in the relative
generation costs of RE and NRE. This uncertainty is known as asset value
volatility, and is equivalent to the volatility of the underlying asset
in financial options.

An option\'s payoff, if exercised at any moment, is the difference
between the underlying asset\'s value at that moment and the present
value of the exercise price. This difference might be considered the
\"now-or-never\" NPV of the investment. If the exercise price is lower
than the asset value, then the payoff from investing, or exercising the
option, will be positive; and if the exercise price is higher than the
asset value, the payoff will be negative. In terms of option pricing,
the former option is "in the money," while the latter is "out of the
money." Out-of-the-money options are equivalent to negative value NPV
projects, only they have holding value due to future uncertainty and
rare events. This is because the downside from holding the option is
limited to the option holding costs, if any (the investment, if it would
be a loss-making venture, need not be undertaken), while the upside is
always available. In other words, even if the future value of an asset
is expected to be very low, a financial option on the asset can still
have value because of the possibility of a future increase in the value
of the underlying asset.

These same observations carry through to real options. In particular,
the downside of RE technologies is the continued expenditures on R&D
while the option is being held, while the upside is the potential value
that RE technologies could generate in a high fossil fuel-price
environment. As with financial options, it is the option to wait and the
volatility of the value of the cost savings that creates hold value
above and beyond the NPV of the option, revealing that the option to
postpone the deployment of RE technologies of uncertain future value has
current value that cannot be measured using DCF techniques. Generally,
the more negative the NPV and the more volatile the environment, the
more valuable the option. Given the current uncertainty regarding the
future costs and availability of fossil fuels, we would expect the RE
R&D option to have considerable insurance value.[^26]

In summary, as an alternative to the DCF approach, we can view RE
deployment as a real option that is currently out of the
money---immediate exercise yields a negative net payoff as indicated by
our "now or never" NPV value of negative \$35.3 billion. However, given
the option to postpone RE deployment, given the option to undertake
cost-reducing RE R&D while waiting, and given the possibility that RE
technologies may produce energy cost savings should NRE electricity
costs unexpectedly rise relative to the cost of RE, having the option to
install the RE technologies in the future can be a valuable insurance
policy. In order to calculate this insurance value, we can, through real
options analysis, weigh up the value of the deployment option versus any
ongoing R&D expenditures.

## The Real Options Model

Starting with the DCF model outlined above, we can model the value of RE
technologies as a real option on energy cost savings. In doing so, we
build upon real options models by Ott and Thompson (1996) and Schwartz
and Moon (2000). To begin the option analysis of the problem, we need to
be more precise about the future uncertainty surrounding electricity and
investment costs. We represent the changes in NRE and RE electricity
costs as the geometric Brownian motions

${dP}_{F} = {\alpha P}_{F}dt + \sigma_{F}P_{F}dz$ (6)

$$${dP}_{R} = \upsilon(M_{0})P_{R}dt + \sigma_{R}(M_{0})P_{R}dw$, *K* \>
0 (7)

${dP}_{R} = {\gamma P}_{R}dt + \theta_{R}P_{R}dw$, *K* = 0 (8)

where d*z* and d*w* are uncorrelated increments to the Gauss-Wiener
process and *σ~F~* and *σ~R~* are the cost volatility parameters.
Equation 6 reflects movement in NRE electricity costs. Equation 7
reflects movements in RE electricity costs during the R&D stage and
switching period, and Equation 8 reflects movements in RE electricity
costs upon completion of the RE switchover period. In Equation 7, the
more R&D spending, the faster the rate of decline in RE electricity
costs over time, up to a limit, before and during the RE switchover
period. This is reflected in the *υ* (*M~0~*) parameter, where *M~0~* is
the current rate of R&D spending (see Table 2). If the marginal
productivity of R&D is high enough, additional R&D spending can add
value to RE technologies. We add the fact that additional R&D spending
decreases the rate of uncertainty as to that rate of decline; the
volatility of changes in the supply cost of the most competitive RE
technology is reduced by the level of current spending on R&D.[^27] The
reasoning here is that all uncertainty about the cost of RE technology
can be resolved given enough R&D spending. On the other hand, when there
is very little R&D spending, there is increased uncertainty about
eventual deployment costs. Additional R&D spending thus increases the
expected payoff from RE technologies by lowering RE electricity costs,
but also decreases the uncertainty surrounding that payoff, each having
an opposite effect on option value. We assume that the Gauss-Wiener
processes for each cost process are uncorrelated, meaning that random
shocks to the cost processes that cause them to deviate from the
expected trends are derived from independent economic events.[^28] Under
these processes, the supply cost of each form of energy can drop, but
can never be zero. Zero costs therefore provide lower bounds to the
processes. The geometric nature of the processes also means that
unexpected deviations are permanent. That is, if there is an R&D
breakthrough that suddenly lowers the cost of producing RE electricity,
we never lose that technological advance.

We take remaining switchover expenditure *K* at any time to be
uncertain, with changes in *K* following the controlled diffusion
process

$$$dK = - \text{iP}_{F}dt + \beta\lbrack\text{iP}_{F}K\rbrack^{0\text{.}5}dx$,
(9)

where, again, *iP~F~* = *I* and d*x* is an increment to a Gauss-Wiener
process that is uncorrelated with the market portfolio and uncorrelated
with either of the two cost processes (Schwartz and Moon 2000).[^29] The
uncertainty in *K* relates to technical uncertainty, whereby the number
of periods in which *I* must be spent is unknown. Costs may be greater,
and the switchover period extended, if unexpended technical, legal, or
legislative impediments to deployment of RE are discovered once RE
switchover is initiated. Notice that if *I* = 0 then *dK* =0, and there
is no resolution of this technical uncertainty over the level of *K*
required to bring RE to market.[^30] This uncertainty is only resolved
once the switchover period is underway. Any successful reduction in *K*
through spending *I* also, by reducing the time to build, brings the
payoffs closer and reduces the uncertainty over the final payoff to be
received once switchover period is completed.

The term *β* is a scale parameter reflecting the uncertainty surrounding
*K*. If *β* = 0, *K* is certain, the usual assumption in DCF analysis.
In Equation 9, the stochastic term
$\beta\lbrack\text{iP}_{F}K\rbrack^{0\text{.}5}dx$has a mean of zero,
meaning that the expected level of *K* is unbiased. Uncertainty
decreases as *K* decreases, reflecting learning with investment; the
variance of *K* is

$\sigma_{K}^{2}(K) = \left( \frac{\beta^{2}}{2 - \beta^{2}} \right)K^{2}$.
(10)

Based on data presented in Pindyck (1993), we assume *β* = 0.5. The
total switchover cost is the integral of actual investments made, which
can only be known at the completion of the switchover period.

This is a compound option, sequential investment problem with technical
uncertainty.[^31] The option is a compound option because of the time to
build nature of the investment problem (Majd and Pindyck 1987, Dixit and
Pindyck 1994, Chapter 10). In a discrete sense, each annual switchover
expenditure, *I*, creates a new investment option on the present value
of cost savings from the installed RE technology, with a diminished
exercise price of *K-I*. The switchover program can be costlessly
suspended, allowing the flexibility of abandoning the switchover, either
temporarily or permanently.

These facets of the model merely formalized the aspects of uncertainty
contained within the valuation problem. As in the DCF analysis above,
the expected present value of the investment opportunity, if undertaken
immediately at the maximum rate, is still as in Equation 4 above. All we
have done is parameterized the nature of the future economic uncertainty
in our model. Having formalized the uncertainties in the problem, we now
proceed to value the portfolio of available RE technologies as a
compound perpetual American call option whose value is affected by the
rate of RE R&D spending.

Let the option value of RE technologies be *Φ*(*P~R~*,*P~F~*,*K*). The
approach to valuing *Φ*(*P~R~*,*P~F~*,*K*) is similar to that used in
Schwartz and Moon (2000). We assume the risk in *P~R~* and *K* is
unsystematic, and the risk in *P~F~* is traded in the market. These
assumptions allow us to use equilibrium arguments to derive the value of
the option *Φ*(*P~R~*,*P~F~*,*K*). Applying Ito's Lemma to
*Φ*(*P~R~*,*P~F~*,*K*), and assuming *Φ* is continuous and twice
differentiable in its arguments, we have

$df = f_{P_{R}}\text{dP}_{R} + f_{P_{F}}\text{dP}_{F} + f_{K}\text{dK} + \frac{1}{2}f_{P_{R}P_{R}}P_{R}^{2}s_{R}^{2}\text{dt} + \frac{1}{2}f_{P_{F}P_{F}}P_{F}^{2}s_{F}^{2}\text{dt} + \frac{1}{2}f_{\text{KK}}\beta^{2}\text{iP}_{F}Kdt$
(11)

Consider an optimally managed portfolio consisting of one unit of the
option and *h* units short in a traded NRE electricity derivative such
as electricity futures that spans the risk in spot electricity
cost.[^32] Let *X* denote the market price of this spanning asset, with

$dX = A(P_{F},t)Xdt + B(P_{F},t)Xdz$.

Since the spanning asset is a traded asset, its rate of drift must
return the required rate for assets of this risk class. The value of the
portfolio at initiation is *π* = (*φ* -*hX*), and, holding the short
position constant, the portfolio's change in value over some small
period d*t* is

$dp = d(f - \text{hX}) = df - hdX$. (12)

If we choose *h* = $\frac{s_{F}P_{F}f_{P_{F}}}{\text{BX}}$, the holder
of the portfolio receives the capital gain

$$\begin{array}{r}
d\pi = \varphi_{P_{F}}({\alpha P}_{F} - \frac{\sigma_{F}P_{F}A}{B})\text{dt} + \varphi_{P_{R}}{dP}_{R} + \varphi_{K}dK + \frac{1}{2}\varphi_{P_{R}P_{R}}P_{R}^{2}\sigma_{R}^{2}dt + \frac{1}{2}\varphi_{P_{F}P_{F}}P_{F}^{2}\sigma_{F}^{2}dt \\
\text{                                                                                                             } + \frac{1}{2}\varphi_{\text{KK}}\beta^{2}\text{iP}_{F}Kdt\text{.}
\end{array}$$

Due to the presence of the terms d*P~R~* and d*K*, the capital gain is
uncertain. The expected capital gain is

$$\begin{array}{r}
E(d\pi) = \varphi_{P_{F}}({\alpha P}_{F} - \frac{\sigma P_{F}A}{B})\text{dt} + \varphi_{P_{R}}{\upsilon P}_{R}dt - \varphi_{K}\text{iP}_{F}dt + \frac{1}{2}\varphi_{P_{R}P_{R}}P_{R}^{2}\sigma_{R}^{2}dt \\
\text{                                                                                      } + \frac{1}{2}\varphi_{P_{F}P_{F}}P_{F}^{2}\sigma_{F}^{2}dt + \frac{1}{2}\varphi_{\text{KK}}\beta^{2}\text{iP}_{F}Kdt\text{.}
\end{array}$$

While holding the portfolio, the investor incurs holding costs
*mP~F~*d*t*, and, when d*K* ≠ 0, investment costs *iP~F~*d*t*. Since the
portfolio only contains unsystematic risk, the expected return on
holding the portfolio, which includes capital gains less side payments,
is, in equilibrium, *r*(*φ* -*hX*)d*t*.[^33] Equating the expected
capital gain, less the portfolio holding and investment costs, to the
equilibrium expected return *r*(*φ* -*hX*)d*t*,

$$\begin{array}{r}
\varphi_{P_{F}}({\alpha P}_{F} - \frac{\sigma_{F}P_{F}A}{B})\text{dt} + \varphi_{P_{R}}{\upsilon P}_{R}dt - \varphi_{K}\text{iP}_{F}dt - \text{mP}_{F}dt - \text{iP}_{F}dt + \frac{1}{2}\varphi_{P_{R}P_{R}}P_{R}^{2}\sigma_{R}^{2}dt \\
\text{                                                  } + \frac{1}{2}\varphi_{P_{F}P_{F}}P_{F}^{2}\sigma_{F}^{2}dt + \frac{1}{2}\varphi_{\text{KK}}\beta^{2}\text{iP}_{F}Kdt = r(\varphi - \frac{\sigma_{F}P_{F}}{\text{BX}}\varphi_{P_{F}}X)dt\text{.}
\end{array}$$

Simplifying,

$\begin{array}{r}
{\upsilon P}_{R}\varphi_{P_{R}} - \text{iP}_{F}\varphi_{K} + \left( \alpha - \sigma_{F}\frac{(A - r)}{B} \right)P_{F}\varphi_{P_{F}} + \frac{1}{2}\varphi_{P_{R}P_{R}}P_{R}^{2}\sigma_{R}^{2} + \frac{1}{2}\varphi_{P_{F}P_{F}}P_{F}^{2}\sigma_{F}^{2} \\
\text{                                                                                         } + \frac{1}{2}\varphi_{\text{KK}}\beta^{2}\text{iP}_{F}K - r\varphi - \text{mP}_{F} - \text{iP}_{F} = 0,
\end{array}$ (13)

or equivalently, under optimal asset management,

$\begin{array}{r}
{uP}_{R}f_{P_{R}} + \left( \alpha - s_{F}\frac{(A - r)}{B} \right)P_{F}f_{P_{F}} + \frac{1}{2}f_{P_{R}P_{R}}P_{R}^{2}s_{R}^{2} + \frac{1}{2}f_{P_{F}P_{F}}P_{F}^{2}s_{F}^{2} \\
\text{                                                          } - rf - \text{mP}_{F} + \underset{0 \leq i \leq i_{\text{max}}}{\text{max}}\left\{ i\left( \frac{1}{2}f_{\text{KK}}\beta^{2}P_{F}K - P_{F}f_{K} - P_{F} \right),0 \right\} = 0
\end{array}$ (14)

The term $\left( \alpha - \sigma_{F}\frac{(A - r)}{B} \right)$ is the
risk-adjusted rate of drift of the cost of NRE electricity, with the
risk adjustment coming from observations of the return characteristics
for the spanning asset. Note that NPV Equation 3, under equivalent
risk-neutral valuation, is a solution to Equation 14 when *σ~F~* =
*σ~R~* = *β* = 0.[^34]

This is a problem with three state variables, the cost of RE
electricity, the cost of NRE electricity, and the remaining investment
required to complete the switchover. Given a rate of R&D spending
response, *m*, there is one control variable, the rate of switchover
response, *i,* which can vary from 0 to *i~max~*. Since Equations 13 and
14 are linear in *i*, the optimal level of deployment will be zero if
($\frac{1}{2}f_{\text{KK}}\beta^{2}K - f_{K} - 1 < 0$)or *i~max~* if
($\frac{1}{2}f_{\text{KK}}\beta^{2}K - f_{K} - 1³0)$.[^35] Thus, when it
is optimal to initiate RE deployment, the value of the option is
represented by the partial differential equation

$\begin{array}{r}
{uP}_{R}f_{P_{R}} - i_{\text{max}}P_{F}f_{K} + \left( \alpha - s_{F}\frac{(A - r)}{B} \right)P_{F}f_{P_{F}} + \frac{1}{2}f_{P_{R}P_{R}}P_{R}^{2}s_{R}^{2} + \frac{1}{2}f_{P_{F}P_{F}}P_{F}^{2}s_{F}^{2} \\
\text{                                                                                      } + \frac{1}{2}f_{\text{KK}}\beta^{2}i_{\text{max}}P_{F}K - rf - \text{mP}_{F} - i_{\text{max}}P_{F} = 0
\end{array}$ (15)

When the option is being held and the deployment response rate is zero,
the relevant differential equation is

$u(m)P_{R}f_{P_{R}} + \left( \alpha - s_{F}\frac{(A - r)}{B} \right)P_{F}f_{P_{F}} + \frac{1}{2}f_{P_{R}P_{R}}P_{R}^{2}s_{R}^{2} + \frac{1}{2}f_{P_{F}P_{F}}P_{F}^{2}s_{F}^{2} - rf - \text{mP}_{F} = 0$
(16)

The value of the RE technologies is indicated by the solution to
Equations 15 and 16 given the appropriate boundary conditions. The free
boundary is (*P*\**~F~*(*P~R~*,*K*)).

This is a stochastic problem in three variables. It is helpful to
simplify the problem using its natural homogeneity. Doubling the current
values of *P~R~*, *P~F~* and *K* merely doubles the expected value of RE
technology installations and the expected cost of investing in RE
installation. As such, the valuation problem is linear homogenous, and
we can again, as in the DCF analysis above, convert the problem into one
of cost ratios by using the cost of NRE electricity as the numeraire. We
can then write

*Φ*(*P~R~*,*P~F~*,*K*) = *P~F~N*(*p*,*k*) (17)

where again *p* = *P~R~*/*P~F~* and *k* = *K*/*P~F~*. The value of the
function *N* is now to be determined, which, when multiplied by the
current cost of NRE-generated electricity, will give the option value of
RE technologies program. Differentiating *Φ*(*P~R~*,*P~F~*,*K*) and
*N*(*p*,*k*) gives

$\begin{array}{r}
f_{P_{F}}(P_{R},P_{F},K) = P_{F}N_{p}(p,k)( - \frac{P_{R}}{P_{F}^{2}}) + P_{F}N_{k}(p,k)( - \frac{K}{P_{F}^{2}}) + N(p,k) \\
\text{                         } = N(p,k) - \text{pN}_{p}(p,k) - \text{kN}_{k}(p,k)
\end{array}$ (18)

$\begin{array}{r}
f_{P_{F}P_{F}}(P_{R},P_{F},K) = N_{p}(p,k)( - \frac{P_{R}}{P_{F}^{2}}) + N_{k}(p,k)( - \frac{K}{P_{F}^{2}}) - \text{pN}_{\text{pp}}(p,k)( - \frac{P_{R}}{P_{F}^{2}}) - \text{pN}_{\text{pk}}(p,k)( - \frac{K}{P_{F}^{2}}) \\
\text{                                } - N_{p}(p,k)( - \frac{P_{R}}{P_{F}^{2}}) - \text{kN}_{\text{kp}}(p,k)( - \frac{P_{R}}{P_{F}^{2}}) - \text{kN}_{\text{kk}}(p,k)( - \frac{K}{P_{F}^{2}}) - N_{k}(p,k)( - \frac{K}{P_{F}^{2}}) \\
\text{                           } = \left( p^{2}N_{\text{pp}}(p,k) + 2\text{pkN}_{\text{pk}}(p,k) + k^{2}N_{\text{kk}}(p,k) \right)\left( \frac{1}{P_{F}} \right)
\end{array}$ (19)

$f_{P_{R}}(P_{R},P_{F},K) = P_{F}N_{p}(p,k)(\frac{1}{P_{F}}) = N_{p}(p,k)$
(20)

$f_{P_{R}P_{R}}(P_{R},P_{F},K) = N_{\text{pp}}(p,k)(\frac{1}{P_{F}})$
(21)

$f_{K}(P_{R},P_{F},K) = P_{F}N_{k}(p,k)(\frac{1}{P_{F}}) = N_{k}(p,k)$
(22)

$f_{\text{KK}}(P_{R},P_{F},K) = N_{\text{kk}}(p,k)(\frac{1}{P_{F}})$
(23)

Substituting conditions (18) through (23) into (15) and (16) and
simplifying, gives the related partial differential equations when
*P~F~* is the numeraire,

$\begin{array}{r}
\frac{1}{2}(s_{R}^{2} + s_{F}^{2})p^{2}N_{\text{pp}}(p,k) + \frac{1}{2}(k^{2}s_{F}^{2} + \beta^{2}i_{\text{max}}k)N_{\text{kk}}(p,k) + \text{pk}s_{F}^{2}N_{\text{pk}}(p,k) - (i_{\text{max}} + yk)N_{k}(p,k) \\
\text{                                                                  } + (u - y)\text{pN}_{p}(p,k) + (y - r)N(p,k) - m - i_{\text{max}} = 0
\end{array}$ (24)

and

$\begin{array}{r}
\frac{1}{2}(s_{R}^{2} + s_{F}^{2})p^{2}n_{\text{pp}}(p,k) + \frac{1}{2}k^{2}s_{F}^{2}n_{\text{kk}}(p,k) + \text{pk}s_{F}^{2}n_{\text{pk}}(p,k) - y\text{kn}_{k}(p,k) \\
\text{                                                                  } + (u - y)\text{pn}_{p}(p,k) + (y - r)n(p,k) - m = 0
\end{array}$ (25)

where $y = \alpha - s_{F}\frac{(A - r)}{B}$. Once again, we note that
the NPV Equation 4, under equivalent risk-neutral valuation, is a
solution to Equation 24.

Unfortunately, we cannot observe the spanning asset's performance since
this information is not publicly available so we cannot estimate *A* or
*B*. From the analysis of Bessembinder and Lemmon (1999), we tentatively
assume that the spanning asset has a zero price of risk, making *A* =
*r*, and $y = \alpha$.

We now have two partial differential equations, 24 and 25, the first
applying to the (*p*,*k*) region in which investment in RE market
conditioning is taking place at rate *i~max~*, and the second applying
to the region where no switchover is taking place. Equations 24 and 25
can be solved for *N* and *n*, and thus for *φ*, given the problem's
boundary conditions. The first boundary is the value of the option at
*k* = 0, when the switchover is complete. The equations for calculating
the option value of the program at *k* = 0 are

$n(p) = E\left\lbrack \text{NPV}(p) \right\rbrack = \frac{q}{(\hat{\alpha} - \alpha)}\left\lbrack 1 + \frac{1}{(\hat{\alpha} - \alpha)} \right\rbrack - \frac{\text{pq}}{(\hat{\gamma} - \gamma)}\left\lbrack 1 + \frac{1}{(\hat{\gamma} - \gamma)} \right\rbrack$,
*p* \> *p*\* (26)

$\frac{1}{2}(s_{R}^{2} + s_{F}^{2})p^{2}n_{\text{pp}}(p) + (u - \alpha)\text{pn}_{p}(p) + (\alpha - r)n(p) - m = 0$,
*p*\**~ab~* \< *p* \< *p*\* (27)

*n*(*p*) = 0, *p \< p*\**~ab~* (28)

where *p*\* is the relative cost level at which RE begins meeting
incremental demand and *p*\**~ab~* is the relative cost level at which
RE technologies are abandoned. There are also value-matching and smooth
pasting conditions at *p*\**~ab~* and *p*\*. Equation 26 states that
once switchover is complete, the value of the RE technologies is the
expected NPV of cost savings given that it is optimal to turn on the
installed RE technologies. Equation 27 is the value of RE technologies
when it is optimal to wait before turning them on. Equation 28 is the
value of abandoned RE technologies. Abandonment may be optimal when
waiting incurs ongoing RE R&D costs.

When *k* \> 0 Equations 24 and 25 are solved using the *k* = 0 option
value and the additional boundary conditions,

*n*(*p*\*~ab~,*k*) = 0 (29)

*n~p~*(*p*\*~ab~,*k*) = 0 (30)

$$*N*(*p*\*,*k*) = *n*(*p*\*,*k*) (31)

*N~p~*(*p*\*,*k*) = *n~p~*(*p*\*,*k*) (32)

$\frac{1}{2}N_{\text{kk}}(p\text{*,}k)\beta^{2}k - N_{k}(p\text{*,}k) = \frac{1}{2}n_{\text{kk}}(p\text{*,}k)\beta^{2}k - n_{k}(p\text{*,}k) = 1$
(33)

Boundary Equations 29 and 30 allow for the costless abandonment of the
program, should the option value, in the light of the holding costs *m*,
become negative, where *p*\*~ab~(*k*) is the abandonment boundary.
Equation 30 is the "smooth pasting" efficiency condition at this
boundary. Equations 31 and 32 are the continuity and smooth pasting
conditions at the (*p*\*,*k*) boundary between switchover and no
switchover. In these boundary conditions, we assume that the switchover
program can be suspended and recommenced at no cost. Equation 33 is the
investment optimality condition at the free boundary (Milne and Whalley
2000).

A numerical method must be employed to produce the values of
*N*(*p*,*k*), *n*(*p*,*k*) and the free boundaries *p*\*(*k*) and
*p*\*~ab~(*k*) given specified parameter values. From this, the actual
current option value *Φ*(*P~R~*,*P~F~*,*K*) can be recovered by
multiplying *N*(*p*,*k*) or *n*(*p*,*k*) by the current value of *P~F~*.
Within the numerical procedure, a finite difference procedure is used to
discretize Equations 24 and 25. The two second-order derivative terms in
these equations are discretized using a standard three-point stencil,
the second-order mixed derivative term is approximated by a four-point
stencil scheme at each rectangular grid point. The two first-order terms
are treated using the standard up-winding finite difference scheme that
ensures a stable and accurate approximation for the governing equations.
The discretized equations are then solved iteratively by a method that
is similar to the Schwarz domain decomposition technique that has been
well studied for elliptic problems and variational inequalities. The
domain decomposition method is a very efficient technique for free
boundary value problems such as the valuation of American options.

## Real Options Parameter Values

In the process of identifying the precise nature of the future economic
uncertainty in our real options model, we introduced several new
important parameters that were not included in the DCF model, Table 1.
The values of the additional parameters included in the real options
model are presented in Table 2.

+---------------+------+--------------+------------------------------+
| **Table 2:    |      |              |                              |
| Real Options  |      |              |                              |
| Model         |      |              |                              |
| Uncertainty   |      |              |                              |
| Cha           |      |              |                              |
| racterization |      |              |                              |
| Parameters**  |      |              |                              |
+===============+======+==============+==============================+
| **Parameter** | **M  | **Value**    | **Notes**                    |
|               | odel |              |                              |
|               | Symb |              |                              |
|               | ol** |              |                              |
+---------------+------+--------------+------------------------------+
| Renewable     | *υ   | *Max(-0.033  | An estimate of the marginal  |
| electricity   | (M~  | \*           | productivity of RE R&D       |
| annual rate   | 0~)* | M~0~,-.04)*  | efforts. This estimate is    |
| of cost       |      |              | consistent with the          |
| reduction as  |      |              | technical goals outlined in  |
| a function of |      |              | the RETC, and the report of  |
| R&D funding   |      |              | the President\'s Committee   |
|               |      |              | on Science and Technology    |
|               |      |              | (PCAST 1997), both of which  |
|               |      |              | indicate that current R&D    |
|               |      |              | funding levels must be       |
|               |      |              | roughly doubled to achieve   |
|               |      |              | program objectives. We       |
|               |      |              | assume that the maximum rate |
|               |      |              | of RE cost decline is 4%/yr. |
|               |      |              | This reflects the our belief |
|               |      |              | that: (1) there is a maximum |
|               |      |              | funding level that can be    |
|               |      |              | absorbed given the current   |
|               |      |              | RE R&D infrastructure; and   |
|               |      |              | (2) even given higher        |
|               |      |              | funding levels it still      |
|               |      |              | takes time to achieve R&D    |
|               |      |              | success.                     |
+---------------+------+--------------+------------------------------+
| Volatility    | *σ~R | *.009/ M~0~* | We assume that additional    |
| surrounding   | ~(M~ |              | R&D spending decreases the   |
| the renewable | 0~)* | *for M~0~    | rate of uncertainty as to    |
| electricity   |      | greater than | that rate of decline. The    |
| annual rate   |      | .6,*         | reasoning here is that all   |
| of cost       |      |              | uncertainty about RE         |
| reduction as  |      | and          | technology can be resolved   |
| a function of |      |              | given enough R&D spending.   |
| R&D           |      | *.045 -      | We also assume that RE R&D   |
|               |      | .05\* M~0~*  | has a downside in that it    |
|               |      |              | locks investors into a       |
|               |      | *for M~0~    | technological pathway and    |
|               |      | less than or | diminishing the ability to   |
|               |      | equal to .6* | adopt alternative RE         |
|               |      |              | technologies. There is an    |
|               |      |              | intercept term (.045) in the |
|               |      |              | volatility equation for      |
|               |      |              | funding levels less than (or |
|               |      |              | equal to) \$0.6 million. We  |
|               |      |              | assume that, at current      |
|               |      |              | budget levels, RE R&D cost   |
|               |      |              | goals have a 95% chance of   |
|               |      |              | being achieved within a      |
|               |      |              | 25-30% band of accuracy. If  |
|               |      |              | R&D funding levels are       |
|               |      |              | doubled, RE R&D cost goals   |
|               |      |              | have a 95% chance of being   |
|               |      |              | achieved within a 10-15%     |
|               |      |              | band of accuracy. This is    |
|               |      |              | consistent with accuracy     |
|               |      |              | bands provided in the RETC   |
|               |      |              | (EPRI 1997).                 |
+---------------+------+--------------+------------------------------+
| Volatility    | *σ   | 0.11         | In the past 20 years,        |
| surrounding   | ~F~* |              | average annual natural gas   |
| the fossil    |      |              | prices have exhibited a      |
| fuel          |      |              | volatility of 0.17. Since    |
| electricity   |      |              | fuel costs represent         |
| annual rate   |      |              | approximately 65% of total   |
| of cost       |      |              | electricity generation       |
| increase.     |      |              | costs, we estimate the       |
|               |      |              | volatility of fossil fuel    |
|               |      |              | generation to be 0.11        |
|               |      |              | (.17\*.65). We use long-run  |
|               |      |              | volatility here (the         |
|               |      |              | volatility of average annual |
|               |      |              | prices) rather than          |
|               |      |              | short-run volatility (the    |
|               |      |              | annualized volatility of     |
|               |      |              | daily prices) because the    |
|               |      |              | emphasis in this analysis is |
|               |      |              | on long-run price trends. We |
|               |      |              | assume arbitrage             |
|               |      |              | opportunities (such as       |
|               |      |              | forward contracts) exist     |
|               |      |              | that allow firms to purchase |
|               |      |              | fossil fuels at average      |
|               |      |              | annual prices. Also, due to  |
|               |      |              | data availability            |
|               |      |              | constraints, we do not       |
|               |      |              | include the natural gas      |
|               |      |              | price increases that started |
|               |      |              | in the second half of 2000   |
|               |      |              | in our volatility            |
|               |      |              | calculations.                |
+---------------+------+--------------+------------------------------+

##  

## Real Options Model Results

Figure 8 shows the current expected NPV of RE technologies given FY 2000
federal R&D funding levels and an estimated annual incremental supply of
37.3 Billion kWh/yr. once RE technologies are deployed. The option value
of RE technologies, given the current RE/NRE electricity cost ratio of
1.29, is \$30.6 billion (2000 dollars). That is, in the presence of
technical and market uncertainty, the future expected cost savings from
eventual installation of RE electricity generation capacity have a
present value of \$30.6 billion dollars, net of ongoing RE R&D expenses.

![](media/image8.png){width="5.999305555555556in"
height="4.107638888888889in"}

For model comparison purposes, Figure 9 shows the value of RE
technologies for the different models presented in this paper: DCF, DCF
with optimal installation timing, and real options. The value of RE
technologies increases by approximately \$66 billion (2000 dollars) when
we move from the DCF model to the real options framework. The increase
in value comes from two sources: optimal timing and insurance value.
Since the expected NPV of RE technologies under an optimal deployment is
\$4.8 billion, we can attribute approximately 40% ((35.3 + 4.8)/(35.3 +
30.6)) of the option value to optimal timing and 60% to insurance value.

In Figure 10, we distinguish between the RE technology value created by
historical federal R&D expenditures (FY 1974 - FY 2000), and future R&D
expenditures. This is important because the decision faced by policy
makers today is whether to continue program funding (i.e. commit future
dollars). We can disentangle the value created by historical and future
R&D expenditures by setting *M~0~*, annual RE R&D funding, to zero.
Because *υ* (*M~0~*)=-0.033\**M~0~*, when *M~0~*=0, *υ*=0. Essentially,
by doing this, we assume that past R&D expenditures were responsible for
lowering the cost of RE electricity to its current price, and that
future R&D expenditures will be required to further lower the cost.[^36]
When we set the level of annual RE R&D funding to zero in our real
options model, we discover that the net value of RE technologies
decreases from \$30.6 to \$26.3 billion (2000 dollars) given the current
RE/NRE cost ratio of approximately 1.29.Thus, the expected net value of
future RE R&D expenditures is \$4.3 billion (2000 dollars).

![](media/image9.png){width="5.996527777777778in"
height="4.100694444444445in"}

## Optimizing R&D Expenditures

We have shown that, given our parameter assumptions, RE technologies
provide positive value to the nation from a narrowly defined
market-based perspective. We now move on to the determination of the
optimal annual RE R&D budget. The objective of this section is to use
the real options model developed above to determine the current funding
rate, *M~0~*, that maximizes the program value. This analysis depends
heavily upon our assumptions about the marginal rate of R&D
productivity, . In particular, we assume that the more R&D spending on
RE technology, the faster the rate of decline in RE electricity costs
over time. As indicated in Table 2, we assume that
$\upsilon(M_{0}) =$*Max(-0.033 \* M~0~,-.04)* where *M~0~* = the annual
RE R&D budget expressed in billions of dollars. This means that, at the
current funding level of approximately \$300 million, the annual rate of
decline in the cost of RE electricity is 1%/yr. (the value we used in
our DCF analysis). We also assume that the maximum rate of RE cost
decline is 4%/yr. This reflects our belief that: (1) there is a maximum
funding level that can be efficiently absorbed in the near to mid-term
given the current RE R&D infrastructure; and (2) given even higher
funding levels, it still takes time to achieve R&D success. Finally, we
assume that additional R&D spending decreases the rate of uncertainty as
to that rate of decline. Additional R&D spending thus increases the
expected payoff from project deployment, but also decreases the
uncertainty surrounding that payoff, each having an opposite effect on
option value.

Throughout its existence, the federal RE R&D program has experienced an
inconsistent level of support from federal decision makers (Figure 2).
The initial program budget in FY 1974 was \$72.6 million (2000 dollars).
Spurred in part by the 1978-1979 petroleum price increases related to
the Iranian Revolution, the federal RE R&D program budget was gradually
increased to a maximum of \$1.56 billion (2000 US) by FY 1980. During
the Reagan-Bush administration (1980-1988), RE R&D program funding was
reduced by nearly 90%. Under the direction of the Clinton-Gore
administration, program funding leveled out in the \$300 million (2000
dollars) range. More recently, however, the Bush Administration
submitted a FY 2002 budget request that reduced RE R&D funding to \$237
million. The question under examination here is whether the current
budget of approximately \$300 million/yr. (2000 dollars) represents the
optimal RE R&D portfolio funding level.

The results of the options analysis on the optimal level of ongoing R&D
yields are presented in Figure 11. Figure 11 shows that increased R&D
spending does indeed pay off. At a deployment rate of 37.3 Billion
kWh/yr. once switchover is completed, the optimal annual rate of R&D
expenditures is approximately \$1.2 billion/yr. At this funding level,
the present value of the RE technologies, net R&D expenses, climbs from
\$30.6 billion to \$50 billion (2000 dollars). This analysis provides
evidence to the effect that FY 1980\'s \$1.56 billion funding level was
more close to the appropriate level than today\'s budget of
approximately \$300 million. This analysis also lends support to the
recommendations of the President\'s Committee of Advisors on Science and
Technology, which asserted that significant RE R&D budget increases
would be required to \"provide good prospects that RE technologies will
be able to make large contributions to global energy during the first
quarter of the next century.\" (PCAST 1997).

We must remark here that the optimal annual funding level of \$1.2
billion (2000 dollars) is a direct result of our decision to cap the
maximum rate of RE electricity cost declines at 4%/yr. \$1.2 billion is
simply the funding level where 0.033\*M~0~ is approximately equal to
0.04. Thus, the real insight provided by our model is that RE R&D
expenditures should be increased to the maximum level that can be
absorbed by the federal RE R&D complex without reducing marginal R&D
productivity. The key question then is, can the federal RE R&D complex
sustain such a large budget increase? The answer, as reflected in the
scientific literature and according to technology researchers, is yes.
[^37] In fact, by examining data on R&D investments and patent records,
Margolis and Kammen (1999) found that energy R&D investments and patents
were highly correlated between 1976 and 1996. They found no evidence of
declining marginal productivity of energy R&D investment during this
period, regardless of the magnitude of R&D funding. According to
Margolis and Kammen, this evidence \"supports the hypothesis that the US
under invests in energy-related R&D,\" and \"illustrates that cut-backs
in energy-related R&D have dramatic impacts on innovation in the energy
sector.\" (p. 579).[^38]

![](media/image11.png){width="5.998611111111111in"
height="4.106944444444444in"}

It is important to underscore the importance of continued federal RE R&D
expenditures within the context of the ongoing restructuring of the U.S.
electric utility industry. The uncertainty surrounding restructuring has
initiated an exodus from energy R&D and long-range strategic planning in
the electricity sector as a whole. This abandonment of R&D is reflected
in recent trends at investment-owned utilities (IOUs). For example,
between 1994 and 1996, IOU investments in R&D decreased by 35%, from
\$650 to \$403 million/yr. (FERC 1997). During the same period, the 10
largest IOU contributors to the Electric Power Research Institute
(EPRI), the electric utility industry R&D consortium, cut back their
funding to EPRI by 47%, from \$130 to \$69 million (FERC 1997). The
transition to a more competitive market is expected to lead to
continuing declines in private-sector investment in energy technology
R&D (Dooley 1998; GAO 1996). In this environment, public energy R&D has
never been more important.

# The Internet Model

In order to allow readers to examine the impacts of alternative input
assumptions, we have developed publicly accessible Internet versions of
the models presented in this paper. We have made these models available
in the context of an ongoing effort at NREL called \"e-Analysis,\" which
is focused on the creation of Internet models that serve as an
interactive complement to our traditional analysis products. Interested
readers are invited to use the online valuation model at
<http://www.nrel.gov/realoptions>, and join in a discussion of this
paper at <http://nrel.communityzero.com/realoptions>.

## Conclusions

We have examined RE technologies from both the traditional DCF economic
valuation perspective, a perspective that does not consider insurance
value or optimal deployment timing, and the real options perspective,
which draws upon insights from financial markets in order to value all
RE technology benefits. We found that RE technologies are economically
attractive from the real options perspective when optimal timing and the
insurance value is included. Further, using real option analysis, we
determined that the optimal level of RE R&D investment is \$1.2
billion/yr., approximately four times the program\'s FY2000 funding
level.

We must, however, temper this result with a statement we made earlier.
The usefulness of this exercise lies in the insights provided by the
real options model, not in our final annual investment recommendation.
The key insight here is that RE technologies hold a significant amount
of value that cannot be detected by using traditional valuation
techniques. Thus, in order to appropriately value these technologies and
the benefits of continued R&D spending, we must adopt a more advanced
valuation perspective such as real options analysis.

Even though our estimates of technology value are a function of our
selected parameter values, which are subject to debate, we still feel
that our estimates of RE technology value have been conservative. For
example, we assume that the decision on future optimal RE R&D levels is
made now, and maintained until the option is completely exercised or
until the option is abandoned. We could allow for dynamically varying
R&D funding levels, which would then create additional value by
optimizing the level of R&D. We also could allow for the switching to RE
to be an increasing function of annual switching expenditures,
permitting a form of adjustment costs, and set no exogenous upper limit
to annual switching expenditures (Childs and Triantis 1999). This could
add further value to RE technologies. Finally, we have not allowed for
any reversibility of the RE technology deployment once it ultimately
replaces fossil-fueled generation. Yet RE generation is partially
reversible and this also would add value to RE technologies through R&D.
In sum, our model could be extended to include optionality that would
further increase the present value of cost savings from future RE
technology adoption.

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[^1]: The views expressed in this paper are those of the authors and do
    not necessarily reflect those of the Colorado School of Mines nor
    the National Renewable Energy Laboratory. We thank Dr. Steven Ott
    for his valuable comments. We also thank Dr. Junping Wang from the
    Colorado School of Mines Department of Mathematical and Computer
    Sciences for assistance with the numerical analysis used in this
    paper.

[^2]: Graham Davis is associate professor of Economics and Business at
    the Colorado School of Mines and can be reached via e-mail at
    gdavis@mines.edu.

[^3]: Brandon Owens is senior analyst at the National Renewable Energy
    Laboratory and can be reached via e-mail at brandon_owens@nrel.gov.

[^4]: Natural gas was thought to be a diminishing resource during this
    period because production was declining and prices were increasing.
    In 1973, domestic dry production was 21.7 million cubic feet (cf)
    and the average wellhead price of marketed production was 19 cents
    per thousand cf. By 1976, production had fallen to 19.7 million cf
    and the average wellhead price had increased by more than 300% to 58
    cents per thousand cf (EIA 2000c).

[^5]: Characterizing the RE R&D program as an insurance policy is a
    reasonable, but not perfect, analogy. In this case, an investment is
    being made to reduce the potential cost of unfavorable future
    outcomes. That is why one takes out insurance, as a hedge against
    such outcomes. When one pays an insurance premium, the policy is
    guaranteed to pay off if the unfavorable outcome occurs. However,
    with R&D expenditures there is no guarantee that the investment will
    payoff, since the outcome of R&D activities are also uncertain.

[^6]: The insurance value of all energy R&D was first estimated by
    Schock et al. (1999) using a probabilistic framework. They estimated
    that the national value of energy R&D as an insurance investment to
    reduce the cost of the risks of climate change, oil price shocks,
    urban air pollution, and energy disruptions is greater than \$12
    billion/year.

[^7]: Since most of the technologies under development within the RE R&D
    program are electricity generation technologies, our model will
    focus on the electricity market.

[^8]: There may be benefits or costs of an RE program that accrue to
    energy wholesalers and retailers. We assume these effects offset,
    such that we only need to focus on the benefits to the industrial
    and residential consumers who ultimately consume the power.

[^9]: Of course, this is a simplification that ignores marginal
    transmission and distribution expenditures. See fn 16 for a
    discussion of how to model this expenditures within this framework.

[^10]: This cost includes all capital expenditures, operating revenue
    and income, taxes and debt, and interest payments.

[^11]: By examining the value of future cash flows, assuming that RE
    technologies are installed today, we are constructing the typical
    "now-or-never" NPV. This NPV does not take into account the optimal
    timing of RE deployment. To evaluate the NPV of the RE program
    assuming that deployment can be made at some time in the future,
    rather than only "now or never," we can adjust our DCF model to
    allow for an arbitrary deployment start time *τ* and adding annual
    R&D costs at rate *M* while waiting to invest. We can estimate the
    optimal time of investment by maximizing our DCF equations with
    respect to start time *τ* ≥ 0, using numerical iterations to find
    the optimal time *τ*\*. We perform this calculation later in the
    paper.

[^12]: The costs include capital recovery, so that the technology can be
    seen as being perpetually installed, allowing the upper limit on the
    integrand to be infinity.

[^13]: In Equation 1, we ignore any tax deductibility of energy costs as
    an input to industrial production, since this is simply a transfer
    between economic parties.

[^14]: Since we abstract from the daily and seasonal trends in wholesale
    power prices, at issue is simply the uncertainty around the expected
    annual drift in electricity costs prior to and after switchover, *α*
    and *γ,* respectively*.*

[^15]: Since there is no autonomous inflation in *K*, our model assumes
    that, as we wait to invest, we gain technological efficiency with
    respect to the real deployment costs, *ceteris paribus*. Since RE
    electricity costs are expected to improve relative to NRE
    electricity costs, this seems a reasonable assumption. Also, *K* can
    represent transmission and distribution and other electricity
    infrastructure expenditures that may be required for large-scale RE
    technology deployment.

[^16]: The intuition behind this functional form is that the maximum
    rate of deployment of RE will be faster when fossil fuel-prices are
    higher.

[^17]: Where there is the option to delay deployment, RE technology
    value is maximized by switching at the maximum possible rate. This
    is because the switching cost is assumed constant, so that the
    present value of switching costs are lower the longer they can be
    delayed. Once immediate deployment becomes optimal, instantaneous
    switching is preferred to a market-conditioning scheme that takes
    time.

[^18]: See Childs et al. (1998). There are, in essence, dynamic external
    informational economies that make the value of optimal management of
    the portfolio greater than the sum of optimal management of the
    individual projects. Even when project values are uncorrelated and
    where, at most, one technology can be deployed, the payoffs to being
    able to execute the more valuable of a suite of projects means that
    the value of the portfolio will be at least as large as the value
    determined here.

[^19]: This need not be the case. RE researchers anticipate the need for
    continued public support after RE deployment has occurred in order
    to stabilize infant RE industries and promote continued
    technological development.

[^20]: We assume that R&D funding continues during the
    market-conditioning phase.

[^21]: The parameter *m* can be thought of as a response coefficient,
    with our functional form reflecting the political reality that RE
    R&D funding has historically been a function of the price of fossil
    fuels. The higher *m*, the more sensitive R&D spending to changes in
    fossil fuel-prices.

[^22]: The annualized rate of increase in daily first nearby futures
    prices from June 28, 1994, through June 28, 2000, is 0.19%, and a
    regression of futures returns on the returns to the S&P 500 Index
    yields an insignificant slope parameter. Both results indicate a
    lack of systematic risk in natural gas prices, from which we deduce
    that changes in levelized costs of natural-gas-fired electricity
    generation is unsystematic.

[^23]: See Berk et al. (1998), Childs et al. (1998), Childs and Triantis
    (1999), Huchzermeier and Loch (1998), Kumaraswamy (1997), Laughton
    et al. (1993), Ott and Thompson (1996), Schwartz and Moon (2000),
    Smit and Trigeorgis (1997), and Willner (1995).

[^24]: If renewable energy turns out to be uneconomic, the
    infrastructure, while perfectly functional, will have no salvage
    value.

[^25]: In fact, in our calculations below, we allow the option to switch
    to RE to stay alive even if the RE R&D is curtailed.

[^26]: We cannot be assured of this, though. Many real options have
    holding costs, which can make their value negative. This is in stark
    contrast to financial options, which are always valuable.

[^27]: We could also allow the rate of learning-by-doing effects,
    reflected in the parameter *γ* in Equation 8, to be affected by the
    type of technology created by a R&D program of expenditure level
    *M~0~*. Increasing *M~0~* would result in the installation of a
    "better" long-run technology (see Majd and Pindyck 1989, fn. 6).

[^28]: Shocks to the supply price of fossil fuel are, for example, a
    result of changing inventories; whereas shocks to the supply cost of
    renewables are related to technological uncertainties. Any
    correlation between the costs of RE and NRE electricity are likely
    to be minor, and including these correlations into the calculations
    greatly increases their complexity. Thus, we choose to ignore them.

[^29]: This is a controlled diffusion process because *i* is a control
    variable. One might envision the level of *K* as being dependent on
    and negatively related to the value of renewable energy. For now we
    assume *K* is the relatively minor cost of a fixed number of "tasks"
    involved with bringing RE online. The cost of these tasks do not
    vary with the level of market penetration of RE. See Berk et al.
    (1998) for a model where *K* varies with the profitability and scale
    of the underlying project.

[^30]: However, as we noted above, since *K* is constant in nominal
    terms, there is some technical learning while waiting to invest. The
    fact that the risk in *K* cannot be replicated, and so learning can
    only take place through capital expenditures, means that markets are
    incomplete.

[^31]: Pindyck (1993) proposes a model where there is also input cost
    uncertainty. In this case, we abstract from input cost uncertainty,
    as it seems to be relatively minor.

[^32]: Electricity trading is confined to wholesale markets, either for
    delivery or cash settlement (Bessembinder and Lemmon 1999). We
    assume that the portfolio holder has access to this market, and that
    the risk in the wholesale market spans the risk in the retail
    market.

[^33]: Even though markets are incomplete in our model, it is still
    possible for the development of this project to leave the
    equilibrium pricing kernel unaffected (Childs et al. 1998).

[^34]: See Dixit and Pindyck (1994), pp. 121-125, for a description of
    risk-neutral valuation.

[^35]: Note that *P~F~* \> 0.

[^36]: This simplification neglects RE cost improvements that may occur
    due to either economy-wide technological improvement or learning
    effects from RE production in the absence of future RE R&D
    expenditures.

[^37]: Per personal correspondence, RE technology researchers at the
    National Renewable Energy Laboratory believe that the federal RE R&D
    complex can sustain significant budget increases provided that the
    revised budgets include resources for R&D capital expansion.

[^38]: Although Ambuj D. Sagar (2000) argues that Marolis and Kammen do
    not present a conclusive case in their analysis.