WPS5516
Policy Research Working Paper 5516
Infrastructure Investments under
Uncertainty with the Possibility of Retrofit
Theory and Simulations
Jon Strand
Sebastian Miller
Sauleh Siddiqui
The World Bank
Development Research Group
Environment and Energy Team
January 2011
Policy Research Working Paper 5516
Abstract
Investments in large, long-lived, energy-intensive of this option. However, the future retrofit option also
infrastructure investments using fossil fuels increase induces more energy-intensive infrastructure choices,
longer-term energy use and greenhouse gas emissions, partly offsetting the direct effect of having the option
unless the plant is shut down early or undergoes on anticipated energy use. Efficient, forward-looking
costly retrofit later. These investments will depend on infrastructure investments have high potential for
expectations of retrofit costs and future energy costs, reducing long-term energy consumption. Particularly
including energy cost increases from tighter controls if energy prices are expected to rise, however, the
on carbon emissions. Simulation analysis shows that potential for reduced energy consumption will be
the retrofit option can significantly reduce anticipated eroded if expectations of energy prices do not include
future energy consumption as of the time of initial environmental costs or future retrofit possibilities and
investment, and total future energy plus retrofit costs. technologies are not adequately developed.
The more uncertain are the costs, the greater the value
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the World Bank to provide open access to its research and make a contribution to development policy discussions around
the world. Policy Research Working Papers are also posted on the Web at http://econ.worldbank.org. The author may be
contacted at jstrand1@worldbank.org.
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Produced by the Research Support Team
Infrastructure Investments under Uncertainty with the Possibility of
Retrofit: Theory and Simulations*
By
Jon Strand
Development Research Group
Environment and Energy Team
The World Bank
Jstrand1@worldbank.org
Sebastian Miller
Research Department
Inter-American Development Bank
and
Sauleh Siddiqui
Department of Mathematics
University of Maryland, College Park
* This paper was prepared as part of a larger research program in the Development
Economics Group on infrastructure and its relation to greenhouse gas mitigation. Financial
support (to Strand) was provided by the Government of Norway; and (to Miller and Siddiqui)
by the Bank's Research Support Budget. The views expressed in the paper are the author's
alone and not necessarily those of the World Bank or its member countries.
1. Introduction
This paper presents an analytical model framework for choice of energy consumption in
relation to energy-demanding infrastructure investments, and simulations to illustrate
properties of this framework. An important premise for the analysis is that infrastructure
investments, sunk at an initial time of decision, "tie up" energy consumption for a long future
period. In our stylized and simple model, time is divided into two discrete periods: an initial
period 1, "the present"; and a "future" period 2, which may be much longer. All relevant
variables are assumed to be commonly known at the start of the period, for that period.
Several key cost variables in period 2, including energy and environmental costs and certain
technological costs (so-called "retrofit" costs, to be explained below), are unknown in period
1, but ex ante statistical distributions, for their period 2 realizations, are assumed to be known
(or knowable) at the time when the initial infrastructure investment is sunk in period 1.1 We
assume that the infrastructure persists throughout both periods, but may be (deliberately, and
at no additional cost) shut down at the start of period 2. The energy consumption tied to the
infrastructure is by assumption based on fossil fuels, at least initially.
The motivation for our analysis is that many types of infrastructure lead to considerable
climate policy "inertia", in that they establish levels of fossil-fuel consumption that may be
difficult to reduce later. As is increasingly recognized in the literature, including Ha-Duong et
al (1996), Wigley (1996), Ha-Duong (1998), Lecocq et al (1998), and Shalizi and Lecocq
(2009), the presence of such an established infrastructure may form a major ex post obstacle
to effective mitigation policy, for a long future period, possibly 50-100 years or more. This
holds regardless of whether the initial infrastructure investment is "optimal" (in an ex ante
sense), or not. The particular problem with infrastructure investment in this context is that
costs of mitigation or abatement, to reduce emissions to desirable levels, may, at least in some
cases, be very high ex post after the infrastructure has been established.
Below we discuss reasons why the infrastructure investment may turn out not to be optimal.
The bias is then typically in the direction of too high fossil energy consumption and carbon
emissions. Potential reasons are many and include systematic under-valuation of future
energy costs; failures to incorporate true (current and future) social carbon emissions costs;
and excessive discounting.
In our analysis we use a very stylized model of infrastructure investment, introducing two
potential mechanisms by which the fossil-fuel consumption can be modified "ex post" (in
period 2).
The first is to "retrofit" the infrastructure in period 2, at a cost. After a "retrofit", we assume,
infrastructure operation causes no emissions of greenhouse gases (GHGs) from then on. A
"retrofit" can be interpreted in several ways. First, infrastructure may be viewed as run
without any use of fossil fuels from then on. This is particularly relevant when fossil fuels are
replaced by alternative (non-fossil) energy sources, and the use of these sources reduces or
eliminates the emissions of GHGs from the "normal" operation of the infrastructure. But this
1
This of course is a simplification at least relative to some presentations where the distribution of future energy
and environmental cost is assumed to be unknown at an initial stage; this can lead to some serious problems of
inference as argued e g by Weitzman (2009). Here, we may use a standard Bayesian approach to justify our
position; namely, as the "best initial assessment" of the distribution given our current knowledge. See also
Geweke (2001) and Schuster (2004) for formal justifications of this approach.
2
interpretation is also relevant when applied to cases where the overall energy demand of the
infrastructure is (dramatically) reduced. In our stylized model, emissions are also in such
cases assumed to be eliminated completely. With either interpretation, the existence of a
potential retrofit option in no way implies that retrofit is necessarily optimal; in many cases,
exercising the option of retrofit would be prohibitively costly.
Alternatively, a "retrofit" could mean that the consumption of fossil fuels in operating the
infrastructure is unchanged, but that the carbon is removed from these fuels (through carbon
capture and storage, CCS, or similar technologies or processes). It can however be a more
problematic interpretation in our model given that the post-retrofit operating period T is
variable, since the retrofit cost is then "periodized" and assumed constant "per time unit"
within period 2. When T is variable, retrofit costs with a given distribution G will correspond
to a variable total retrofit cost (as it would be proportional to a variable T). When the retrofit
cost in fact represents a given sunk cost initially in period 2, the G function will need to be
amended when T changes.
Both "retrofit" and basic energy/climate cost in period 2 are assumed to be uncertain in period
1, but with initially known joint distribution, assuming that the two costs are non-negatively
correlated. Retrofit costs could easily be more uncertain than energy and climate costs, since
the nature of future (period 2) retrofit technologies is likely to be unknown at the start of
period 1 when initial infrastructure investments must be made.
The second way to avoid energy consumption related to existing infrastructure in period 2 is
to abandon it, "close it down". Since this investment is then rendered worthless, this is a
wasteful, and painful, option since the initial infrastructure investment is costly to establish. It
can still be attractive and rational ex post, given that energy and/or retrofit costs of continued
operation both turn out to be both very high: the lower of the two costs must be higher than
the utility value of continued operation. The abandoned infrastructure will then, presumably,
be replaced by alternative, less energy-intensive, infrastructure in period 2. The probability of
such a simultaneous occurrence is greater when energy and retrofit costs are positively
correlated. Our closedown alternative represents a "benchmark" case with zero emissions and
energy consumption; and we do not specify any particular replacement alternative.2
We characterize ex ante strategies for establishing energy and emissions intensity associated
with the initial infrastructure investment; ex post strategies for retrofitting and operating the
infrastructure at a "later" stage (in period 2); and interactions between these strategies. We
characterize optimal infrastructure investment, and (important from a climate policy
standpoint) identify factors behind inefficient (too energy and emissions intensive)
infrastructures being established. We also study whether, and to what extent, an initially high
energy intensity level can be modified in later periods through retrofit or closedown, in cases
where energy and environmental costs are high.
"Optimal" infrastructure choice is defined for given current prices, and given distributions of
future prices. "Optimality" can be established either for a private agent making the
infrastructure decisions, or for a social planner; which if these will be invoked in the
following will depend on the context. The decision-making agent could be private, but in
most cases a public-sector entity (a local or national government). A social planner, taking a
2
One possible interpretation of this case is that energy consumption and emissions in the closedown alternative
serve as a reference "zero" point, relative to the "business-as-usual" and retrofit alternatives.
3
national, regional or global perspective, will tend to incorporate prices, costs, discount rates
etc. given at the respective (national, regional or global) level, and, we assume, optimally
from that particular point of view. A fundamental problem with this, in a climate policy
context, is that such a view tends not to be correct, even when formulated at the national level.
As a key feature of the GHG emissions control problem, a global view is needed, where the
marginal externality cost at the global level is incorporated. Local decision makers are likely
not to behave in a globally optimal way, except when international agreements dictate that
globally optimal (emissions and energy) prices be applied. Little today indicates that such
"optimal" prices will be applied in the near or intermediate future; thus a discrepancy between
the ideal, global, social planner and the practical decision maker will be the order of the day.
One objective of our analysis is to study how much such a decision maker is likely to deviate
from a "socially optimal" decision (from a global point of view).3
Increased availability of retrofit and closedown options affect expected fossil-fuel
consumption and GHG emissions over the lifetime of a given infrastructure in two opposing
ways. First, they reduce expected fossil-fuel consumption and emissions through the option to
avoid such consumption and emissions ex post, in states where emissions and energy costs are
particularly high, where the infrastructure can then instead be retrofitted or closed down. On
the other hand, an anticipated increased availability of such options serves to increase the
chosen energy intensity embedded in the infrastructure when established. A greater
availability of the two additional options make it less risky for the decision maker to choose a
high initial (fossil-fuel) energy efficiency.
Simulations, in section 3 below, show that a higher variance on retrofit and/or energy costs
(for given unconditional expectations of these costs) reduces expected future costs, both in
energy terms, and (energy plus retrofit) costs terms. With more uncertainty about both energy
and retrofit costs (for given unconditional expectations), the retrofit option is exercised in
more cases, and to greater benefit (more states with "gainful" retrofit).4 Greater variances are
beneficial as they open up for more low-cost alternatives, which are exploited under an
optimal ex-post policy.5 The gains from greater cost variability are reduced when costs are
positively correlated (assumed in some of our simulations).
The distribution of retrofit costs in period 2 depends on technological retrofit possibilities,
which in turn are affected by R&D efforts to develop such technologies. While R&D efforts
are not modelled explicitly in the paper, we study exogenous shifts in the distribution for such
costs. When such cost reductions are correctly anticipated at the time infrastructure
investments are made, they lead to increased energy intensity of the initial infrastructure
investment choice. Overall expected energy consumption over the lifetime of the project may
then either increase or decrease, depending on which of two effects is stronger: this initial
effect on infrastructure energy intensity, which results in higher energy consumption in
"business-as-usual" states (from the higher energy intensity of the initially established
infrastructure); or the lower energy use ex post, due to "business-as-usual" states being fewer
and the infrastructure expected to be retrofitted in more states (in period 2 of our model).
Simulations, in section 3 and appendix B, show that the latter factor tends to dominate in our
model; except possibly in a special case where the initial energy consumption is very sensitive
3
See Strand (2009) for further elaboration on these issues.
4
This conclusion holds when the decision maker is risk neutral as assumed in our presentation; it may need
qualification under risk aversion.
5
This result holds when decision makers are risk neutral, which is assumed here. Under risk aversion, the utility
effect of greater uncertainty could here go either way.
4
to future energy costs (when the utility function for infrastructure services takes a Cobb-
Douglas form). Overall expected energy use and climate impacts are reduced with more
energy technology R&D, resulting in lower expected retrofit costs. Perhaps equally
reasonably, however, (major) reductions in expected retrofit costs will not be fully anticipated
at the time infrastructure investments are made (or, at least, are not fully built into these
infrastructure decisions). In such cases, reductions in expected infrastructure costs will reduce
expected energy costs by more, as they are not accompanied by offsetting effects (in terms of
higher energy intensity) for the initially established infrastructure.
Overall expected retrofit costs may in such cases increase or decrease. The factor contributing
to a decrease is the very drop in cost. The factor contributing to an increase is the retrofit
option being exercised in more states of the world (thus avoiding energy expenditures in more
ex post states). Typically here also, the first factor seems to dominate.
More interesting from a policy perspective is what happens in response to an upward shift in
the ex ante distribution of period 2 energy costs, here taken to include any climate or other
environmental costs resulting from energy use; a key issue in a climate policy context. We
study the system's response to increases in the future cost of carbon emissions and/or energy
that may or may be not fully anticipated at the time when major infrastructure investments are
made. Our model provides answers to both the following two questions: What is the fossil
energy use (and carbon emissions) response, when these future cost increases are fully
anticipated, and expected to be incurred? And what is the response when they are not
expected to be incurred (as may be the case for some climate-related costs)?
When an increase in expected future energy cost is not anticipated at time of investment, the
embedded energy intensity in these investments will be excessive. Our simulations here
indicate that when a doubling of the energy/environmental cost can be anticipated, and this is
not considered in making the infrastructure investment, it could result in an overall increase in
fossil energy consumption by 30-60 percent, even when the ex post response to the energy
price increase is optimal. It is here still the case that expected energy use and emissions will
be reduced ex post in proportion to the increased use of the retrofit option. Most of this
reduction is realized in states with high energy costs, thus leading to (perhaps substantial)
expected energy cost savings.
When the increase in future expected energy cost is correctly anticipated at time of
infrastructure investment, lifetime expected energy use and emissions are reduced by more
than when higher energy costs are not anticipated. The additional factor is that the energy
intensity of the initial infrastructure is reduced. We show, mainly through simulations, that the
compounded effect of these two factors can be substantial, and that greater uncertainty about
energy and retrofit costs adds to this effect.
When an increased future energy and climate cost is not expected to be incurred (but correctly
anticipated) by the decision maker (be it a national or local government, or a private party), it
has no effect on decisions, neither ex ante nor ex post. This implies policy failures in two
different respects: both making the infrastructure excessively energy intensive; and allowing
for too few retrofits ex post. When again there is a doubling of the energy price which is not
considered, our simulations here show (see table 1 in the final section 4) that the degree to
which energy consumption is excessive could be dramatic, and could exceed 200 percent,
when compounding the effect of excessively energy-intensive infrastructure ex ante, and
excessive energy consumption ex post.
5
The analytical and quantitative literature dealing with such issues is small. Arthur (1983),
David (1992) and Leibowitz and Margulis (1995) provide background by defining and
discussing the issue of path dependency and its implications for future actions. The more
specific topic of infrastructure choice and its implications for mitigation policy is discussed
only recently. Shalizi and Lecocq (2009) provide a discussion of infrastructure costs and
constraints which is more applied and intuitive than that provided here. The persistent effects
of infrastructure choice on energy consumption and carbon emissions are discussed also by
Brueckner (2000), Gusdorf and Hallegatte (2007a,b), and Glaeser and Kahn (2008). In
particular, Gusdorf and Hallegatte (2007a) study the energy intensity of urban infrastructure
for given population density. They focus in particular on inertia resulting from established
urban structure, in response to "low" initial energy prices, which may later rise. They show,
through simulations, that a permanent energy price shock leads to a transition period that is
long (20 years or more) and painful (with high energy costs, and carbon emissions), but that
energy consumption eventually tends to fall toward a substantially lower steady-state level.
Glaeser and Kahn (2008) by contrast focus on energy consumption implications of differences
in population density, both within urban regions and when comparing urban and rural
population patterns. In this context they seek to quantify relationships between energy
consumption and spatial patterns of cities in the U.S. They find, in particular, much lower per-
capita energy consumption, and carbon emissions, in central cities than in suburbs. This bears
on our analysis as it indicates that "compact" infrastructure (as found e g in central cities) is
less energy demanding than "less compact" (found e g in suburbs).
The option to retrofit already established infrastructure, by removing either the initial energy
requirement, or the carbon emissions associated with it (via CCS technology or replacing
fossil fuels by renewables), reduces the inertia associated with infrastructure. This is a focus
in this and two related papers, Strand (2010) and Framstad and Strand (2010). Strand (2010)
considers different utility function representations and their implications, in a similar setting.
Framstad and Strand (2010) study optimal infrastructure investment when future energy
prices follow a continuous stochastic process, where a delayed retrofit decision has a positive
option value. Implications of retrofit possibilities and costs are further discussed analytically
by Jaccard (1997) and Jaccard and Rivers (2007). The latter paper studies three more specific
types of demand-side infrastructure: urban structure; buildings; and equipment. The authors
argue, based on simulations (and using a discount rate of 3 percent), that for buildings, and
even more for urban structure, it is generally advantageous to make strong considerations for
future emissions even when emissions prices start low and increase strongly over time (while
this is often not the case for equipment where natural turnover provides sufficient flexibility).
Shalizi and Lecocq (2009) provide a broader and more practically oriented discussion, with
examples from both energy demand and supply; their overall argument is that energy-
intensive infrastructure involving supply is generally more rigid than that involving demand;
but sometimes (but not always) more prone to complete retrofit.
Other literature connections need mention. One is the "low-carbon society" and ways to
achieve it, treated by many; among others Strachan et al (2008a, b), and Hourcade and
Cerassous (2008).6 This basic idea is, as here that achieving a society with low GHG
emissions (necessary for efficiency in the long run) requires a high concern for infrastructure
investment design. Another connection is two World Development Reports, the WDR 2003
6
A very early champion of this line of thinking and discussion was Amory Lovins; see, in particular, Lovins
(1977).
6
(Sustainable Development in a Dynamic World; World Bank (2003)), and the most recent
WDR 2010 (Development and Climate Change; World Bank (2009)). In both, "inertia in
physical capital" (echoing our analysis of infrastructure) is a main theme.
2. Two-Period Model with Uncertain Retrofit Costs
2.1 The Basic Optimality Problem
Consider a world existing for two "periods". Infrastructure investment is made at the start of
period 1, and can be "retrofitted" at the start of period 2.7 As long as it is operated and not
retrofitted, a given infrastructure gives rise to a given energy consumption per unit of time,
determined at the time of initial investment. Energy supply costs and environmental/climate-
related costs are uncertain at the time of establishment in period 1, but are revealed at the start
of period 2. We assume that when retrofitted, the infrastructure is purged of all fossil-fuel
energy content and/or all its carbon emissions. The infrastructure however still provides the
same utility services to the public as it did before the retrofit. "Retrofits", we assume, are not
available in period 1: they represent a new technology, developed and available at the start of
period 2.
Period 1 has unit length, while T is the "length" of period 2. T can in principle be given two
alternative interpretations. First, it could simply be interpreted as the time elapsing during
period 2, relative to the initial (unit) period. To invoke this interpretation in the following, it
would need to be coupled with an assumption that decision makers choose a zero discount
rate. Alternatively, T could embed discounting, in which case it would represent the
discounted value of period 2 relative to that of period 1.8 Under this interpretation, heavier
discounting would lead to reduced T for given period length.
We also assume that the infrastructure can in principle be shut down at the start of period 2.
Such action will be taken when the total utility of operating the infrastructure is less than the
minimum of the energy cost of operation, and the retrofit cost, in period 2.
In period 1, the unit energy cost is q1 (given and constant).9 The policy maker decides on an
infrastructure investment with given capital cost K. For simplicity and to focus on other issues
than investment size, assume that all relevant infrastructure projects have the same investment
cost. Infrastructure type is identified by a given energy intensity H, where we assume that all
energy consumption associated with the infrastructure is fixed once the infrastructure is
established, and until it is possibly retrofitted. Considering only economically viable projects,
we focus on one particular trade-off only: an infrastructure project with higher energy content
must give higher immediate utility, but will be more costly to operate (as long as not being
7
In the model as it otherwise stands, the assumption that a retrofit can be done only at the start of period 2, and
not during this period, is no limitation as, we assume, no new information (nor any new or better retrofit
technology) will be forthcoming during period 2.
8
More precisely, when unity represents the present discounted value of a current income flow of one dollar
throughout period 1, T would in this case represent the present discounted value of a current income flow of one
dollar throughout period 2, as evaluated from the start of period 1.
9
In the continuation, when we say "energy cost", we mean the combined energy and environmental cost
associated with (fossil-fuel) energy use. This would be unproblematic when all environmental costs are charged
to energy use in the form of energy taxes and quota prices. It is more problematic when this is not the case; this
issue is elaborated in the final section.
7
retrofitted) due to its greater fossil-fuel energy requirement. Denote the current (per time unit)
utility flowing from the infrastructure when being operated by U(H), where U'(H) > 0, U''(H)
< 0. We assume that U(H) is given and constant and the same in both periods (and thus not
subject to uncertainty).
Three alternative actions may be chosen in period 2:
1. No new action, proceed with "business as usual." In this case the full energy cost will
be incurred in period 2. This is the optimal strategy when the energy cost in period 2
turns out to be lower than both the retrofit cost, and the period-2 utility of the
infrastructure.
2. Retrofitting the infrastructure. This is the optimal strategy when the retrofit cost in
period 2 turns out to be lower than either the energy cost, or the period-2 utility of the
infrastructure.
3. Infrastructure closedown. This is optimal when environmental and retrofit costs both
turn out higher than the period-2 utility of the infrastructure. Closedown is a drastic
measure. It will typically require that other infrastructure be supplied, to replace the
services lost by project closedown. This is not explicitly modelled here. Implicitly our
model embeds such effects, via the absolute value of the utility flow provided by the
infrastructure (which should be defined relative to a situation where the utility is
missing; thus a "relatively drastic" alternative).
The problem of a decision maker in establishing the infrastructure in period 1 is to select an
energy investment intensity H so as to maximize
(1) EW (1) U ( H ) q1 H EW (2)
where E is the expectations operator and W(2) is the (optimized) value function associated
with the infrastructure in period 2 (embedding the optimal action, among alternatives 1-3
above).
EW(2) embeds the decision maker's optimal responses at the start of period 2 (assuming that
no further changes occur during period 2). Define F(q,y) = Fqy as the (continuous ex ante,
when viewed from period 1) cumulative bivariate distribution over q and y levels to be
realized in period 2, with support [0, qM] × [0, yM], where qM and yM could be large.10
Possibly, the marginal distribution Fq for period 2 is shifted up by increased emissions in
period 1. Here, y represents retrofit costs (per unit of energy capacity to be retrofitted) in
period 2. Retrofit costs cannot be negative, but could in principle be small in period 2,
depending on the technology available for substituting out the fossil-fuel energy consumption
or purging carbon from fossil fuels at that time. We assume that an infrastructure project, after
a retrofit, incurs no energy costs nor any other current costs in period 2, apart from the retrofit
cost itself (which in the model is "periodized" in the same way as energy cost).11 In the
10
In simulations below we assume that F is bivariate log-normal, in which case F is not bounded above (it is
however "thin-tailed").
11
Alternatively, the retrofit cost could be interpreted to include some energy cost. This is unproblematic as long
as the retrofit cost can be periodized.
8
analytical presentation, period 2 realizations of energy cost and retrofit cost are not assumed
to be independent.12
Consider the choice between the three alternatives lines of action 1) 3) in period 2. We start
with action 3), project closedown. Define total utility per energy unit for installed
infrastructure by U(H1)/H1 = y*, where H1 is energy intensity associated with the
infrastructure investment chosen in period 1. Action 3) will then be chosen when the cost per
unit of energy q, and the retrofit cost y, both exceed y*. The probability of this event, when
viewed from period 1, is
(2) P (3)
q y* y y*
f (q, y )dqdy .
Assume 0 < y* < min { qM, yM }, and 0 < F(y*, y*) < 1, implying P(3) > 0.
The probability P(1) of inaction (action 1), is given by the following expression:
y*
(3) P(1) f (q, y)dqdy .
y q q 0
The probability P(2) of retrofit (action 2) is complementary (equal to 1 P(1) P(3)), but can
also be found in a similar way as P(1), as follows:
y*
(4) P ( 2) f (q, y)dydq .
q y y 0
Note that we can switch the order of any double integrals with an application of Fubini's
theorem.13 Define the following (unconditional) expected unit cost variables in period 2,
related to energy cost and retrofit cost respectively, by
(5) Eq (2) f (q, y)dydq
q 0
q
y 0
(6) Ey (2) f (q, y)dqdy .
y 0
y
q 0
These are costs that would be realized given that, in the first place, "business as usual" energy
use is applied in all states in period 2; and in the second place, retrofit is applied in all states in
12
Correlation of energy and retrofit costs is often realistic. Such correlation could in principle be either positive
or negative. Negative cost correlation could occur when the energy cost in period 2 is anticipated during the
period of R&D efforts to develop new retrofit technologies. A high anticipated energy cost could then make the
development of retrofit technologies more urgent, and more effort expended for that purpose. This endogenous
process could then lead to negatively correlated costs. On the other hand, common drivers may affect both costs
in the same direction. This is relevant e g when energy cost is correlated with general production cost; when a
retrofit involves some use of fossil energy; or when the subsequent use of renewable energy whose marginal
production cost is positively correlated with the cost of fossil fuels. In such cases the two cost variables would
tend to be positively correlated. This corresponds to our main assumption, for the simulations below.
13
See e g Royden (1988).
9
period 2. These are not realized costs and instead merely benchmarks (both excessive), for
two main reasons: the lower of the cost alternative is applied given that one of them is
applied; and in some states the third alternative (closedown) implies lower costs.
Considering now actual realized costs, the expected "per time unit" period 2 energy and
retrofit cost as viewed from period 1, given an optimal strategy for period 2, are respectively
y*
(7) E CH (2) q f (q, y )dydq H 1 Ech(2) H 1 ,
q 0 y q
y*
(8) E CR(2) y f (q, y )dqdy H 1 Ecr (2) H 1 .
y 0 q y
E[CH(2)] expresses energy costs per "time unit" in period 2, while E[CR(2)] similarly
expresses retrofit costs when similarly periodized (counted per period unit), under an optimal
decision rule for ex post factor choice ("normal operation"; retrofit; or closedown). H1
denotes (per-period) energy consumption associated with the infrastructure as established in
period 1. Define E[C(2)] = E[CH(2)] + E[CR(2)], as well as Ec(2) = E[C(2)]/H1. In
particular, we must then have
Ec(2) < (1-P(3))min[Eq(2), Ey(2)],
as long as all three ex post policy alternatives: "business as usual", retrofit, and closedown,
are actually exercised in period 2.
A factor not apparent from this derivation is that two alternative interpretations of retrofit
costs are likely to have somewhat different implications for the analysis. The first is to view
retrofitting simply as replacing a fossil fuel with a non-fossil fuel which gives rise to no GHG
emissions. Retrofit costs are then incurred currently in the same way as regular (fossil) fuel
costs. Under the second interpretation, retrofit costs represent an investment to remove the
fuel need tied to the infrastructure, or the emissions associated with the fossil fuel. In (8), this
implies that E[CR(2)] must be "periodized" (given that T is different from unity), and spread
evenly across T time units in period 2.
Expected (discounted) net utility from the infrastructure when operated in period 2 is denoted
EW(2), and equals the gross utility of the infrastructure, TU(H1) = Ty*H1, in states where it is
not closed down in period 2 (thus with probability 1-P(3)), minus total combined expected
energy and retrofit costs, T{EC(2)} (= T{E[CH(2)] + E[CR(2)]}), over states where the
infrastructure is operated (without, or with, retrofitting). We then have
(9) EW (2) { y *[1 P(3)]H1 E[CH (2)] E[CR(2)]}T .
The first-period decision problem is formulated as maximizing the expected utility of the
infrastructure investment in period 1, considering an optimal strategy in period 2. Define
(10) EW (1) U ( H1 ) q1H1 EW (2) ( y * q1 ) H1 EW (2) .
10
Assume for now that the distribution of period 2 energy (including environmental) costs is
exogenous (and not affected by emissions arising from the infrastructure). The solution to the
maximization problem in period 1 takes the form
EW (2)
d
(11)
dEW (1)
U '( H1 ) q1
EW (2)
[U '( H1 ) y*] H1 .
dH1 H1 dy *
From (9) we find
EW (2)
d
H1
(12) [1 P (3)]T .
dy *
Using the definition of EW(2), and setting the derivative in (11) equal to zero, we find the
following implicit expression for the optimal energy intensity of the infrastructure:
E (CH (2) E (CR (2))
q1 T
H1
U ' (H1 )
1 1 P(3)T
y* y*
(13) q1 q f (q, y )dydq y f (q, y )dqdy T .
q 0 y q y 0 q y
y* y*
1 f (q, y )dqdy f (q, y )dydq T
y q q 0 q y y 0
1+[1-P(3)]T equals the expected number of time units that the infrastructure will be operative
during period 2. {E(CH(2))+E(CR(2))}T = {EC(2)}T is the ex ante expected (energy plus
retrofit) cost in period 2, which divided by H1 is measured relative to energy intensity as
established in period 1. {EC(2)}T represents total energy costs plus retrofit costs during the
period 2 expected operation time [1P(3)]T. The expression in the curled bracket in the
numerator of (13) then denotes expected energy plus retrofit cost per unit of energy
consumption defined by the established infrastructure. (13) can then be given a simple
interpretation: the marginal utility of increased energy intensity associated with the
infrastructure investment (the left-hand side of (13)) should equal the average energy plus
retrofit costs incurred over the lifetime of the infrastructure capital (the right-hand side).
The optimal energy intensity is chosen according to average energy cost in operating the
infrastructure, over its expected period of operation. Perhaps surprisingly, the extent of the
operation period as such is not very important for the chosen intensity.14 For e.g. a
hypothetical case where the infrastructure is shut down in period 2 for certainty; the energy
intensity would be determined simply by U'(H1) = q1. When average energy cost in period 2
exceeds q1, U'(H1) > q1, and H1 lower (since the required utility return per unit of
infrastructure must be higher which requires a lower infrastructure mass).
14
This result is closely related to our initial assumption, that the size of the infrastructure investment is
exogenously given.
11
For y* (= U(H1)/H1) we find
dy * 1
(14) yH * ( y * U ') .
dH1 H1
Here y*-U' must be positive and more so the more curved the utility function is in the
neighborhood of H1. Thus we can expect yH* < 0.15 This implies that a more energy-
demanding infrastructure will have a lower threshold for operation, and will be "closed down"
(or rather, replaced) ex post in more cases in period 2, when energy and retrofit costs increase.
This may appear reasonable; remember that in our model all infrastructure projects are
considered to be "equally large" in the sense of requiring the same initial investment cost.
What distinguishes projects is the ex post energy requirement for their operation (or put
otherwise, their energy intensity).
Appendix A1-A2 deals with comparative statics for the above model, when there are period 2
shifts in the distribution of energy costs (A1), and in the distribution of retrofit costs (A2), for
a simplified case with independent costs.
2.2 Analytical Specification of the U Function
We will in this sub-section consider the main class of the utility function specification, U(H1),
that will be applied in the simulations in section 3 below. Our basic assumption is that U(H1)
belongs to the class of constant relative risk aversion utility (CRRA) functions. The (vNM)
utility function then takes the following general form:
H11
(15) U ( H1 ) A K,
1
where A and are positive parameters, and K is a (non-negative) scaling parameter ensuring
that U only takes non-negative values on its relevant ranges.16 The (Arrow-Pratt) measure of
relative risk aversion is given by . Define [q+Ec(2)T]/[1+(1-P(3))T] = Q as the expected
total unit cost of operating the infrastructure over its lifetime, per expected time unit of
operation (where T is the number of time units within period 2, and 1-P(3) is the probability
of operation in period 2), and per unit of established infrastructure, H1. We have defined Ec(2)
= E(C(2))/H1 (as the average expected realized cost per time unit in period 2, per unit of
invested H1), which from (7)-(8) is constant for given distributions of q and y. Maximizing ex
ante net utility associated with the initial infrastructure investment, given by U(H1) Q, we
derive the first-order condition
(16) AH1 Q
Here all magnitudes except H1 can be viewed as constants. Consider the effect on H1 when the
cost variable Q changes. This is found as
15
Another way of interpreting this can be found by considering that y* is an expression of the average utility of
infrastructure per unit of H1. yH* < 0 then simply expresses that average utility of infrastructure must exceed its
marginal utility at the point of indifference between operation and closedown.
16
This issue is relevant only for case c) below, where U would otherwise take only negative values.
12
dH1 1 H1
(17)
dQ Q
From (17), the "elasticity of demand for energy intensity", H, is found as
dH1 Q 1
(18) H .
dQ H1
Thus the energy intensity demand elasticity with respect to total operating cost per unit of
established infrastructure (but where the initial infrastructure cost is not included) is a
constant.17 All possible values of from zero to infinity are allowed by this formulation. The
effect of an increase in Q on total expected costs (for energy and retrofit combined) is
d ( H1Q) dH 1
(19) H1 Q 1 H1 .
dQ dQ
We can now easily distinguish between three distinct classes of cases for (15), namely
a) 0 < < 1: U takes a Cobb-Douglas form. Then d(H1Q)/dQ < 0. Overall expected ex ante
(total) operating costs, including the response of the initial energy intensity (and in terms of
both expected energy costs, and expected retrofit costs), are then reduced in response to an
increase in expected ex ante unit cost. The reason is that the energy intensity chosen in
response to a change in the cost variable is then reduced relatively more than the change in
unit costs.
b) = 1: U takes a logarithmic form. Then d(H1Q)/dQ = 0. Overall expected ex ante (total)
cost, including the response of the initial energy intensity, is constant in response when
expected ex ante unit operating cost increases. The energy intensity is reduced in response to
a change in the cost variable, so as to exactly offset the increase in unit expected ex ante cost.
c) > 1: U is exponential. Then d(H1Q)/dQ > 0. Overall expected ex ante operating cost,
including the response of the initial energy intensity, here increases in response to an increase
in expected ex ante unit operating cost. The energy intensity is reduced but by relatively less
than the change in energy costs.18
Under this class of utility functions we have the following general expressions for y*
(expressing the ex post marginal cost beyond which the infrastructure will be shut down in
period 2), and its change with H1:
A
(20) y* H1 KH11
1
17
Remember here our initial assumption, that all possible infrastructure projects have the same given initial
investment cost; infrastructures differ only in their dimensioning of H1, and their utility (which is higher for
higher H1).
18
Note also that the utility function in this case takes only negative values for K = 0 in (15), and tends to minus
infinity as H1 tends to zero. A meaningful application of the model to this case (in particular, a meaningful
definition of the `closedown" limit operation price y*) would require K > 0 and that we restrict attention to H
levels
13
dy * 1 K
(21) yH * A H1 .
dH1 H1 1 H1
As noted, under case c) one would here need to impose a minimum positive value of K.
The CRRA class, embedding all cases a) - c) where the coefficient of relative risk aversion is
allowed to vary from zero to infinity, can be viewed as sufficiently wide for our purposes. In
particular, it embeds all relevant possible values of the (crucial) elasticity of demand for
energy intensity in infrastructure with respect to expected operating cost. The only effective
constraint on this class is the constancy of this elasticity, for any chosen value of .
2.3 Optimality, Deviations from Optimality, and the Relation to This Analysis
"Optimality" will here be defined alternately by a (global) social planner, or by the local
decision maker choosing the initial infrastructure investment (who is typically not a global
planner). It is here useful to identify different sources of deviations between the two concepts,
with resulting inefficiency in deciding the initial infrastructure investment, when decisions are
otherwise "optimal" (from the point of view of the decision maker). We next wish to identify
implications of "non-optimal" choices by the actual (local) decision maker, or choices made
under incorrect information. We distinguish between the following five issues (spelled out in
more detail in Strand (2010)). This classification is particularly useful for interpreting the
simulation results in section 3 below.
A) The initially expected distribution of energy costs, relevant for making current
infrastructure decisions, is lower than (or more precisely, down-shifted relative to) the
true (correct) distribution. This could occur e.g. when the infrastructure decision is
based on an expectation of something close to current energy cost on average in
period 2, while the correct distribution implies higher average energy costs.19 In this
case we may expect the infrastructure energy intensity to be chosen at a too high level,
and fossil-fuel consumption in period 1 to be excessive. The period 2 realized
expected energy consumption is however not obviously higher than optimal in
consequence. This is because the closedown and/or retrofit options could then in
response be exercised in more cases.20
B) The distribution of energy costs facing the policy maker is correctly anticipated, but is
down-shifted relative to the "optimal" energy cost distribution. This is relevant
whenever the authorities, in the economy in question, implement emissions prices that
are lower than "globally correct" prices. This case is highly relevant: as of today,
hardly any country implements what most analysis would agree are "globally correct"
emissions prices; nor seems to be willing to do so in the foreseeable future. In this case
we can expect, unambiguously, excessive fossil-fuel consumption in both periods.
19
We argue that this could occur even in cases where the entity making the infrastructure decisions would face a
"true" energy/environmental cost. One such case is where the administrative procedure for making public
investments involves incorporation of future costs and benefits for a limited period (say, 20 years), while a
correct investment decision would require a much longer period (say, 50 years or more). An increasing energy
price (beyond the 20 year term) would add to the bias involved in such investment.
20
In our simulations below we however do not allow for this possibility at least not with respect to retrofit costs
which are throughout assumed to be proportional to energy costs, ex post.
14
C) Future retrofit costs are incorrectly anticipated. In particular, with a too optimistic
view, the anticipated retrofit cost distribution will be lower than the correct
distribution. This will bias the infrastructure decision in the direction of excessive
energy intensity for the initially established infrastructure. On the other hand, when
the distribution of future energy costs is too optimistic, the retrofit cost distribution
will be less important for the energy intensity decision, since the prior expectation is
then that retrofits are necessary in fewer cases. Instead of "technology optimism" one
could however have "technology pessimism" (a too pessimistic view of the
distribution of future retrofit costs), which would tend in the opposite direction.
D) Future retrofit costs are correctly anticipated, but are higher than socially optimal
costs. Such a case is where the R&D effort in developing new energy technologies is
suboptimal, including those for retrofits. Overall implementation costs (in particular,
minimum of energy and retrofit costs) would then be shifted upward in period 2.
Given correct perceptions in period 1, the choice of energy intensity of the
infrastructure is in response lower than optimal (efficient energy intensity is greater
than the one chosen), while the probability of ex post business-as-usual operation of
the infrastructure is higher than optimal (as there are more states where the energy
cost is below retrofit cost). This follows straightforwardly from the fact that the
business-as-usual alternative will be chosen in more cases in period 2, and the retrofit
alternative in fewer cases. With no other distortions, the energy intensity of the
initially chosen infrastructure would then in fact be suboptimal. On balance, the
second factor is likely to dominate, leading to socially excessive energy consumption
in period 2.
E) The policy-relevant value of T, call it T1, is less than the optimal value, call it T0.
Reasons for this may be either excessive discounting (for the case where T is
interpreted as a discounted value), or that the initial policy decision undervalues the
length of period 2. Since T, as noted at the start of this section, can be interpreted as a
discounted value of period 2 relative to period 1, we may have a discrepancy between
the "socially correct" value T0 and the value T1 used when deciding on H1. T1 < T0
could then reflect excessive discounting. When average costs per operation period are
greater in period 2 than in period 1 (as might be expected), this leads to a lower
average operation cost for the infrastructure, and a more energy-intensive
infrastructure.
In four of these cases (all except D), the overall expected fossil-fuel energy consumption (and
GHG emissions) over the potential lifetime of the infrastructure would tend to be excessive
from a social point of view, in the sense that it is higher than the expected fossil-fuel
consumption and emissions for the case where all global externalities are optimally
considered and anticipated.21
21
One might in addition argue that case D is less likely in practice: when actually realized retrofit costs tend to
be higher than their socially optimal level, this will, probably, tend not to be anticipated. If not, the end result
will be excessively energy intensive infrastructure, and overall excessive emissions.
15
3. Simulations
We will in this section illustrate properties of our model through four sets of simulations: 1)
ex ante expected probability of energy use or retrofit in period 2; 2) ex ante expected energy
and retrofit costs in period 2; 3) energy intensity of the infrastructure chosen in period 1; and
4) overall expected energy use, which combines the energy intensity of infrastructure (3) with
the probability of subsequent energy use (1). Under our (we think, reasonable) numerical
cases the closedown option turns out to play a very small role for the overall results, and will
not be emphasized in the following; but must still be kept in mind.22 All simulations are made
under the assumption that energy and retrofit costs are jointly log-normally distributed, and
calculations are done using the scientific program Matlab. Most of the simulations for
expected energy cost in period 2 (section 3.2) depart directly from the expected cost
expressions (7), and (8) for expected retrofit cost in period 2. For probabilities of using energy
and retrofitting respectively (section 3.1) we use (3) and (4), and for energy intensity of
infrastructure (section 3.3), we use (13). For expected overall energy use section 3.4), we
apply the solution value for H1 together with the ex ante probability of energy use, (3).
In all cases where nothing is otherwise stated, the (unconditionally) expected energy and
retrofit cost (per unit of energy use and retrofit investment respectively) are both kept
constant, at levels E(q(2)) = 2, and E(y(2)) = 3. (E(q(2)) is the unconditionally expected
energy/environmental cost in period 2; it would be the actual expected cost given no retrofit
or closedown. A similar interpretation holds for E(y(2)).) While both energy and retrofit costs
are uncertain, ex ante energy costs are thus generally lower in expectation. The distributions
of energy and retrofit costs are both assumed to be jointly log-normal, and either independent,
or have a positive correlation coefficient of 0.5. In sections 3.3-3.4, simulations are done for
all variants a) c) of the utility function (15).
3.1 Ex Ante Probability of Energy Use or Retrofit in Period 2
This first set of simulations considers implications of various parametric changes in the
model, for the expected value of (per-period) energy costs during period 2. It is useful to first
study period 2 behavior; this will be a benchmark for subsequent analysis of period 1
investment and overall energy use, in sections 3.3-3.4 below.
Figure 1 describes the probability that the firm in period 2 will continue using energy at the
level initially established in period 1, given a parametric change in expected energy cost, for
different degrees of uncertainty about both energy and retrofit cost.23 In all examples used in
figure 1, Ey = 3. We see, as should be obvious, that a low (high) q implies a high (low)
22
Two other sets of assumptions in the following also potentially be attacked. One is the log-normal distribution
assumption. Log-normality is however a rather robust assumption in this context; see e g Schuster (1984). The
other is our assumption on the utility function of energy intensity in infrastructure, where we assume a set of
functions parameterized by the demand elasticity with respect to price (thus constant for a given specified
function; the CRRA class of functions). While this general class of functions can be viewed as robust, criticism
can be raised in particular to the Cobb-Douglas and log specifications; see below.
23
Note that the probability of closedown is also accounted for in these calculations, but its probability turns out
to be too small to matter for the results. (Throughout, the value of continued use is set at 10, implying that this
option will be exercised only when min(q, y) > 10.)
16
probability of energy use. Also, this tendency is greater when variances are smaller.24 This is
because, with greater variances, there will be more attractive options on average, to substitute
out one variable for the other; and thus a smaller propensity to rely only on the factor that is
less expensive in expectation (which is energy for Eq 3, and retrofit for Eq > 3).
Figure 1: Probabilities of energy use and retrofit as functions of unit energy cost, for
independent costs
24
While this is shown in the figures only for simultaneous changes in both variances, the same basic result holds
when only one of the variances at the time is changed.
17
Figure 2: Probabilities of energy use and retrofit as functions of unit retrofit cost, for
independent costs
Figure 2 above similarly tells us how these probabilities change when expected energy cost is
kept constant and expected retrofit cost instead changes parametrically. A very similar pattern
emerges except that it is, of course, the probability of energy use that increases when the
expected value increases: the two figures are, in important ways, mirror images of one
another.
In appendix B (figures1.1-1.2) we consider similar calculations as those shown in figures 1-2,
except that the distributions for q and y are assumed to be positively correlated, with
correlation coefficient 0.5 (and not independent as here). We then find the distributions to be
quite similar, with the (small) change that P(1) is higher for low q, and lower for high q (in
figure 1.1). This is because positive cost correlation leaves less scope for gainful substitution,
of a cheaper alternative for a more costly, ex post.
The results in the appendix also show that the probability of using energy increases as the
variance of energy costs increases; and is reduced (and thus the probability of retrofitting
increased) when the variance of retrofit cost increases. This result follows because energy
costs are lower than retrofit costs in expectation; increased (reduced) uncertainty of retrofit
costs will lead to greater (smaller) probability that energy use is substituted out through a
retrofit.
18
Figure 3: Probabilities of energy use and retrofit, different degrees of uncertainty, as
function of degree of cost correlation
Figure 3 above describes effects of a parametric increase in the correlation coefficient for the
jointly lognormal distribution of energy and retrofit costs, going from zero to unity, for the
case of Eq = 2, and Ey = 3. We see that a more correlated cost structure generally implies that
the probability of energy use increases. This is intuitively reasonable as, in our example, with
perfect correlation we would have q < y always and thus energy use always preferred. With
less than perfect correlation, there will exist states where q > y and thus retrofit instead used;
and more such states when variances are higher. We however see, for the examples simulated,
that in no event is the probability of energy use (P(1)) below 0.8.
3.2 Ex Ante Expected Energy and Retrofit Costs in Period 2
We now consider simulations of expected costs in period 2, as viewed from period 1 and
where Ey = 3 (figure 4), and Eq = 2 (figure 5). Figure 4 considers parametric changes in Eq
under these assumptions, while figure 5 considers similar parametric changes in Ey.
Interpreting the figures requires some care. Note that, as Eq in figure 4 is increased
parametrically from values below 3, to values above 3, energy ceases to be the more
economical alternative on the average. Thus, these two curves always cross at Eq = 3 (in
figure 4), and Ey = 2 (in figure 5). Similarly in figure 5, as Ey is reduced below 2, retrofit
there takes over as the more efficient alternative on the average. Also, total expected cost will
always be an increasing function of both Eq (figure 4) and Ey (figure 5); and the highest value
the total expected cost curve can attain is 3 (figure 4) and 2 (figure 5). We see that total
expected cost is kept well under these levels, and more so for higher variances. This is again
intuitive: high variances give great option for ex post cost minimization thus lowering average
expected cost. We also see that, for the high-variance alternative in figure 4 (var(q) = var(y) =
19
9), conditional expected energy cost keeps increasing well beyond the crossing point 3 (as
there are "sufficiently many" states in which energy costs are chosen, so that the cost-
increasing effect of Eq going up dominates the overall expected cost picture).
Figure 4: Ex ante expected energy/retrofit costs in period 2 as function of unit energy
costs, for different variances, independent costs
Figure 5: Ex ante expected energy/retrofit costs in period 2 as function of unit retrofit
costs, for different variances, independent costs
20
Figures in appendix B describe how conditional expected energy, retrofit and total costs vary
with changes in var(q) and var(y). Under certainty no retrofits would be incurred in period 2
(since then by assumption y = 3, higher than q = 2). Under uncertainty, additional options
open up, as particularly high (energy, and retrofit) costs can be avoided, and cases with low
costs implemented. As a result, overall conditional costs will be reduced, and more so with
greater uncertainty, represented here by the variances of q and y. In figure 5, this feature is
seen to hold for partial increases in both variances. In particular, when both variances are
about 2.5, approximately half of (unconditional) energy cost is avoided, while half as much is
added in the form of retrofit cost. The total overall factor cost saving is then about one fourth.
Note also, as a general feature of the results from the simulations, that the factor with the
lower unconditional expected cost has the higher conditional expected cost. The reason is,
obviously, that when the unconditional expectation is lower, the respective alternative will be
applied in more cases (and the opposite alternative in fewer cases).
Figure 6 below describes how expected costs (for energy, retrofit and in total) vary with the
degree of correlation between the two cost variables, q and y. All values are here as fraction of
maximal expected cost (which equals 2, the minimal of Eq and Ey)). We see that some cost
avoidance is possible (in the sense that overall costs drop below unity in the figures), and
more so for low correlation.
Figure 6: Ex ante expected energy/retrofit costs in period 2 as fractions of maximal
expected cost (=2), under different degrees of cost uncertainty, and as function of degree
of cost correlation
21
3.3 The Choice of Infrastructure Investment
We now simulate the initial infrastructure investment decision, H1, as given in (13) (jointly
with factors determining the joint distribution of q and y). The reported simulations all assume
T = 5. We are here thus assuming that period 2 is 5 times as long as period 1 (in expected
discounted value terms). Throughout we assume that the probabilities of period 2 action are
not affected by H1.25 The number of time periods is however found to affect H1 (the balance
between periods 1 and 2 then changes, and expected costs are generally different in the two
periods).
The form of the utility function U(H1), going into (13), here matters greatly. Figures 7-8
report such results for our three main alternative utility function specifications (all based on
CRRA assumptions): a) The Cobb-Douglas case (with exponent = 0.5). In this case the
elasticity of H1 with respect to total expected ex post cost (after investment is sunk) is H =
1/ = 2. b) The log-linear case, in which H = 1. c) The exponential specification exponent 0.5;
here we assume = 1.5, and thus H = 2/3. Among these specifications, the exponential
function provides the least sensitive choice of H1 in response to an increase in Eq; and the
Cobb-Douglas function the most sensitive, with the log-linear specification as an intermediate
case. This follows naturally from section 2.2 above, where we showed that in the log-linear
case, an increase in ex ante unit cost (combining energy and retrofit costs) leaves total ex ante
cost constant; the reason is that a reduction in H1 exactly offsets the expected unit cost
increase in this particular case. In the Cobb-Douglas case the offsetting change in H1 is
greater; and in the exponential case, smaller.
While our simulations thus cover a range of possible cases, the realism of some of the
alternatives may be questioned. In particular, for some alternative specifications it might be
argued that the scope for change in H1 is, perhaps, too great. In particular, under the Cobb-
Douglas specification this scope may appear excessive relative to what is seen empirically in
most cases. The exponential specification (with H = 2/3 under our parametric example) may
then perhaps seem as the most realistic case. This is useful to have in mind in interpreting our
numbers in the following.
25
Conceivably, there may be such effects, which will in case modify our conclusions somewhat; see section 4
below for a discussion.
22
Figure 7: Energy intensity of infrastructure investment as function of expected energy
cost in period 2, for different utility functions and variances, non-correlated costs
Figure 8: Energy intensity of infrastructure investment as function of expected retrofit
cost in period 2, for different utility functions, variances, non-correlated costs
23
The chosen energy intensity is in all cases a decreasing function of both Eq (figure 7), and of
Ey (figure 8) with the greatest rates of reduction in the Cobb-Douglas case, and the smallest in
the exponential case. In the intermediate logarithmic case, the slope is similar in the three
variance cases, but the level of H1 (for any given Eq in figure 7) is higher with higher
variances. This is due to conditional expected costs being lower with higher variances. Such
anticipated future cost savings are in the Cobb-Douglas case exactly counteracted by a higher
energy intensity of the infrastructure. This effect is magnified in the Cobb-Douglas case, and
weakened in the exponential case. In the (perhaps more realistic) exponential case the
response of infrastructure energy intensity to cost increases is smaller.
Figure 9: Energy intensity of infrastructure investment for different degrees of
uncertainty in period 2, as function of degree of cost correlation
Figure 9 above describes effects of increased cost correlation on H1, for Eq = 2 and Ey = 3. In
all cases there is a strong tendency for H1 to fall with greater cost correlation, which gives less
room for ex post cost reductions. Utility function levels are here less meaningful (and need to
be calibrated); only relative levels (along the different curves) are. We see that the scope for
cost avoidance is not dramatic: in the log and exponential cases expected costs are reduced by
about 10 percent when the cost correlation is reduced from 0.8 to 0.
3.4 Ex Ante Expected Energy Consumption over the Project Lifetime
In this final sub-section containing simulations, we compute the (per-period) ex ante expected
energy consumption over the project's lifetime, found by multiplying the expected per-period
energy consumption, where period 2 consumption is found by multiplying the probability of
energy use by the intensity of the infrastructure. As before we assume T=5. The infrastructure
24
is operated in a "business as usual" way (and energy used) with probability 1 in period 1, and
with probability P(1) in period 2.
Energy consumption is here strongly decreasing in Eq, and, typically, more strongly so than
what was found for H1 in the section above. The reason is that ex ante energy consumption
depicted in figure 10 compounds two separate effects on energy consumption, and both go in
the same direction: first, H1 (the energy intensity of infrastructure at establishment) is
reduced; and secondly, the probability of using energy ex post, P(1), is reduced. This provides
for a, potentially large, reduction in energy consumption in response to a (correctly and
rationally anticipated) future increase in expected energy costs, given that the infrastructure
decision, fully and rationally, takes such future cost increases into consideration.
The energy consumption response in figure 10 is "efficient" in the sense that the infrastructure
decision is fully considered, and fully rational using "correct" values for future energy costs;
and embedding the future retrofit possibilities optimally. In such cases, we can expect energy
use to be highly responsive to changes in the level of these future expected costs.
Figure 10: Expected per-period energy consumption over the project's lifetime, as
function of expected energy cost in period 2, different utility functions,
independent costs
This is of course a tall order in terms of being a description of reality. When instead
infrastructure is not (at all) responsive to future expected costs, the simulations in section 3.1
above (ex post probability of energy use only) represent the only way in which energy
consumption can respond to changes in energy prices (which, we should recall, by assumption
contain all relevant climate and environmental costs). In addition, the retrofit possibilities
considered here can be viewed as representing some "optimal" development of such
25
possibilities; when these are not forthcoming the ex post probability P(1) cannot change
(much); and energy use will be essentially stuck, regardless of energy cost or price. Finally,
the response of H1 to changes in total ex post operation cost could be excessive under our
example, and is, at least, likely to be excessive under the Cobb-Douglas and log-linear utility
specifications.
Figure 11 tells another interesting story, namely, what happens to ex ante energy consumption
over the lifetime of the infrastructure, in response to ex ante, fully and rationally expected
increases in expected retrofit costs in period 2. Here also two effects on energy consumption
go in different directions. The ex post effect is to increase energy consumption as energy is
used in more states ex post when Ey increases. The ex ante effect implies that the initial
energy intensity of the infrastructure is reduced, in response to the increase in (ex ante)
expected costs in period 2. The most interesting result here is that, when variances on costs
are high, the latter effect may dominate as the former effect is small (good substitution
possibilities ex post imply that expected ex post costs increase by little when Ey rises).
Figure 11: Expected per-period energy consumption over the project's lifetime, as
function of expected retrofit cost in period 2, different utility functions,
independent costs
Figure 12 below describes effects of different degrees of correlation between energy and
retrofit costs, on overall energy consumption over the project's lifetime. Again, only relative
figures are meaningful when the correlation coefficient changes. Higher cost correlation leads,
in almost all cases, to less ex ante energy consumption in particular as energy intensity, H1,
drops (as already seen from figure 6), and as the probability of energy use, while seen to
increase from figure 2, increases only slightly. This latter effect may however in some cases
be sufficiently strong for energy consumption to increase overall in some cases (as under the
Cobb-Douglas utility function, when correlation is already high).
26
Overall, some striking conclusions, highly relevant for energy and climate policy, can be
drawn from these simulations. Given that the expectation of unconditional energy cost E(q(2))
is set at a given level (in our case then, equal to 2), the conditional or actual expected energy
cost related to the infrastructure in period 2 is lower in all cases. The difference is greater
when variances (of both q and y) are larger. This is because the decision maker alternatively,
and optimally, exercises the better of three options ex post, energy use, retrofit, or closedown.
Energy costs are avoided in states of the world where such costs are high, but also when they
are low but retrofit costs are even lower. These (cost-avoidance) effects are stronger, the more
variable energy and retrofit costs are (for given unconditional expectations).
Focusing on energy costs, (conditional) expected energy costs are reduced when costs become
more variable (for given unconditional expectations), due to three separate reasons. It then
becomes more likely that a) a given retrofit cost is lower, and that b) a given utility value of
continued operation is lower than actual realized energy cost. Besides, c) a more variable
retrofit cost increases the likelihood that the retrofit cost (with given expectation) is lower
than any given energy cost. All these factors tend to ameliorate the overall effect of the initial
"tying up" of energy costs associated with a given infrastructure.
Figure 12: Expected per-period energy consumption over the project's lifetime, for
different levels of uncertainty, and function of degree of cost correlation
The figures illustrate a further feature of the theoretical analysis, namely that when expected
(unconditional) retrofit cost is greater than expected energy cost (for given var(q)), expected
conditional or actual energy cost is greater than expected retrofit costs. This is because when
a given expected cost is high, the respective alternative tends to be exercised in fewer cases.
27
A few cautionary notes must be recognized when interpreting these simulations. In particular,
the model has the (perhaps unrealistic) feature that when the closedown option is exercised,
there is no energy consumption or any emissions. The same holds when the retrofit option is
exercised. This assumption could however, without much loss of generality, be weakened by
assuming a certain (minimum) level of energy use and emissions in this case.
4. Summary and Final Comments
The main purpose of this paper has been to analyze the implications of two interacting sets of
decisions concerning infrastructure investments with long lifetimes and that commit society to
potentially high levels of energy use, and carbon emissions, for a long future period after the
investments have been made. One of these decisions is the (basic and initial) energy intensity
of the infrastructure investment. In section 2 above we studied factors behind such
investments. With particular reference to the class of so-called constant relative risk aversion
(CRRA) utility functions for characterizing the relationship between energy intensity and the
public's utility from the infrastructure, we noted that the energy intensity of the initially
established infrastructure can respond more or less strongly to a change in its future expected
cost of operation (considering different infrastructure types that all have the same initial
investment cost). We discussed three classes of cases: a) one where initial energy intensity is
increased more than proportionately to a given relative increase in ex post operation costs
(when the utility function takes a Cobb-Douglas form); b) the case where energy intensity
increases proportionately to the reduction in overall cost (the logarithmic case); and c) a set of
cases where it increases less than proportionately (the exponential case). We indicated that
case c), which implies that the price elasticity of demand for energy through infrastructure
energy intensity is less than unity in absolute value, seems as the most empirically reasonable
of the three cases. We indicated that the implication of a change in ex post costs for initial
energy intensity, and as a result overall expected lifetime energy costs and carbon emissions,
could be substantial.
A second main aim of the paper has been to study impacts of two types of "ex post" policy
interventions, that may be applied after an infrastructure investment has been sunk. The first
is a "retrofit" of the infrastructure (by making an additional later investment that then removes
the energy demand and/or emissions while retaining its utility value to the public). The second
policy option is simply to close the infrastructure facility down (which is a more drastic
alternative, as the utility value of the infrastructure is then is lost, together with energy use
and emissions). Most of the focus here has been on the retrofit alternative, and on its ability to
reduce subsequent (energy and environmental) costs, and its effect back on energy intensity of
the initial infrastructure.
These objectives of the paper must be viewed against the 5 types of market failure, discussed
in section 2 above, that can result in inefficient infrastructure choices. Roughly, these
explanations can be classified into two groups. The first group is related to insufficient or
faulty general climate-related or energy policies (including insufficient emissions pricing and
technology support); while the second group is related to inefficiencies and incompleteness
related to expectations about future policies. Points B and D fall largely into the former
category, while points A and C fall mainly into the latter. Point E might conceivably fall into
either category.
28
It is easy to understand why policies of individual governments hosting infrastructure projects
are often faulty: namely, the basic lack of incentives of governments to address the problems
of mitigation, at least in the absence of a comprehensive and binding agreement requiring the
governments to behave in an optimal fashion. The second group of explanations has a more
diverse set of explanations, which are however all related to either policy incompleteness or
"irrationality" (or "behaviorism") in the policy process.
We now discuss the results from our simulations, presented in section 3 above and in
appendix B, in light of these 5 points.
A) The initially expected distribution of energy costs, relevant for making current
infrastructure decisions, is down-shifted relative to the true (correct) distribution that
will be realized in period 2. This causes infrastructure decisions to be incorrect, and
chosen with too high energy intensity. Figure 7 in section 3.3 then represents the
degree of distortion (as the scaling of the initial infrastructure, which is proportional to
final lifetime energy demand for the infrastructure, is in this case not optimal and
generally excessive). Ex post decisions for given, already established, infrastructure
are however assumed to be socially optimal. The energy consumption choice, in
response to an increase in "true" energy cost which is not anticipated nor ignored, is
then valid only for period 2. It is represented by simulations in figure 1 (figure 1.1 in
appendix B for correlated costs). In our examples, the scope for ex post avoidance of
costs could be substantial as expected energy costs increase (most for the Cobb-
Douglas specification and least for the exponential specification). The fully optimal
response, involving also optimal ex ante infrastructure choice, is characterized by
figure 10 and is generally greater (in some cases far greater). To provide an example,
assume that the correct value of Eq is 4, while the assumed value is 2. Focusing now
on the log and exponential specifications, we find that the chosen energy intensity of
infrastructure is in the range 30-50 percent higher than the optimal level in this case.
This is then also the degree of excessive carbon emissions in this case.
B) Energy (including environmental) costs facing the policy maker are lower than
globally optimal costs. The distribution of energy costs facing the policy maker is
however correctly anticipated. The energy cost distribution applied by the policy
maker is then down-shifted relative to the "optimal" energy cost distribution (the
charge to the project, for combined energy and environmental costs, is lower than the
global cost); and project energy costs are not responsive to increases in true
environmental costs. In this case, there is (in the limit with no increase in the local
energy cost in response to a "true" cost increase) no response whatsoever to an
increase in energy/environmental costs, neither ex ante nor ex post. The degree of
distortion to lifetime energy consumption can then be found directly from figure 10, in
section 3.4. This is the maximal distortion to this variable found here; both the initial
energy intensity, and the ex post choice of energy consumption versus retrofit, are
distorted upward for a large effect. The distortive effect can be read out of figure 10 by
comparing values at "correct" versus "accounted for" values of Eq. Considering the
same example as under case A (where the true expected energy cost is 4, while the
expected cost facing policy makers is 2; and again considering the log and exponential
specifications), the degree of excess consumption of fossil fuels is now more dramatic,
between 2 and 3.5 times the optimal level. This reflects the inefficient choices at two
levels, namely both the initial investment level, and the ex post level at which the
investment is applied.
29
C) Expectations about future retrofit costs are incorrect at the stage of infrastructure
investment. Such mistakes could go both ways (retrofit costs could be either over- or
under-estimated), and thus lead to either excessive or inefficiently low overall energy
consumption. Assume first that true retrofit costs are under-valued. Referring to our
simulations, we must then consider the infrastructure decision which is biased by this
distortion. Figure 8 (and figure 3.2 in the appendix, for the correlated cost case)
provides the clue to answering this question. In particular, under the log-linear or
exponential utility cases, with an under-valuation of the ex ante expected retrofit cost,
set at 2 instead of a correct value of 3, infrastructure is too energy-intensive, but not
dramatically so: energy intensity is excessive by about 10 percent or less in all
examples. With an excessive assessment of the retrofit costs, the mistake goes in the
opposite direction (so that infrastructure has too low energy intensity), but again only
slightly.
D) Future retrofit costs are correctly anticipated, but are higher than socially optimal
costs. The idea here is that future retrofit costs are endogenous and affected e g by
investments made in technology to reduce these costs; and such investments may be
under-provided. In this case we need to consider lifetime energy consumption (both
initial investments, and ex post energy use), and its response to changes in Ey,
described in figure 11 (and figure 4.2 in the appendix). Two factors then affect overall
energy use, and in different directions. First, the higher than optimal expected future
cost (implied by the higher retrofit cost) leads to an inefficiently low ex ante energy
intensity of the initial infrastructure. Secondly, the ex post probability of energy use is
excessive, since retrofit is used in too few cases. We find, from figure 11, that the
balance between these two factors depends heavily both on the degree of cost
uncertainty, and on the utility function specification. We first note that total energy
consumption is rising with the level of the retrofit cost, in all cases of the simulations
under the log and exponential utility specifications. We also note that when cost
uncertainty is low, the ex post effect, representing the ex post probability of continued
energy use, always dominates, and overall expected energy use increases under all
utility function specifications. With greater variances on the cost components,
however, the overall probability of energy use changes much less; and the two factors
more or less cancel out in the simulations.
E) The policy-relevant value of T, T1, is below its optimal value, T0. Such effects are less
directly represented by our simulations. It may be represented via a too low weight to
period 2 and its ensuing costs, when making the infrastructure decision. The main
effect is that the energy intensity of the initial infrastructure, and in consequence
energy consumption, is excessive. There is however less reason why the ex post
retrofit decision should be distorted.
Policy incompleteness (applying to point E above) could arise as an issue when policies are
based on ad hoc rules that are outcomes not of deliberate optimisation, but instead a
simplified process that may lead to systematic biases in a climate context. One such case is
when the rate of discount for evaluation of public projects with climate impacts is determined
administratively for large classes of projects (typically at a high rate), and not aligned with
optimality rules relevant for (long-run) climate-related projects. Another case of policy
incompleteness is when the returns to public projects are accounted for only over a limited
horizon (say, 20 years), or the project is not based on explicit cost-benefit calculations.
30
Considering implications for initial infrastructure design of problems related to categories A-
E above, we have seen, supported by our simulations, that all categories except D
overwhelmingly tend to illustrate cases where energy consumption is excessive over the
lifetime of an established infrastructure. In cases A-B, this occurs in two complementary
ways: through excessive energy intensity of the initial infrastructure; and through excessive
ex post "business-as-usual" operation of the investment in period 2. Under case C, initial
energy intensity is excessive, while "business-as-usual" operation is not affected. On balance
expected energy use is excessive. The fourth category (D) is different in that it tends to make
energy intensity of the initial investment lower than optimal; while the retrofit option is used
ex post in too few cases. Our simulations indicate that these two factors may balance out for
overall energy consumption; or (when variances on energy and retrofit costs are low) lead to
excessive overall energy consumption as the latter factor dominates. The fifth category, E, is
similar to C as the ex ante infrastructure is biased in the direction of too high energy intensity,
while the ex post decision is not generally biased.
Arguably, cases A-B are keys to understanding the implications of (perceived and actually
realized) energy prices on the path of future energy consumption following from particular
infrastructure investment projects. Table 1 below sums up some main results from our
simulations in section 3, which are done under the assumption of uncorrelated energy and
retrofit costs. We present two sets of figures. A) Those based on "policy error ex ante only",
which can be found from figure 7. These are excessive rates of fossil energy consumption
over the lifetime of the infrastructure, that result from the investment decision itself being
wrong; but where adjustments to correct realized energy price are made ex post. The only
source if excessive energy consumption is thus that the infrastructure investment is based on
wrong expectations about future energy prices.
Disregarding the Cobb-Douglas case (which seems less realistic than other cases), our
simulations show that when the actual energy/environmental cost has expectation 3 (while the
ex ante anticipated expectation is 2), the degree of excessive energy consumption due to
"wrong" infrastructure is roughly between 20 and 35 percent; when the correct expectation is
4, this degree of excessive energy consumption is 30 and 60 percent.
When there is in addition a policy error ex post, so that the correct energy/environmental price
is not implemented but instead the lower price is used, the overall error is far greater.
Focusing also now on the log and exponential cases, the overall policy error (taking the
compounded effect of the excessively energy intensive infrastructure, and the excessive
energy use ex post) is now from about 65 percent to more than 100 percent of the efficient
level when the correct Eq level is 3 (while the energy price actually applied is 2). When
correct Eq = 4, this error is dramatically higher, most so for the case of low variances.26
26
This particular result has a natural explanation, namely, that for these simulations we assume throughout that
Ey = 3. Since it here makes a big difference for realized energy consumption whether Eq is smaller or greater
than Ey, there is a big jump in realized energy consumption as Eq goes from a level below to a level above Ey.
This jump could be unrealistically high in particular when variances are small.
31
Table 1: Relative policy error, in the form of excessive carbon emissions over the lifetime
of the infrastructure, in percent as a result of incorrect a) ex ante expected
energy/environmental cost (=2); and b) ex post expected realized energy/environmental
cost (=2), for different correct Eq, based on simulations in figures 7 and 10.
Type of Utility Low variances Medium variances High variances
policy function Eq= Eq=3 Eq=4 Eq= Eq=3 Eq=4 Eq= Eq=3 Eq=4
error specification
2.5 2.5 2.5
Policy Cobb- 39 75 124 40 79 146 40 81 158
error ex Douglas
ante only Logarithmic 18 32 50 18 34 57 18 35 61
Exponential 12 21 31 12 21 35 12 22 37
Policy Cobb- 68 177 693 64 154 446 62 147 403
error Douglas
both ex Logarithmic 42 109 430 39 90 248 37 84 213
ante and Exponential 34 91 363 31 72 199 30 66 168
ex post
A main objective of this paper has been to study impacts of options to retrofit an established
infrastructure on total expected costs, expected energy (including environmental) costs, and
initial energy intensity of infrastructure. Our simulations provide a rich set of examples of
how overall realized (conditional) expected costs vary in response to variations of certain key
parameters (expectations and variances), and given that ex post energy and retrofit costs are
both log-normally distributed (and independent). They indicate that some fraction of expected
future energy use related to infrastructure can always be avoided by optimally exercising
either the retrofit or close-down option at a later stage, given that exercising such options is ex
ante optimal. This degree of cost avoidance can be large, under plausible assumptions; and
larger when there is more uncertainty about energy and retrofit costs. In certain parametric
cases more than half of the (ex ante expected) potential energy consumption is avoided
through optimally exercising the retrofit alternative ex post. Expected total (energy plus
retrofit) costs are then reduced, in some cases substantially.
Note that with a long expected time from infrastructure investment to availability of a relevant
alternative option (retrofit, or closedown), and when the decision maker discounts too heavily
(excessively), the options will also be discounted too heavily and given too little weight in the
infrastructure decision problem. This factor would then serve as a partial counterweight to
those emphasized here, that tend to reduce energy (and climate) cost below socially efficient
levels, and lead to too energy intensive infrastructure choices.
Several limitations of our analysis must be pointed out. First, it is based on ex ante
distributions of energy and retrofit costs that are both log-normal and known. Secondly, we
assume only two periods, "the present", and "the future", which allows for only one decision
point beyond that of infrastructure investment (at the start of period 2). Our choice of
assumptions here was guided by a concern for generality in the distributional assumptions,
while still permitting a tractable analysis. An extension of the current framework to three or
more periods would make the analysis less tractable, but ought to be pursued in follow-up
work. Relevant costs (of energy, emissions, and related to the retrofit technology), as well as
benefits (the current utility value of the infrastructure technology), all in reality evolve
32
continuously through time making the two-period framework less accurate. An obvious
development would then be to assume that retrofits could be carried out at several points of
time; and with separate developments for energy and retrofit costs. Such extensions are being
considered in a companion paper to the current one, Framstad and Strand (2010), where
energy and environmental costs are assumed to evolve continuously over time; in other
respects however assumptions are much simpler in that paper, in particular, a fixed retrofit
cost is assumed.27
Our analysis, both theoretical and for simulations, assumes that the scaling of the initial
infrastructure does not interact with the ex post decision to retrofit. This follows from our
basic assumption that retrofit costs are proportional to energy consumption. When comparing
overall retrofit costs with overall energy costs ex post for a given project, project size then
does not matter; only energy and retrofit prices matter. This is, admittedly, a special case,
which could be generalized. With such a generalization, the ratio of overall retrofit costs to
overall energy costs would likely be a decreasing function of project size (unit retrofit costs
would be reduced for larger projects). Additional effects would then arise. In particular, larger
ex ante projects would be favoured more as these would, effectively, permit more cost
avoidance later on (through more frequent retrofitting). It is less clear how overall energy
consumption over the project's lifetime would be affected; the initial scale-up of projects, and
the subsequent more frequent retrofitting, would work in opposite directions.
Our choice of utility functions for simulations might need some comment. One of our applied
specifications, the Cobb-Douglas formulation, can be argued to yield unrealistic results as the
response of energy intensity in infrastructure, to changes in expected lifetime project costs, is
too great (with an elasticity for energy intensity in infrastructure to ex post cost which is
greater than one in absolute value). This formulation is thus likely to be less credible than the
other two utility function formulations, in yielding respectively a unitary (the log formulation)
and less than unitary price elasticity (the exponential formulation).
A further extension would be to consider (partial) retrofits where neither fossil energy use nor
carbon emissions are reduced to zero, which may be more cost efficient in some cases. Also,
in our model, upon a possible infrastructure closedown, "nothing happens". A more
satisfactory analysis would involve a replacement infrastructure taking over the flow of
services lost by closedown; which would require specific assumptions about costs (of new,
future, infrastructure investment) and benefits (flowing from the new, instead of the old,
infrastructure). We seek to pursue such extensions in future work.
27
A basic result here is that continuous development of costs produces an "option value" of waiting which
serves to delay the retrofit decision; this lowers expected total costs but increases environmental costs.
33
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36
Appendix A: Further Analytical Results
A1. Shifts in the Energy Cost Distributon with Cost Independence
We will now study some implications of changes in the distribution of energy costs in period
2. Throughout appendix A we consider a general utility function specification, and general
distribution functions F and G. The only main limitation is that we here focus on the
simplified case where q and y are independent.28 Consider a downward shift in the
distribution of energy costs, that leaves all other parameters unchanged. Set Fq = F and Fy =
G. Call the new distribution F(q) = F(q+) (so that F is shifted up relative to F by a constant
amount for any given q). This is formally the same as the entire distribution of q being
shifted down, but (invoking the assumption of independence between the two cost variables)
retaining the distribution function F. Energy costs, in consequence, fall on average. In
particular, since the distribution G is unaltered, the following new definition of E[CH(2)] then
applies:
y*
(7a) E[CH (2)] [1 G (q )] f (q )qdq H1
q 0
y*
(8a) E[CR (2)] [1 F ( y )]g ( y ) ydy H1
y 0
Differentiating the expressions for P(1), P(2), E[CH(2)] and E[CR(2)] with respect to then
yields (assuming that P(3) is not "significantly" altered by any resulting change in y*):29
y*
dP (1) dy *
(A1) [1 G (q )] f '(q )dq [(1 G ( y*)] f ( y*) .
d q 0
d
y*
dP (2) dy *
(A2) f ( y ) g ( y )dy [(1 F ( y*)]g ( y*)
d y 0
d
Recall the definition Ec(2) = E[C(2)]/H1 as the ex post realized cost (per time unit in period
2) per established energy unit. Define now also Ech(2) + E[CH(2)]/H1, and Ecr(2) +
E[CR(2)]/H1. We then derive the following general expressions:
dE[CH (2)] dEch(2) E[CH (2)] dH1
(A3) H1
d d H1 d
dE[CR (2)] dEcr (2) E[CR(2)] dH1
(A4) H1 .
d d H1 d
28
A similar comparative-static analysis in the more general case, with dependent distribution functions, turns out
not to yield tractable and easy to interpret expressions. For that reason we are here focusing on the case of
independent distributions.
29
This requires, under the quadratic utility specification in particular, that the coefficient b in (14a) is small, and
that H1 in response changes "relatively much" compared to y*.
37
P(1) here increases (as f' is positive for low G values): the probability of "business as usual"
increases. This is intuitive: when energy costs fall, the likelihood that the (business-as-usual)
energy cost option is exercised in period 2 increases, "everything else equal". The probability
that the retrofit option is exercised in period 2 drops unambiguously. The increase in the
former is greater so that P(1)+P(2) increases (i.e. the closedown option, is exercised in fewer
cases). Ignoring first effects via changes in H1, we find that the effect of a shift in on unit
energy costs (represented by the first term on the right-hand side of (A3)) is ambiguous. Two
factors go in different directions: a greater P(1) implies that energy costs are incurred in more
states, leading such costs to increase. On the other hand, unit energy costs drop for any given
state, which tends to reduce costs. The effect on expected unit retrofit costs is however
unambiguously negative. This is intuitive: the only thing that happens to retrofit costs is that
such costs are applied in fewer states, thus reducing overall expected retrofit costs.
Note here that while dP(1)/d is positive, dE[CH(2)]/d can be either positive or negative,
depending on the distributions. Consider now the CRRA case (with relative risk aversion
coefficient = ). We then find
dH1 1 TH1 dEc(2)
(A5)
d q TEc(2) d
dy * Ty * dEc(2)
(A6)
d q TEc(2) d
Since (as generally found below) dEc(2)/d is generally negative, the change in H1 is positive,
as expected: a lower overall expected cost in period 2 makes it attractive to set a higher
energy intensity for the initial infrastructure. But the cut-off point y* for ex post costs (beyond
which closedown will be selected) is reduced. This serves to somewhat increase the frequency
with which the closedown option will be exercised ex post. Note that y* falls in H1, rather
generally and more specifically for our utility function specification (36).
Differentiating Ech(2) and Ecr(2) we find:
dEch(2)
y*
Ty * dEc(2)
(A7) (1 G (q )) f '(q)qdq [1 G ( y*)] f ( y*) y *
d q 0 q TEc(2) d
dEcr (2) y* Ty * dEc(2)
(A8) ( f (q ) g (q )qdq [1 F ( y*)]g ( y*) y *
d q 0 q TEc(2) d
(A7)-(A8) solve simultaneously for these two derivatives, noting that dEc(2) = dEch(2) +
dEcr(2). In considering this system, we find that meaningful solutions (where both derivatives
have the same sign as the respective first expressions on the right-hand sides) requires the
following condition to hold:
38
T ( y*) 2
(A9) 1 [1 G ( y*)] f ( y*) [1 F ( y*)]g ( y*) 0
q TEc(2)
This condition puts bounds on f(y*) and g(y*) (densities of the respective distribution
functions in the neighbourhood of y*): these cannot be too large at an equilibrium solution.
In general, we cannot unambiguously determine the sign of dEch(2)/d, but it must have the
same sign as the first term on the right-hand side of (A7). This has an intuitive interpretation:
ex ante unit energy costs may go down if the cost shift factor () dominates; but it may go up
if the "probability factor" (representing a higher likelihood that "business as usual" energy use
will be chosen in period 2) dominates. dEcr(2)/d must, by contrast, be negative: the only
thing that happens to effective retrofit costs in this case is the likelihood of a retrofit is
reduced, implying that effectively applied costs are reduced.
dEc(2)/d must in general be negative (which is obvious as unit costs are generally reduced).
What happens to overall costs as a function of , when also the effect on H1 is accounted for?
The effect of the shift in the cost distribution on overall expected cost, E[C(2)], is given as the
sum of the terms from (A3) and (A4). Assuming still CRRA, we find
dE[C (2)] dEc(2) 1 TEc(2)
(A10) H1 1 .
d d q TEc(2)
Here, as noted, dEc(2)/d < 0 (from the system (A7)(A8)). Thus, dE[C(2)] < 0 as long as the
last parenthesis is positive. This requires that
TEc(2)
(A11) .
q TEc(2)
Consider here the case where = 1, and the utility function logarithmic. Then dE[C(2)] < 0
insofar as q > 0, and more so the larger q is in relation to TEc(2) (in other words, when first-
period costs are a larger fraction of expected overall operating costs over the lifetime of the
facility). In particular (and as discussed in section 2.2), when q = 0 (first-period costs can be
ignored), a negative shift in the distribution of ex post energy costs leads to an decrease
(increase) in total expected overall costs when also scaling the initial energy intensity is
considered, given > (<) 1. Thus in particular when < 1, any initial cost reduction from the
positive shift to is more than eliminated through a higher choice of energy intensity,
associated with the initial infrastructure investment.
Consider next the effects on carbon emissions from a shift in . The expression for expected
ex ante carbon emissions, denoted E, is given by
(A12) E H1[1 TP (1)] .
Note here in particular that carbon emissions equal H1 whenever the "business as usual"
alternative is being pursued; this alternative is pursued with probability 1 in period 1, and by
ex ante probability P(1) in period 2. Thus:
39
dE dH dP (1)
(A13) [1 TP(1)] 1 H1T
d d d
dP(1)/d is here positive, and so is dH1/d. Thus carbon emissions increase, for two separate
reasons; first, energy intensity of the infrastructure, H1, increases; and secondly, the
probability of the ex post business as usual alternative (with normal energy consumption),
P(1), increases. Note however that P(1) increases less as a result of y* being reduced in
response to the higher H1.
More specifically in the CRRA case we have the following expression:
(A14)
y*
dE 1 TH1 dEc(2)
TH1 [1 G (q )] f '(q )dq [(1 TP (1)] T [1 G ( y*)]g ( y*) y *
d q 0 q TEc(2) d
A2. Shifts in the Retrofit Cost Distribution with Cost Independence
This subsection derives some comparative-static results regarding impacts of changes in costs
of retrofitting in period 2. Again we focus on the case of independent energy and retrofit costs
(q and y). We study impacts on outcomes, from both marginal changes in retrofit costs, and
from the retrofit option being at all available.
y*
(7b) E[CH (2)] [1 G (q )] f (q )qdq H1
q 0
y*
(8b) E[CR (2)] [1 F ( y )]g ( y ) ydy H1
y 0
Differentiating the expressions for P(1), P(2), E[CH(2)] and E[CR(2)] with respect to then
yields
y*
dP (1) dy *
(A15) f ( y ) g ( y )dy [1 G ( y*)] f ( y*)
d y 0
d
y*
dP (2) dy *
(A16) [1 F (q )]g '(q )dq [1 F ( y*)]g ( y*) .
d q 0
d
dE[CH (2)] y* E[CH (2)] dH1
(A17) f ( y ) g ( y )dy H1
d H1 d
q 0
dE[CR (2)]
y*
E[CR (2)] dH1
(A18) (1 F (q )) g '(q )qdq H1
d H1 d
q 0
40
Interpretations are similar to the case of energy cost changes. When increases, the
distribution of retrofit costs (in analogous fashion to the energy cost distribution, in section
A1) shifts downward, and average retrofit costs fall. The probability of "business-as-usual"
energy consumption (P(1)) then decreases unambiguously, while the probability of retrofit
(P(2)) increases unambiguously. The increase in the latter is also now in general greater, so
that P(1)+P(2) increases. Thus expected energy intensity of the infrastructure falls
unambiguously (as represented by the integral on the right-hand side of (A17)), while the
change in expected retrofit cost per established energy unit is ambiguous (the integral on the
right-hand side of (A18)). Their sum, EC(2), falls unambiguously.
Consider now, in the same way as in section A1 above, changes in energy intensity of the
infrastructure (H1), and energy use and carbon emissions. Focusing again on the CRRA case,
the expression for effects on H1 is still given by (A5) except that is replaced by . This
means that the effect on H1 is quite similar to that in section 3, for a shift in energy costs. The
expressions for the effects on Ech(2) and Ecr(2) are also similar and given by
y*
dEch(2) Ty * dEc(2)
(A19) f ( y ) g ( y )dy [1 G ( y*)] f ( y*) y *
d q 0
q TEc(2) d
y*
dEcr (2) Ty * dEc(2)
(A20) (1 F (q )) g '(q )qdq [1 F ( y*)]g ( y*) y *
d q 0
q TEc(2) d
Moreover, a similar condition to (A9) must hold for stability of the system (42)-(43) to give a
meaningful simultaneous solution to Ech(2) and Ecr(2) and thus Ec(2).
The main difference from section 3 is that the effect on carbon emissions of a parameter shift
now is different. The effect on E is still given by (A13) (only replacing by ). The main
difference arises as P(1) is now affected differently. The expression for the effect on E takes
the form
y*
dE 1 TH1 dEc(2)
(A21) TH1 f ( y ) g ( y )dy [(1 TP(1)] T [1 G ( y*)]g ( y*) y *
d q 0 q TEc(2) d
Here the first main expression (indicating the main effect on probability of business as usual
as this alternative now is replaced by the lower-cost retrofit alternative) is now negative, while
the second (which mainly indicates the effect via higher initial energy intensity) is positive as
in (A14). In general we cannot say which of the two terms dominates. It is however clear that
when is greater than unity (and the response of H1 to an expected change in future costs is
relatively small), overall (ex ante expected) energy use is likely reduced, as the drop in ex
post energy use is relatively large in this case.
Consider here an alternative case where the retrofit option is no longer available (NR denoting
the "no retrofit" case)30. We have the following probability of closedown in period 2:
30
One interpretation of such a case is that the lower bound of the retrofit distribution, y0, is higher than the
average total value (per unit of energy consumed) of the infrastructure in period 2.
41
(A22) PNR (3) 1 F ( y*),1 PNR (3) F ( y*) .
In this case the probability of (energy-demanding) infrastructure operation in period 2, PNR(1),
is given simply by 1-PNR(3). For given y*, the probability of closedown is smaller when the
retrofit option is available, (P(3)), than when it is not (PNR(3)), by a factor (b-y*+y0)/b. The
probability of operation (with or without retrofit) is correspondingly greater when a retrofit
option is available. The probability of energy-demanding operation is smaller with the retrofit
option, by a factor (1-(y*+q0)/2b).
To study how a lack of retrofit option changes the initial energy intensity of the infrastructure,
H1, assume a further simplified case with no energy cost in period 1 (q1 = 0). The objective is
then simply to compare the expected per-unit combined energy and retrofit cost in period 2 in
the two cases. This cost equals (EC(2)/H1)/(1-P(3)) in the case where the retrofit option is
included, and (ECNR(2)/H1)/PNR(1) in the case where the retrofit option is not included. These
are the respective expressions for average overall costs per operation time, or probability of
operation in period 2. In either case this expression is to be set equal to U'(H1), for an optimal
H1 level to be achieved.
A3. The Value of the Closedown Option
We will now, rather briefly, consider some effects of the closedown option on the solution,
initial energy intensity, and on cost variables. The closedown option is practically irrelevant
when total utility per unit of energy consumed for the chosen infrastructure, y* = U(H1)/H1, is
so high that closedown is "almost never" used (i.e., [1-Fq(y*)][1-Fy(y*)] is "very small").
This may apply to cases where the infrastructure involves a high sunk cost relative to energy
consumption (such as, perhaps, for urban structures including housing and transport systems).
The expected energy and retrofit cost, and ex ante probabilities of "business-as-usual"
operation and retrofit in period 2, are still given by (2)-(6). With no closedown option, P(1) +
P(2) = 1. Expected ex ante utility of second-period operation is now
(9a) EW (2) { y * H1 E[CH (2)] E[CR (2)]}T .
The first-period decision problem can now be formulated as maximizing EW(1), from (8). The
resulting solution for optimal energy intensity of the infrastructure is now found from the
following condition:
E[CH (2)] E[CR (2)]
q1 T
H1
(13a) U '( H1 ) .
1 T
(13a) can be compared to the case with closedown, where H1 is determined from (13). There
are two main differences between (13) and (13a). First, E[CH(2)] and E[CR(2)] are greater in
(13a). Secondly, [1-P(3)] in the denominator of (13) is missing in (13a). As a result, overall
expected costs are greater, and the weight to second-period costs versus first-period costs is
greater. Thus, when expected second-period costs "per period" are greater, this also tends to
increase the overall expression on the right-hand side of (13). Overall, U'(H1) is higher, and
42
H1 reduced. Having a closedown option increases the energy intensity of the original
infrastructure investment. This is intuitive: the option will be used only in states where both
retrofit and energy costs are very high, and eliminates costs in these states, which in turn
provides incentives to raise the infrastructure's initial energy intensity.
A factor in relation to closedown is the curvature of the U function in particular with
"significant" degrees of risk aversion. A higher H1 then reduces U(H1)/H1, perhaps
substantially. The threshold for ex post closedown, when this decision is compared to the
higher of energy and retrofit costs, is then reduced. Closedown is exercised in more states of
the world, the higher is the initial energy intensity. This is reasonable, and tends to dampen
the overall energy and carbon footprint of an initially energy-intensive infrastructure.
We have assumed that, after closedown, no energy expenditure is incurred whatsoever. This is
unrealistic since the closed down infrastructure will in practice need to be replaced by some
alternative that is likely to be in itself energy-demanding (although presumably less so than
that initially established; this would tend to follow since replacements occur in states with
high energy costs in period 2).
A4. Endogeneity of Retrofit Costs
The retrofit options available in period 2 are likely to follow at least in part from technology
developments over the course of period 1, and these may in turn be influenced by R&D
efforts. This section considers possible effects of such efforts. Again, the explicit focus is on
cost independence which simplifies the analysis considerably.
Influencing R&D efforts with the purpose of mitigating GHGs has emerged as a core theme in
the climate policy debate, from several angles. One is how an optimal climate and energy
policy (in the form e g of emissions or energy taxes) can depend on the presence of R&D; this
has been discussed e g by Goulder and Schneider (1999), Goulder and Mathai (2000),
Bonanno et al (2003), Greaker and Pade (2008), Acemoglu et al (2009), and Golombek,
Greaker and Hoel (2010). A separate issue is that while it may be very difficult to reach an
international agreement to reduce GHG emissions directly using policy instruments such as
emissions taxes and caps, some analysts claim that reaching an agreement to support
emissions-reducing technological progress may be easier.31
Here we simply assume that the infrastructure investment decision maker may also carry out
R&D activity to reduce the costs of retrofitting this particular infrastructure in period 2, and
not for other units nor more generally. We assume that the entire distribution function for
period 2 retrofit costs can be given a constant vertical shift (in similar fashion as in section 4
below) through additional R&D effort in period 1.32 This upward shift in distribution is the
same as a downward cost shift, as in section 4, and was there given from (A18) in appendix
A, as a result of changes in a shift parameter for this distribution.
Here, consider the following modified discounted utility as viewed from period 1:
31
See in particular Barrett (2006, 2009).
32
This is of course highly unrealistic. In practice, R&D efforts will affect retrofit costs more generally, and also
for other projects, and thus imply positive externality effects for the latter. This issue is not discussed fully here.
For further discussion see e g Golombek and Hoel (2005, 2006), Golombek, Greaker and Hoel (2010).
43
(A23) EW (1; R ) ( y * q1 ) H1 EW (2) R
where EW(2) is given from (7), R is the first-period R&D cost, the Fy function is shifted, with
shift parameter , and where the size of is a positive function of R. Focusing on the case of
cost independence, we then derive the following general optimality condition with respect to
R:
(A24)
dEW (1; R ) dH dP (1) dP(2) dEC (2)
y * q1 y *( P(1) P(2))T 1 y * TH1 T R 1 0
dR d d d
where R is the derivative of with respect to R.33 The partial derivatives in (A24) are found
from (A19)-(A22), plus (11) differentiated. While (A24) looks complicated, its essence is that
the total derivative of EW(1) with respect to R consists of three marginal benefit terms inside
the curled bracket, classified by how model variables are affected: 1) effects via the increase
in H; 2) via increase in the joint probability of period 2 operation, P(1)+P(2); and 3) via
operation costs (energy costs plus retrofit costs) in period 2. These three terms are traded off
against, and at the optimum set equal to, the unit cost of R&D investment.
An important parameter here is R. Presumably, the marginal effect of R&D costs diminishes
with greater costs (as will, rather generally, be required for a unique internal optimum to exist
for the problem (A24)).34 Consider a second-order Taylor expansion
(A25) (R) = R R2
where and are positive constants, so that the first- and second-order derivatives of the
function are given by
(A26) R = 2R > 0
(A27) ''(R) = RR = 2 < 0
The main point here is that when the curled bracket in (A24) is large, R will be small, and R
correspondingly large. A large overall positive utility effect of a given shift in the retrofit
function then leads to a large optimal R&D effort R.
We have assumed that this investment only affects costs for one particular infrastructure
facility. More often, such R&D expenditures are likely to have effects also on costs for other
projects.35 The marginal social benefit of R&D is then much greater than the "private" benefit
for the infrastructure project sponsor. Assume that (A25) correctly represents the overall
social impact of R on retrofit costs, while the private impact is only a fraction h (< 1) of the
33
A second-order condition here also needs to be fulfilled. A sufficient condition here is that R is decreasing in
R.
34
One way to visualize this is to consider R&D projects carried out in sequence by their likelihood of success;
when only a few projects are funded these are the most promising.
35
Another way to express this effect is that there is likely to be a high degree of "technology spillovers"
associated with R&D for development of new retrofit technology; see discussions of such spillovers e g by
Golombek and Hoel (2005, 2006), Golombek, Greaker and Hoel (2010).
44
social impact. In this case, the marginal change in when R changes, as perceived privately,
is also a fraction h of the social impact given by (A26), and thus
(A28) '(R;h) = h( 2R) > 0.
When h is smaller, R must be smaller to fulfil (A24).36 As a result, the R&D activity will be
(perhaps much) lower than optimal when most of the overall returns to private R&D accrue to
others. One then faces an obvious problem of policy coordination across countries, which in
principle could be as serious as that for regular mitigation policy. High appropriability of rents
to developers of new technology will tend to reduce this coordination problem.37
36
With this formulation, when h is small, no such non-negative R can be found. Then no R&D investments will
be undertaken by private agents.
37
One way of securing this is strong patent laws. But this has negative side effects, in particular, the markets in
which the newly developed technologies are applied will not be competitive; see e g Greaker and Pade (2008).
45
Appendix B: Additional Simulations
1: Simulated ex post probabilities of energy use and retrofit
Figure 1.1: Probabilities of energy use and retrofit as functions of unit energy cost, for
positively correlated costs.
Figure 1.2: Probabilities of energy use and retrofit as functions of unit retrofit cost, for
positively correlated costs.
Figures 1.3-1.4: Probabilities of energy use and retrofit under variable energy cost
uncertainty, for independent and positively correlated costs (low variances)
46
47
Figures 1.5-1.6: Probabilities of energy use and retrofit under variable uncertainty
(medium variances)
48
Figures 1.7-1.8: Probabilities of energy use and retrofit under variable uncertainty
(high variances)
49
2: Simulations of ex post energy and retrofit costs
Figure 2.1: Ex ante expected energy/retrofit costs in period 2 as function of expected unit
energy costs, for different variances, positively correlated costs
Figure 2.2: Ex ante expected energy/retrofit costs in period 2 as function of expected unit
retrofit costs, for different variances, positively correlated costs
50
Figures 2.3-2.4: Ex ante expected energy/retrofit costs in period 2 as function of
variances (low and medium variances), independent costs
51
Figures 2.5-2.6: Ex ante expected energy/retrofit costs in period 2 as function of
variances (low and medium variances), positively correlated costs
52
Figures 2.7-2.8: Ex ante expected energy/retrofit costs in period 2 as function of
variances (high variances)
53
3: Infrastructure investments
Figure 3.1: Size of infrastructure investment as function of expected energy cost in
period 2, for different utility functions, variances, positively correlated costs
Figure 3.2: Size of infrastructure investment as function of expected retrofit cost in
period 2, for different utility functions, variances, positively correlated costs
54
Figures 3.3-3.4: Size of infrastructure investment, for different utility functions and
variances in period 2 (one variance meduim; and one low), independent costs
55
Figures 3.5-3.6: Size of infrastructure investment for different utility functions and
variances in period 2 (one variance medium; and one low) positively correlated costs
56
Figures 3.7-3.8: Size of infrastructure investment for different utility functions and
variances in period 2 (one variance high)
57
4: Energy consumption per period
Figure 4.1: Expected per-period energy consumption over the project's lifetime, as
function of expected energy cost in period 2, different utility functions, correlated costs
Figure 4.2: Expected per-period energy consumption over the project's lifetime, as
function of expected retrofit cost in period 2, different utility functions, correlated costs
58
Figures 4.3-4.4: Expected per-period energy consumption over the project's lifetime, for
different utility functions and variances in period 2 (one variance low)
59
Figures 4.5-4.6: Expected per-period energy consumption over the project's lifetime, for
different utility functions and variances in period 2 (one variance medium)
60
Figures 4.7-4.8: Expected per-period energy consumption over the project's lifetime, for
different utility functions and variances in period 2 (one variance high)
61