Optimal renewable energy capital deployment
Andrew Leach∗
Linda Nøstbakken†
May 9, 2012
(Incomplete draft)
Abstract
We study optimal capital deployment when agents are price and technology takers,
and we focus on renewable energy such as wind. We develop a model where land is
scarce and heterogenous in its suitability for wind energy generation. In this framework, we show how an agent, such as a government, should first install wind-farms on
lower quality locations. Later, as the technological progress slows down relative to the
discount rate, it becomes optimal to develop higher quality locations first. Hence, the
optimal policy involves a switch where we instead of investing in wind energy on lower
quality land, should start developing the best land first.
JEL codes: Q0
Keywords: Renewable energy; Wind power; Capital deployment; Technological progress.
∗
Alberta School of Business, University of Alberta, Edmonton, AB, T6G 2R6, Canada. Email: Andrew.Leach@ualberta.ca
†
Alberta School of Business, University of Alberta, Edmonton, AB, T6G 2R6, Canada. Tel: +1 (780)
248-1266. Email: Linda.Nostbakken@ualberta.ca
1
1
Introduction
California was the first state to develop wind farms in the US and made large investments in wind energy capacity in the 1980s and 1990s. However, the growth in energy
capacity in California has been relatively modest since 2000. Texas, on the other hand,
had little generating capacity in 2000, but is today the largest wind power producer
in the US. From 2000 to 2011, Texas experienced an average annual growth of 69.9%
in installed capacity, compared to an average annual growth of 6.4% in California over
the same period. Considering that the cost of producing wind power according to the
Electric Power Research Institute (EPRI) has decreased nearly fourfold since 1980, it
seems relevant to ask whether California developed their best wind sites too early using
too expensive technologies.
The focus of this study is on optimal renewable energy capital deployment in
economies such as Texas or California. In contrast to the previous literature, we do
not model this as a social planner problem but instead focus on a local decision maker,
such as a government, who develops and implements renewable energy policies given
the constraints faced. The government must decide on when, where, and how much
to invest in wind power. The capital is durable, which means that any investment
in wind farms will lock them into current technologies for many years. Furthermore,
we account for the fact that available areas for wind-power generation are scarce and
varying in quality. A country has a finite area of land and water that can be used for
wind-power generation, but some locations are far more suitable than others. Hence,
it is not just a question of the optimal timing of capital deployment, but also of what
locations to use and when. Should we use the best locations first, when this means
using less efficient technology in the best sites for many years to come? Or should we
save the best locations for later when technological progress has made the capital we
invest in more productive? These are highly relevant questions faced by governments
and private developers worldwide as the world is searching for renewable energy options
to replace fossil fuels.
Ongoing technological progress is a key part of our model. As opposed to most
previous work, the decision maker in our model is a technology taker. Hence, we
model technological progress as something that takes place regardless of what the
decision maker in the model does. To justify this assumption about technological
progress, consider the two economies Texas and China. Texas is a relatively small
global player in renewable energy, while China is large and getting larger. If Texas
does not invest in renewable energy for a while, the technological progress is likely to
continue basically at the same rate as before. If China decides to step out of renewable
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energy for a period, this would likely result in technological progress slowing down,
although there would still be some progress driven by innovations in other countries.
Hence, as long as some countries innovate, and new innovations are shared globally,
technological progress does not depend critically on the actions of the local decision
maker.
We show that with exogenous technological growth, the agent should first install
wind-farms on lower quality locations. Under the assumptions that wind technologies
improve at a decreasing rate towards its potential, and that the relative change in
technology initially is sufficiently high relative to the interest rate, we should develop
lower quality locations first and save the best locations for when the technology is more
efficient. Later, as the technological progress slows down relative to the discount rate,
it becomes optimal to develop higher quality locations first. Hence, under optimal
deployment a switch of policy occurs when we instead of working our way up from
lower to higher land qualities, start developing the best land and instead work our way
down from higher to lower qualities.
Our work is related to several strands of the economic literature. In urban economics, there are a number of studies that analyze optimal land development when
there are differences in land quality (e.g. Capozza & Li, 2002; Clarke & Reed, 1988;
Cunningham, 2006). Studies in which future profits are subject to uncertainty typically rely on real options theory to derive optimal land development policies (Dixit &
Pindyck, 1994). While the investment decision in Dixit & Pindyck (1994) and other
related studies typically is a one-time opportunity, we model investment possibilities
as a series of options to invest. Our hypothesis for renewable energy development on
a given site is similar. When technology allows the profitable generation of electricity
on a given site, you develop unless you expect technology to improve sufficiently to
make further delay more profitable. We add in the notion that more investments can
be made until all available sites are developed. Hence, by modifying the traditional
investment under uncertainty model, we can capture the most important features of
renewable resource investments.
Another study that shares many of the same features as our work is Boyce &
Nøstbakken (2011). They develop a theoretical model with scarce land and analyze
optimal exploration and development of oil on that land. Their model is deterministic
and a key feature is that production on a unit of land depends on the size of the oil
reserve as well as when production started from that field. Hence, the timing of the
development of any unit of land and the initial reserve determine the profit stream
from that land. This is similar to our study, since we also have scarce land for which
we must determine the optimal time of development, where the profit stream from any
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unit of land will depend on when the land was developed (the state of the technology
at time of investment) and the quality of that land. Hence, we use a similar modeling
framework in our study.
Significant work has been done on capital deployment in the renewable energy industry under technological progress. However, this work generally does not take into
account differences in land quality or the fact that land that is suitable for renewable
energy production is scarce. An example is Kumbaroğlu et al. (2008), who study optimal investment in renewable energy when there are learning-by-doing effects and price
uncertainty. Another strand of the literature focuses on geographical diversification of
wind farms, taking into account geographical differences in wind power potential, but
disregarding land scarcity (Fuss & Szolgayová, 2010; Roques et al., 2010).
Finally, there have been some studies of vintage capital models in energy systems.
Vintage capital is an ideal model for most renewable energy technology (e.g. solar
panels or wind turbines) since, once erected, the capital is generally used in its initial
form until it is replaced. Perez-Barahona & Zou (2006) examine vintage capital in a
model of energy efficiency. Lesser & Su (2008) use the vintage of producing capital
(i.e. the age of a wind turbine) as a determinant of the optimal feed-in tariff regime
for renewable energy. This type of payment structure occurs naturally as countries
such as Germany reduce the tariffs they pay for electricity produced by new generating
units over time. The fact that technological progress does not generally affect the
productivity of the existing capital stock leads to important interactions with land
scarcity: once capital is deployed on a tract of land, the productivity of that land is
locked-in for a long period of time, unless the producing unit is scrapped.
The paper is organized as follows. We start out by developing the basic model.
Next, we analyze the deterministic problem, before analyzing the implications of stochastic technological progress in section 4. In section 5 we discuss the policy implications
of our results, before we conclude in section 6.
2
Model
Assume there is a total of N units of land and that land units differ in their appropriateness for wind power production (quality), which we denote Qi . There are large
geographical differences in solar and wind energy potential. In addition, land quality
in our model accounts for transmission costs, which means that even if there is a lot of
wind in an area, the large distance to consumers would imply high transmission costs,
both in terms of energy loss, and building and maintaining the necessary transmission
4
capacity.
We assume exogenously technological progress, which increases the amount of electricity we can generate from the wind. Hence, the production of wind power from a
unit of land depends on the quality of that land and the state of the technology at the
time of investment: x(si )Qi , where x(si ) > 0 is an index that gives the productivity of
the technology available at the time of investment on land i, si . Let us further assume
that there is a fixed cost ci of installing wind generators on a unit of land, which may
differ between different areas, while we assume no variable costs and a unit price of
electricity.1 Finally, to keep things tractable, in the base model we assume that once a
wind generator is installed, it operates infinitely. Based on this, the net present value
of a wind project installed on land i is:
−rsi
N P Vi = e
x(si )Qi
− ci ,
r
(1)
where r is the discount rate and the first term in parentheses is the net present value
at time si of an infinite net revenue stream of x(si )Qi .
Next, we assume that the technology index x follows a stochastic mean reverting
process (Ornstein-Uhlenbeck):
dx = η (x̄ − x) dt + σdz,
(2)
where the first term represents the expected change in x, the instantaneous drift rate,
while the second term represents the variance. dz is the increment of a Wiener process
√
so that dz = t dt, where t is a normally distributed random variable with zero
mean and unit standard deviation. We assume that the initial value of x is x0 < x̄,
hence, x exhibits a positive, deterministic trend representing increased efficiency in
wind generating technology. The deterministic trend approaches x̄ over time. x̄ is the
long-run mean level of x, which is reached when innovation has created wind generators
that convert as much energy from the wind as is technically and physically feasible, at
the lowest possible cost. The parameter η > 0 is the reversion speed and represents
the expected rate of technological progress towards the long-run mean level x̄.
Note that because of the stochastic term in (2), the technology index can increase
and decrease. While technological progress over time is easy to justify, we should
perhaps motivate the possibility of technological regress. Recall that the technology
index follows an increasing trend that approaches a long-run mean level. Hence, we
1
According to IEA (2011), the cost of operation and maintenance is negligible relative the the installation
cost, which includes the cost of the turbine, grid connection and reinforcement, and development.
5
can only experience technological regress in the short run. In the wind industry, this
can be explained by increases in the price of input factors, such as new wind turbines,
which depend on the price development of raw materials, and other factors. Indeed,
as illustrated in the study by Junginger et al. (2005), the price of new wind turbines
has shown fluctuation over time. This can explain temporary drops in x.
3
Solving the deterministic problem
To maximize the net present value of wind energy production, we must decide what
land to invest on and when. Since each wind project is small and does not affect
the rate of technological progress, we can treat each unit of land independently. Let
us start out by analyzing the deterministic optimal capital deployment problem, by
assuming that σ = 0 in (2). We then face the following dynamic optimization problem:
max N P V =
{si }
N
X
e
−rsi
i=1
x(si )Qi
− c0i ,
r
(3)
subject to (2) with σ = 0. Note that if the bracketed term is negative, it will never
be optimal to develop land i. With deterministic technological progress, we know that
x ≤ x̄. Hence, we can define the following threshold on land quality:
Qi ≤
rc0i
≡ Q̄i .
x̄
(4)
To keep the analysis tractable and without loss of generality, let us assume that c0i = c0
for all i in what follows. This implies that Q̄i = Q̄ for all i in (4). The first order
conditions for the optimization problem (3) are:
c0 r − x(si )Qi +
x0 (si )Qi
= 0,
r
(5)
for all land units i that satisfies Qi ≥ Q̄. Implicit differentiation of the first order
condition (??) with respect to land quality Q yields:
0
x(s) − x r(s)
∂s(Q)
i.
= h 00
∂Q
Q x r(s) − x0 (s)
(6)
The partial derivative of s with respect to Q tells us what land to develop first and last.
If the partial derivative is positive, the optimal time of land development increases as
we consider higher quality land. Hence, if (6) is positive it is optimal to develop the
6
best land last and the worst land first, provided that the land should be developed at
all (Qi ≥ Q̄, cf. equation 4). If instead (6) is negative, the highest quality land should
be developed first.
From (2) we have that x0 (s) = η(x̄−x) ≥ 0. Hence, we know that x00 (s) = −ηx0 (s) ≤
0. Consequently, the denominator in (6) is negative. The sign of
∂s
∂Q
thus depends on
the sign of the numerator, which can be positive or negative depending on the current
level of the technology index x. More specifically, we have that
x̄
r
< − 1,
η
x
and negative if
r
η
>
x̄
x
∂s
∂Q
> 0 if
(7)
− 1. Over time, as x → x̄, the right hand side (RHS) of (7)
approaches zero. Hence, as long as the discount rate r is positive, the condition (7)
will eventually be violated and the partial derivative (6) must be positive as t → ∞.
It follows that for a sufficiently small initial value of x, the sign of (6) will shift from
negative to positive over time.
To identify when and if this switch occurs, let us first calculate the value of x for
which condition (7) holds with equality:
xT =
ηx̄
.
r+η
(8)
This is the technology index for which the switch occurs, provided that x0 < xT . From
the technology dynamics equation (2), with σ = 0 and noting that x(0) = x0 , we can
solve for x(t) as follows:
x(t) = x̄ + (x0 − x̄) e−ηt .
(9)
By combining equations (8) and (9), we can solve for the time T at which x(T ) = xT .
This yields the following switching time:
T =
1
[ln (r + η) + ln (x̄ − x0 ) − ln (rx̄)] ,
η
(10)
provided that x0 < xT .
Our results thus far have shown that poor quality land (low wind potential and/or
far from electrical grids) will never be used. Furthermore, we have established that for
a sufficiently low initial technology level (x0 < xT ), the optimal capital deployment
strategy involves starting by installing wind generators on lower quality land given that
Qi ≥ Q̄. As time goes by and the technology becomes better, at some point it becomes
optimal to switch to a different strategy. From this point onwards, we should develop
the best quality land first and then work our way down towards lower quality lands.
7
The switch occurs because the technological progress is faster the further away from x̄
we are. Hence, the technological progress slows down more and more as we approach
the long-run level x̄. For this reason, it is optimal early on to save the best land for
when the technology is better. During this period, we should instead develop lower
quality land. At some point, the gains from waiting for further technological progress
are exceeded by the cost of waiting, in terms of lost revenues. These revenues losses are
larger the higher the potential for wind power in an area (i.e., the higher the value Qi ).
Hence, the deployment policy switch is a result of the trade-off between lost revenue
and access to better technology if we delay investment.
Let us now extend the base model by allowing for replacement of old windmills.
Assume the fixed cost of scrapping and replacing old wind capital on land i is c1i > 0.
Let sik denote the timing of the kth investment in wind energy on land unit i. Note
first that it can only be optimal to scrap old turbines and reinvest on a piece of land
if the following inequality holds:
x(sik ) < x̄ −
rc1i
,
Qi
(11)
where c1i is the (fixed) cost of upgrading the wind turbines on land i to the most
recent technology, while sk indicates the time when the current turbines were installed
so that x(sk ) is the currently installed technology. The inequality (11) states that
for reinvestment to be optimal, the difference between the current technology and the
technology cap (x̄) must be sufficiently large. Furthermore, the more expensive it is
to upgrade the turbines and the lower the quality of the land, the larger the difference
must be between the current technology and a new technology for investment to be
profitable.
Let us now focus on a unit of land for which condition (11) holds so that it is optimal
to install wind turbines and then upgrade the turbines at least once.2 To solve the
investment timing problem, note that since we are considering an infinite time horizon
the problem is always the same. The only thing that changes over time is the efficiency
of the available technology, x(t). Hence, we can treat each investment decision on a
unit of land independently and do not need to solve for the full chain of investment
times simultaneously. It follows that at every point of time t and for every land unit,
the current investment problem can be expressed as:
max N P Vt = e
sk
2
−r(si −t)
Q
[x(sk ) − x(sk−1 )] − c1 ,
r
(12)
Since we can analyze each unit of land independently, we drop the land unit subscript (i) in what follows.
8
subject to (2) with σ = 0, where we assume x(s0 ) ≡ 0, which indicates that no previous
investment has been made on the land if k = 0 and hence there is no production. When
reinvesting on land that is already in production, we must deduct the lost revenues
from scrapping the old wind turbines to make room for new ones. This is why we
deduct the term x(sk−1 ) in (12).
Before we proceed to solve the optimization problem, note that it is only profitable
to invest on the land if the quality of the land satisfies the following constraint:
Q≤
c1 r
≡ Q̂.
x(sk ) − x(sk−1 )
(13)
This condition reduces to the condition we found for the case with no capital replacement (4) when k = 1 so that x(sk−1 ) = 0.
Given that (13) holds, we can find the optimal investment time sk from the first
order condition of the problem:
c1 r − [x(sk ) − x(sk−1 )] Q +
x0 (sk )Q
= 0.
r
(14)
Let us again carry out implicit differentiation of the first order condition (14) with
respect to land quality, Q:
0
x(sk ) − x(sk−1 ) − x (sr k )
∂sk (Q)
h 00
i .
=
k)
∂Q
0 (s )
Q x (s
−
x
k
r
(15)
As discussed above, the denominator of this expression is negative. Hence, the sign of
∂sk
∂Q
depends on the sign of the numerator. Specifically, it can be shown that
r
x̄ − x(sk )
<
,
η
x(sk ) − x(sk−1 )
and negative if
r
η
∂sk
∂Q
> 0 if
(16)
is larger than the RHS expression. Over time, as x(sk ) → x̄, the
RHS of (16) approaches zero and the condition must eventually be violated. Hence, for
a sufficiently low initial technology level x(sk−1 ) so that condition (16) holds initially,
the sign of (15) will shift from negative to positive over time.
We could now easily find the optimal time(s) of investment given the technological
progress (2) as we did when analyzing the base model above. The only difference is
that we now must consider whether an upgrade is profitable in the future once an
investment is made. As the technology index x approaches x̄ over time, such upgrades
becomes less and less likely. This is captured by condition (13) where the difference
9
between x(sk ) and x(sk−1 ) approaches zero as time goes to infinity.
This analysis has shown that even if we allow for technological upgrades on land
that has already been used for wind energy production, the optimal investment policy
can involve a shift as the rate of technological progress falls relative to the discount
rate. The optimal policy is to invest on land of relatively low qualities first while
saving high quality lands for when the technology is better. At some point we switch
and begin developing the best land units first, working our way down towards worse
quality land.
4
Solving the stochastic problem
The stochastic dynamic optimization problem we face, can be treated as a real options
problem. Each land unit represents an option to invest, and we can solve the problem
by finding the critical value of the technology index x∗i such that it is optimal to invest
on land i once x ≥ x∗i .
The opportunity to invest in wind capacity on land i does not generate any costs or
revenues until the investment is undertaken. Hence, the only return on the option is its
capital appreciation. Therefore, for values of x < x∗ , that is, for x in the continuation
region, the Bellman equation is:
rV (x)dt = E (dV (x)) .
(17)
The Bellman equation states that the expected return on the option to investment
per time increment rV dt, must equal the expected rate of capital appreciation from
holding the option E(dV ).
Since x evolves according to the stochastic process (2), we must use Itô’s lemma to
find dV :
1
dV = V 0 (x)dx + V 00 (x)(dx)2 .
2
(18)
Substituting in for dx from (2) and noting that E(dz) = 0 we obtain the following:
1
E(dV ) = η (x̄ − x) V 0 (x)dt + σ 2 V 00 (x)dt.
2
(19)
Hence, we can rewrite the Bellman equation (17) as follows:
1
η (x̄ − x) V 0 (x) + σ 2 V 00 (x) − rV (x) = 0.
2
10
(20)
Table 1: Parameter values
Parameter
1
ρ = 1+r
c0
x̄
σ
Value
0.9
0.5
0.75
0.01
Description
Discount factor
Cost of investment
Long-run average, technology index
Variance, technology index
In addition, the value function V (x) must satisfy the following boundary conditions:
x∗ Q
r
Q
V 0 (x∗ ) = ,
r
V (x∗ ) =
(21)
(22)
where x∗ is the level of the technological index that should trigger investment. Condition (21) is the value matching condition, which is included to ensure that the value
function is continuous in the point where we exercise the option to invest in wind generation on the unit of land. Condition (22) is the smooth pasting condition, which together with the value matching condition, ensure that the value function joins smoothly
(meets tangentially) at the exercise point x∗ (Dixit, 1993).
4.1
Numerical analysis
To do the numerical analysis, we convert the problem to discrete time. The technology
dynamics (2) then becomes xt+1 = xt + η (x̄ − xt ) + σt . We assume parameter values
as given in table ??.
More...
5
Policy implications
To be written.
6
Conclusion
To be written.
11
References
Boyce, J. & Nøstbakken, L. (2011). Exploration and development of us oil and gas
fields, 1955-2002. Journal of Economic Dynamics and Control, 35(6), 891–908.
Capozza, D. & Li, Y. (2002). Optimal land development decisions. Journal of Urban
Economics, 51(1), 123–142.
Clarke, H. & Reed, W. (1988). A stochastic analysis of land development timing and
property valuation. Regional Science and Urban Economics, 18(3), 357–381.
Cunningham, C. (2006). House price uncertainty, timing of development, and vacant
land prices: Evidence for real options in Seattle. Journal of Urban Economics, 59(1),
1–31.
Dixit, A. (1993). The art of smooth pasting. Fundamentals of Pure and Applied
Economics. Amsterdam: Hardwood Academic Publishers.
Dixit, A. & Pindyck, R. (1994). Investment under uncertainty. Princeton, New Jersey:
Princeton University Press.
Fuss, S. & Szolgayová, J. (2010). Fuel price and technological uncertainty in a real
options model for electricity planning. Applied Energy, 87(9), 2938–2944.
IEA (2011). IEA Wind, 2010 Annual Report. Technical report, International Energy
Agency.
Junginger, M., Faaij, A., & Turkenburg, W. (2005). Global experience curves for wind
farms. Energy policy, 33(2), 133–150.
Kumbaroğlu, G., Madlener, R., & Demirel, M. (2008). A real options evaluation model
for the diffusion prospects of new renewable power generation technologies. Energy
Economics, 30(4), 1882–1908.
Lesser, J. & Su, X. (2008). Design of an economically efficient feed-in tariff structure
for renewable energy development. Energy Policy, 36(3), 981–990.
Perez-Barahona, A. & Zou, B. (2006). A comparative study of energy saving technical
progress in a vintage capital model. Resource and Energy Economics, 28(2), 181–191.
Roques, F., Hiroux, C., & Saguan, M. (2010). Optimal wind power deployment in
Europe — a portfolio approach. Energy Policy, 38(7), 3245–3256.
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