WPS5208
Policy Research Working Paper 5208
Climate Cost Uncertainty, Retrofit Cost
Uncertainty, and Infrastructure Closedown
A Framework for Analysis
Jon Strand
Sebastian Miller
The World Bank
Development Research Group
Environment and Energy Team
February 2010
Policy Research Working Paper 5208
Abstract
Large and energy-intensive infrastructure investments excessive energy intensity in such investments. Thus great
with long life times have substantial implications for care must be taken when choosing the energy intensity of
climate policy. This study focuses on options to scale the infrastructure at the time of investment. Simulations
down energy consumption and carbon emissions now indicate that optimally exercising the retrofit option,
and in the future, and on the costs of doing so. Two when it is available, reduces ex ante expected energy
ways carbon emissions can be reduced post-investment consumption relative to the no-option case. Total energy
include retrofitting the infrastructure, or closing it plus retrofit costs can also be substantially reduced, the
down. Generally, the presence of bulky infrastructure more so the larger is ex ante cost uncertainty. However,
investments makes it more costly to reduce emissions the availability of the retrofit option also leads to a more
later. Moreover, when expected energy and environmental energy intensive initial infrastructure choice; this offsets
costs are continually rising, inherent biases in the some, but usually not all, of the gains from options for
selection processes for infrastructure investments lead to subsequent retrofitting.
This paper--a product of the Environment and Energy Team, Development Research Group--is part of a larger effort in
the department to study analytical aspects of climate change. Policy Research Working Papers are also posted on the Web
at http://econ.worldbank.org. The author may be contacted at jstrand1@worldbank.org.
The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development
issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the
names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those
of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and
its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent.
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Climate Cost Uncertainty, Retrofit Cost Uncertainty,
and Infrastructure Closedown:
A Framework for Analysis
By
Jon Strand
The World Bank
Development Research Group
Environment and Energy Team
and
Sebastian Miller
Inter-American Development Bank
Southern Cone Department
Jstrand1@worldbank.org
1. Introduction
This paper presents a simple analytical model of infrastructure choice, and simulations to
illustrate properties of the model framework. An important starting point is that infrastructure
investments, sunk at an initial time of decision, "tie up" energy consumption for a long future
period. In our model time is divided into two discrete periods: an initial period 1, "the
present"; and a following period 2 ("the future"), which may be much longer. For each of the
two periods, all relevant variables are assumed to be commonly known at the start of the
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 their ex ante statistical distributions for period 2 are assumed to be known when
the initial infrastructure investment is made 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 focus of our analysis is that many types of infrastructure lead to considerable climate
policy "inertia", in establishing levels of fossil-fuel consumption that may be difficult to
reduce later. As 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 is the case
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 be very high ex post
after the infrastructure has been established.
Below we discuss reasons why the infrastructure investment may be suboptimal, and then
typically biased in the direction of too high fossil energy consumption and carbon emissions.
The 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 simple model of infrastructure investment, where we introduce
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 operated 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
interpretation is also relevant when applied to cases where the overall energy demand of the
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
infrastructure is simply reduced. In our stylized model, emissions are then assumed to be
eliminated completely. In either interpretation, the existence of a potential retrofit option in no
way implies that retrofit is necessarily an economically optimal choice; our model includes
cases in which exercising the option of retrofit would be prohibitively costly.
An alternative interpretation of "retrofit" is 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). This will however not
be our main interpretation in the following. It can be a somewhat problematic interpretation in
our model, in particular when the post-retrofit operating period T is a variable, since the
retrofit cost here is "periodized" and assumed constant "per time unit" within period 2. When
T is variable, retrofit costs with a given distribution G will then 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" costs and basic energy/climate cost are assumed to be uncertain from the point
of view of period 1, but with initially known joint distribution. In our basic presentation and
simulations, the two costs are assumed to be uncorrelated; this is however often unrealistic
and we comment later on implications of positively correlated prices. As for the degree of
uncertainty, retrofit costs could easily be more uncertain than energy and climate costs, since
future (period 2) retrofit technologies are mostly 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 simply abandon it, "close it down". This is wasteful in the sense that the initial
infrastructure investment was costly to establish, and this investment is then made worthless.
It is a painful option, but can still be attractive and rational ex post, in states where energy
and/or retrofit costs of continued operation both turn out to be very high at the same time. The
required condition is that the lower of these costs is 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. Our model however does not specify any
particular replacement alternative. In effect, our closedown alternative represents a
"benchmark" case with zero emissions and energy consumption. 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. An
important issue is to study optimal infrastructure investment; both characterization, and (even
more important from a climate policy standpoint) factors behind inefficient (too energy and
emissions intensive) infrastructures. Another key issue is 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 distributions of future
prices. "Optimality" can be established either for a private agent making the infrastructure
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
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 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 this 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 optimality point of view). 3
Increased availability of retrofit and closedown options affect expected fossil-fuel
consumption and GHG emissions, in two opposing ways. For one thing, 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, and
instead retrofit or close the infrastructure down. On the other hand, an anticipated increased
availability of such options serves to increase the chosen energy intensity embedded in the
infrastructure. 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.
We find in simulations, in Section 7 below, that a higher variance on retrofit and/or energy
costs (for given unconditional expectations of these costs) generally reduces expected future
costs, both in terms of energy use, and in terms of total (energy plus retrofit) costs. 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 (as a higher variance implies a
larger set of states where the retrofit option is "very gainful". 4 This is due to the general result
that greater variances are advantageous, as they lead to the availability of more low-cost
alternatives, which are exploited under an optimal ex-post policy.
The distribution of retrofit costs in period 2 depends on technological retrofit possibilities,
which in turn are affected by any R&D effort to develop such technologies. Greater R&D
effort, leading to reduced retrofit costs in period 2, increases the energy intensity of the initial
infrastructure investment choice. Overall expected energy consumption may then either
increase or decrease, depending on which of two effects is stronger: the increase due to
greater energy consumption in "business-as-usual" states (resulting from the higher energy
intensity); or the reduction due to the "business-as-usual" solution being chosen in fewer
states (in period 2 of our model). Simulations indicate that the latter factor may in some cases
dominate. 5 As a consequence in such cases, overall expected energy (and climate) costs are
reduced with more energy technology R&D. Expected retrofit costs may also increase or
3
See Strand (2009) for further elaboration of these issues.
4
This conclusion holds when the decision maker is risk neutral; it may need qualification under risk aversion.
5
Note the restriction on these conclusions, that the simulations to which we refer, are throughout based on the
assumption that energy cost and retrofit costs in period 2 are lognormal, and independent.
4
decrease. The factor contributing to a decrease is the very drop in cost. The factor that could
lead to an increase is that the retrofit option is exercised in more states of the world (thus
avoiding energy expenditures entirely in such states). Typically here also, the first factor
seems to dominate. Similarly, when the retrofit and closedown options are both available, an
upward shift in the distribution of energy costs may increase or reduce expected energy costs,
as well as expected retrofit costs; but, most likely (as apparent from our simulations), both
will increase. Total expected costs then also clearly increase.
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 will 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), may potentially reduce the inertia associated with the
infrastructure. This is a focus in this paper, and also two further papers, Strand (2009) and
Framstad and Strand (2009), which deal with complementary issues. Strand (2009) considers
different utility function representations and their implications, in a similar setting. Framstad
and Strand (2009) 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 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.
5
A further issue in the literature, related to the main themes of this paper, is the concept of a
"low-carbon society" and ways to achieve it; see Strachan et al (2008a, b), and Hourcade and
Cerassous (2008). 6 The overriding idea here is rather similar to that in the other literature
cited: namely, that achieving a society with low GHG emissions (necessary for efficiency in
the long run) requires a high concern for the design of current infrastructure investment. We
also note that two World Development Reports, the WDR 2003 (World Bank (2003)), and the
most recent WDR 2010 (World Bank (2009)), both have "inertia in physical capital" (echoing
our analysis of infrastructure) as main themes.
2. Two-Period Model with Uncertain Retrofit Costs
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. "Retrofitting", we assume, is not
available in period 1: it represents 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 in principle may 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
6
A very early champion of this line of thinking and discussion was Amory Lovins; see, in particular, Lovins
(1977).
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 more in the final section.
6
cost. Infrastructure type is in the model 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 over its
lifetime due to its greater fossil-fuel energy requirement. Call the current (per time unit) utility
flowing from the infrastructure when being operated 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 either the retrofit cost, or the average 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 average period-2
utility of the infrastructure.
3) Infrastructure closedown. This is optimal when environmental and retrofit costs both
turn out to be higher than the average period-2 utility of the infrastructure. Note here
however that closedown is a drastic measure. It will typically require that other
infrastructure is supplied, to replace the services lost by project closedown. This is not
explicitly modelled here. Implicitly, however, we may take our model to embed 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) as the (continuous ex ante, when
viewed from period 1) distribution over q levels to be realized in period 2, with support [0,
qM], where qM could be large. 10 Possibly, the F distribution for period 2 is shifted up by
increased emissions in period 1.
We assume a perfectly continuous distribution over retrofit costs, y, in period 2 given by G(y),
with support (0, yM), where yM could also be large. 11 Retrofit costs cannot be negative, but
10
In simulations below we assume that F is log-normal, in which case F is not bounded above (it is however
"thin-tailed").
11
Thus total retrofit costs in period 2 are given as Ty.
7
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 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). 12 In the analytical presentation, period 2 realizations of energy cost
and retrofit cost are assumed to be independent. 13
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) [1 - F (
P(3) = y*)][1 - G ( y*)] .
Assume 0 < y* < min {qM, yM}, and 0 < F(y*), G(y*) < 1, implying P(3) > 0.
The probability P(1) of inaction (action 1), is given by the following expression:
y*
(3) =
P(1) [1 - G(q)] f (q)dq .
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) [1 - F ( y)]g ( y)dy
y =0
The expected "per time unit" period 2 energy and retrofit costs as viewed from period 1, given
an optimal strategy for period 2, are, respectively
y*
(5) E[CH= [1 - G (q )] f (q )qdq H1
(2)]
q =0
12
Alternatively, the retrofit cost could be interpreted to include some energy cost. This is unproblematic as long
as the retrofit cost can be periodized.
13
Independence of costs in this context is not obvious. It may however be considered as realistic in cases where
the processes by which the two are affected, are quite different. But cases where the two costs will be correlated
are also possible, and are often more realistic. Such correlations could be either positive or negative. Negative
cost correlation may 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 may then make the development of retrofit
technologies more urgent, and more effort expended for this purpose. The two costs might then be negatively
correlated. On the other hand, common drivers may affect both costs. This is relevant e g when energy cost is
correlated with general production cost; when a retrofit involves some use of fossil energy, or 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.
8
y*
(6) E[CR (2)] [1 - F ( y )]g ( y ) ydy H1
=
y =0
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). H1 denotes (per-
period) energy consumption associated with the infrastructure as established in period 1.
Two alternative interpretations of retrofit costs have slightly 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 (6), 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 being operated in period 2 is
denoted EW(2), and equals the gross utility of the infrastructure, TU(H1) = Ty*H1, in the
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
(7) EW (2) = *[1 - P(3)]H1 - E[CH (2)] - E[CR(2)]}T .
{y
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
(8) EW (1) =U ( H1 ) - q1 H1 + EW (2) =( y * -q1 ) H1 + EW (2) .
Assume for now that the distribution of period 2 energy 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
(9)
dEW (1)
= U '( H1 ) - q1 +
EW (2)
+ [U '( H1 ) - y*] H1 .
dH1 H1 dy *
From (7) we find
EW (2)
d
H1
(10) = [1 - P(3)]T
dy *
Using the definition of EW(2), and setting the derivative in (9) equal to zero, we find the
following implicit expression for the optimal energy intensity of the infrastructure:
9
E (CH (2)) + E (CR(2))
q1 + T
H1
U '( H1 ) =
1 + [1 - P(3)]T
(11) y* y*
q1 + [1 - G (q )] f (q )qdq + [1 - F ( y )]g ( y ) ydy T .
= 0= 0 q y
=
y* y*
1 + [1 - G (q )] f (q )dq + [1 - F ( y )]g ( y )dy T
= 0= 0 q y
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 [1 P(3)]T. The expression in the curled bracket in the
numerator of (11) then denotes expected energy plus retrofit cost per unit of energy
consumption defined by the established infrastructure. (11) can then be given a simple
interpretation: the marginal utility of increased energy intensity associated with the
infrastructure investment (the left-hand side of (11)) 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 In 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 is higher, U'(H1)
is proportionately higher, and H1 lower.
For y* (= U(H1)/H1) we find
dy * 1
(12) -
yH * = ( y * -U ')
dH1 H1
Here y*-U' is positive and more so the more curved the utility function is in the neighborhood
of H1. Thus we can expect yH* < 0. 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).
To consider a more specific example, take the simple case where the (vNM) utility function is
quadratic: 15
14
This result is closely related to our initial assumption, that the size of the infrastructure investment is
exogenously given.
15
As usual the quadratic representation is a sufficiently precise (Taylor) approximation to any true utility
function U as long as the changes in H1 are not large.
10
(13) =
U ( H1 ) aH1 - bH12
so that
(14) U '( H1 )= a - 2bH1
Inserting into (12), this yields
(12a) yH* = -bH1.
In this case, the absolute value of yH* is directly proportional to the curvature coefficient b in
the utility function. Thus a relatively "flat" utility function implies a small change in the cut-
off value y*, for given H1. A relatively "curved" function implies a large change in y*.
Note also that with this specification
(15) U''(H1) = -2b.
From (11), and given quadratic utility, the coefficient b plays a major role in defining the
response of both H1 and y* to exogenous parameter changes. When b is small, H1 will
respond strongly and y* weakly to such changes. The opposite holds when b is larger. The
interpretation of small or large b is thus of some importance. Intuitively, when b is small, the
economy has several ways in which to design its infrastructure that are considered as "almost"
equivalent for a given set of relevant parameters. Small parameter changes can then provoke a
relatively large response for the "optimal" infrastructure as initially perceived.
Our concept of "optimality" is here defined with respect to the decision problem as set up. It
is useful to identify different sources of inefficiency in deciding the initial infrastructure
investment. We will distinguish between the following five points.
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
base on an expectation of something close to current energy cost on average in period
2, while the correct distribution implies higher average energy costs. 16 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, as the closedown
and/or retrofit options will be exercised in more cases (as we will see below).
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 case is relevant
whenever the authorities, in the economy in question, implement emissions prices that
are lower than "globally correct" prices. We will argue that it is a highly relevant case:
16
We argue that this could occur even in cases where the entity making the infrastructure decisions would face a
"correct" overall energy price (including the "true" costs of emissions). One such case is where the
administrative procedure for making public investments involves incorporation of future costs and benefits for
only a limited period (say, 20 years), while an appropriate investment decision would need to involve a much
longer period (say, up to or exceeding 50 years). An increasing future energy price (also beyond the 20 year
term) would add to the bias involved in such an investment procedure.
11
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.
C) Future retrofit costs are incorrectly anticipated. Such costs are likely to depend on the
rate of technological progress for retrofit technologies, and are inherently unknown at
the investment stage in period 1. A too optimistic view of this development so that the
anticipated retrofit cost distribution is lower than the correct distribution will then lead
to bias in the direction of excessive energy intensity for the initially established
infrastructure. Note however that 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.
On the other hand, instead of "technology optimism" one could have "technology
pessimism" (a too pessimistic view of the distribution of future retrofit costs), which
would work in the opposite direction.
D) Future retrofit costs are correctly anticipated, but are higher than socially optimal
costs. This could be the case when there is a too low R&D effort in developing new
energy technologies, including those for retrofits. This would leads to a higher than
optimal distribution of retrofit costs. This would increase overall implementation costs
(in particular, the minimum of energy and retrofit costs) in period 2. Given correct
perceptions in period 1, the choice of energy intensity of the infrastructure is now
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). On
balance the second factor is likely to dominate, and, thus, socially excessive energy
consumption in period 2. In such cases, and assuming no other distortions, the energy
intensity of the initially chosen infrastructure would, in fact, be suboptimal. On the
other hand, ex post energy costs in period 2 for given infrastructure are overoptimal.
This follows straightforward 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.
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 is 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.
12
3. Impacts of Shifts in the Energy Cost Distribution
We will now study some implications of changes in the distribution of energy costs in period
2. Consider then a downward shift in the distribution of energy costs, that leaves all other
parameters unchanged. Call the new distribution F1(q) = F(q+) (so that F1 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 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*
(5a)
E[CH (2)] = [1 - G (q )] f (q + )qdq H1
q =0
y*
(6a) 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 y* is not significantly altered): 17
y*
dP(1)
(16)
d
= [1 - G(q)] f '(q)dq .
q =0
y*
dP(2)
(17) = - f ( y ) g ( y )dy
d y =0
dE[CH (2)]
y*
E[CH (2)] dH1
(18) (1 - G (q ))
= f '(q )qdq H1 +
d H1 d
q =0
dE[CR (2)] y* E[CR (2)] dH1
(19) - f ( y ) g ( y )dy H
= 1+ .
d H1 d
q =0
dH1/d is found totally differentiating (11) with respect to H1 and .
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 (18)) 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
17
This requires that the coefficient b in (12a) is small, and that H1 changes "relatively much" compared to y*.
13
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.
A positive shift in shifts overall unit energy plus retrofit costs down. From (11), H1 then
increases. Intuitively, lower overall expected operating (energy plus retrofit) costs in period 2
leads to selection of infrastructure with a higher energy intensity. The response of H1 to
changes in unit costs could in principle be large. Overall expected energy and retrofit costs
could then easily increase when the distribution of energy costs shifts down, and this is
correctly anticipated in period 1. Quantitative effects here require specifying functional forms
in more detail.
For the particular case of a quadratic utility function (12a) in infrastructure energy intensity,
y* falls when H1 increases. (12a) and (18) however show that when the response of H to cost
is high (and b is low), y* changes little in response to energy cost changes.
4. Impacts of Shifts in the Retrofit Cost Distribution
This section considers some impacts of changes in costs of retrofitting in period 2. By this we
mean to study impacts on outcomes, from both marginal changes in retrofit costs, and from
the retrofit option being at all available.
y*
(5b) E[CH (2)] = [1 - G (q + )] f (q )qdq H1
q =0
y*
(6b)
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)
(20) = - f ( y ) g ( y )dy
d y =0
y*
dP(2)
(21)
d
= [1 - F (q)]g '(q)dq .
q =0
dE[CH (2)] y* E[CH (2)] dH1
(22) - f ( y ) g ( y )dy H
= 1+
d H1 d
q =0
dE[CR (2)]
y*
E[CR (2)] dH1
(23) (1 - F (q ))
= g '(q )qdq H1 +
d H1 d
q =0
Interpretations are in this case similar to those in the case of energy cost changes. When
increases, the distribution of retrofit costs is (in analogous fashion to the energy cost
14
distribution, in Section 3) is shifted 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
(22)), while the change in expected retrofit cost per established energy unit is ambiguous (the
integral on the right-hand side of (23)). Their sum, EC(2), falls unambiguously.
Consider an alternative case where the retrofit option is no longer available (NR denoting the
"no retrofit" case) 18. We have the following probability of closedown in period 2:
(24) 1- F
PNR (3) =( 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.
5. The Value of the Closedown Option
We now study the way in which the very availability of the closedown option affects the
overall solution, initial energy intensity, and cost variables. The closedown option could be
viewed as "unavailable" when the total utility per unit of energy consumed for the chosen
infrastructure, y* = U(H1)/H1, is so high that closedown is never a realistic option (i.e., [1-
F(y*)][1-G(y*)] is "very small"). It is here easiest to think of cases where the infrastructure
involves a high sunk cost relative to energy consumption (such as, perhaps, for urban
structures including housing and transport systems). We may then consider the limit as y*
tends to infinity.
Expected energy consumption in period 2 is then simply P(1) multiplied by TH1. Availability
of the retrofit option, for given H1, now reduces expected energy consumption by P(2)TH1 in
period 2. 19
18
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.
19
Of course, H1 will not in general be given but determined endogenously; H1 will tend to be greater when the
retrofit option is available since expected costs are then lower.
15
The expected energy and retrofit cost, and ex ante probabilities of "business-as-usual"
operation and retrofit in period 2 are now still given respectively by the general expressions
(2)-(6). Note however that we now have P(1) + P(2) = 1. Expected ex ante utility of second-
period operation is now
(7a) EW (2) = * H1 - E[CH (2)] - E[CR(2)]}T .
{y
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
(11a) U '( H1 ) = .
1+ T
This can now be compared to costs when closedown is an option. There are two main
differences between (11) and (11a). First, the respective expressions E[CH(2)] and E[CR(2)]
are now greater, relative to the ex ante probability of operation and even more absolutely.
Secondly, the term [1-P(3)] in the denominator of (11) has vanished. 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 (11). Overall, U'(H1) is increased,
and H1 reduced. Having an effective closedown option thus increases the energy intensity of
the original infrastructure investment, relative to the case with no such option. This is of
course unsurprising and intuitive: when the option to close down is available, it will be used
only in states where both retrofit and energy costs are very high. Exercising the closedown
option eliminates costs in these most expensive operation states, which in turn provides
incentives to raise the infrastructure's initial energy intensity.
Notably, energy consumption and expenditure related to the infrastructure in period 2 are
lower when the closedown option is available, than when it is not. This conclusion may
however be deceptive: we have here assumed that when closing down no energy expenditure
is incurred whatsoever. This is unrealistic since the closed down infrastructure will need to be
replaced by an alternative that will likely demand energy (although presumably less than that
initially established; in particular since replacements tend to occur in states with high energy
costs in period 2).
6. Endogeneity of Retrofit Costs
The retrofit options available in period 2 are likely to follow at least in part from technology
developments over period 1, and these may in turn be influenced by R&D efforts. The idea in
this section is to study possible effects of such efforts.
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
16
has been discussed e g by Goulder and Schneider (1999), Goulder and Mathai (2000),
Bonanno et al (2003), and Greaker and Pade (2008). A separate issue is that while it may be
very difficult to reach an international agreement to effectively reduce GHG emissions, 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. 20
Here we simply assume that the "decision unit" that carries out the initial infrastructure
investment, may also carry out R&D activity to affect the options, and costs, of retrofitting
this particular infrastructure in period 2. Assume that such investment only affects retrofit
costs for this particular infrastructure, 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. 21
This upward shift in distribution is the same as a downward cost shift, as in section 4, and was
there given from (23), as a result of changes in a shift parameter for this distribution.
Here, consider the following modified discounted utility as viewed from period 1:
(25) 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 G function is shifted, with
shift parameter , and where the size of is a positive function of R. We then derive the
following general optimality condition with respect to R:
(26)
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. 22 The partial derivatives in (26) are found
from (20)-(23) plus (11) differentiated. While (26) 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 be
found under the problem (26)). 23 Consider here again a standard second-order Taylor
expansion (and thus quadratic formulation)
20
See in particular Barrett (2006, 2009).
21
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).
22
A second-order condition here also needs to be fulfilled. A sufficient condition here is that R is decreasing in
R.
23
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.
17
(27) (R) = R R2
where and are positive constants, so that the first- and second-order derivatives of the
function are given by
(28) '(R) = 2R > 0
(29) ''(R) = 2 < 0
The main point here is that when the curled bracket in (26) is large, the first derivative '(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 here assumed that the investment in question only affects costs for one particular
infrastructure facility. In practical cases of individual R&D projects that affect future
infrastructure costs, such R&D expenditures are likely to have most of their effects on costs
for other projects. 24 In the context of our model, the marginal social benefit of R&D is much
greater than the "private" benefit for the infrastructure project sponsor. Consider here a case
where (27) correctly represents the overall social impact of R on retrofit costs, while the
private impact is only a fraction h (< 1) of the 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
(28), and thus
(28a) '(R;h) = h( 2R) > 0.
When h is smaller, R must be smaller to fulfil (26). 25 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. 26
7. Simulations
We will now illustrate some basic properties of this model, through simulations of energy and
retrofit cost distributions, under some simplified assumptions. The simulations will focus on
the period 2 cost structure, and how conditional energy, retrofit and total expected cost is
affected by expectation and variance of the two cost items, given an optimal ex post strategy
to minimize these costs. 27 Most of these simulations depart directly from the expected cost
expressions (5) (expected energy cost in period 2) and (6) expected retrofit cost in period 2).
We generally take H1 to be given (i e we do not study the optimal H1 decision) and equal to
24
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).
25
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.
26
One way of securing this is strong patent laws. But this has other 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).
27
Simulations have been done using Matlab.
18
unity (in other words, the basic energy intensity of the infrastructure is unity), and thus study
changes in unit energy and retrofit costs only.
In the simulations, if nothing is otherwise stated, the (unconditionally) expected energy and
retrofit cost are both kept constant, setting E(q(2)) (= gf(q)dq) = 2, and E(y(2)) (= yg(y)dy) =
3. (E(q(2)) is the (unconditional) 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, energy costs are thus, in the
"benchmark" case, assumed to be lower in expectation. The distributions of energy and
retrofit costs, F(q) and G(y), are both assumed to be log-normal, and independent. 28 The cut-
off level for costs (beyond which the infrastructure will be abandoned) is, unless otherwise is
stated, set at y* = 10 (= 5 times unconditionally expected energy costs), for all simulations.
Figure 1 describes conditional expected energy costs (blue), retrofit costs (green), and total
costs (red), and how they vary with changes in the variances of energy cost (q) and retrofit
cost (y). Thus, under certainty no retrofits would ever take place in period 2 (since retrofit cost
would be higher than the costs of normal, non-retrofitted, operation), and period 2 cost would
simply equal 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 lower, and more so the greater is uncertainty, represented here by the
variances of q and y. In the figure, 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).
Equally important here is the ability to avoid energy costs as such, as a function of increased
uncertainty. We see from Figure 1 that, in many cases, at least half of potential energy cost is
avoided ("conditional Eq" being less than unity).
Figures 2 and 3 repeat similar exercises except for cases where the respective variances are,
respectively, much lower, and much higher, than those shown in Figure 1. From Figure 2,
when the variances are very low, the solution tends to one with only the basic energy cost
option with leads to an energy cost equal to 2, and no retrofit cost (as there are very few cases
where the retrofit alternative outcompetes the energy cost alternative). We see however, in the
lower figure of Figure 2, that there is some more substantial substitution of retrofit cost for
energy cost when the variance on y (retrofit cost) increases beyond 0.2. Overall cost however
remains close to 2 in these cases.
28
The independence assumption can be questioned, as noted above, and will be altered in subsequent work.
19
Figure 1: Expected energy, retrofit and total costs, and the variance of q and y.
2 2
V(q)=1.13
V(y)=2.55
1.5 1.5
Costs
Costs
1 1
0.5 0.5
0 2 4 6 8 0 5 10 15 20
Variance q Variance y
2
V(q)=2.52
Energy Costs
1.5
Retrofit Costs
Costs
Total Costs
1
0.5
0 5 10 15 20
Variance y
Figure 2: Expected energy, retrofit and total costs, and the variance of q and y.
(Low variances)
2 2
1.5 V(y)=0.07 1.5 V(q)=0.09
Costs
Costs
1 1
0.5 0.5
0 0
0 0.05 0.1 0.15 0.2 0 0.1 0.2 0.3 0.4
Variance q Variance y
2
1.5
V(q)=0.07
Costs
1 Energy Costs
Retrofit Costs
0.5 Total Costs
0
0 0.2 0.4 0.6 0.8
Variance y
20
Figure 3: Expected energy, retrofit and total costs, and the variance of q and y.
(High variances)
1.2 1.2
1 V(y)=15 1 V(q)=6.8
0.8 0.8
Costs
Costs
0.6 0.6
0.4 0.4
0.2 0.2
0 100 200 300 0 200 400 600
Variance q Variance y
1
0.8 V(q)=24
Costs
Energy Costs
0.6
Retrofit Costs
Total Costs
0.4
0.2
0 500 1000 1500
Variance y
In Figure 3 the situation is rather different. Here variances on both q and y are assumed to be
much larger than in the Figure 1 alternative. The scope for substitution of the two factors, and
the corresponding scope for cost avoidance, is then much greater. Both energy and retrofit
costs are here incurred about proportionately in the various alternatives, and more than half of
potential total cost is avoided in many cases; and in some cases as much as 75 percent of total
potential energy cost is avoided. Clearly, the amount of energy cost that is effectively tied up
by the initial infrastructure investment is then rather small.
Figures 4-6 show an alternative set of simulations, where the expectations of energy and
retrofit costs are changed parametrically, for given variances. In Figure 4, variances are in an
"intermediate" range, both equal to 2.5 (and again, when nothing else is indicated, Eq = 2, and
Ey = 3). We find, as already commented, that an increase in expectation for one cost item
leads to an initial increase (roughly, up to the expected value of the other cost item) and
subsequently a decrease in conditional expected cost associated with that item. We also see
that, at most, about one third of the overall potential cost is avoided by optimal substitution
given these variances (when both expectations are equal; either both are 2 or both are 3).
Figure 5 deals with the case where the variances are "low". Overall, there is then little scope
for cost avoidance; total expected cost is very close to the overall lower cost alternative. There
are here also more dramatic changes in conditional expected q and y when expectations
increase beyond the value of the opposite cost parameter; this is of course due to the almost
complete phasing out of retrofit cost whenever the expectation exceeds that for energy cost;
and vice versa.
21
In Figure 6, by contrast, variances are "high". In this case, there is no apparent strong
tendency for either one of the cost items to be phased out when its expectation increases. The
reason is that, even when the expected energy (retrofit) cost is high, there is still a high
probability that the actual energy (retrofit) cost is low, and at the same time the retrofit
(energy) cost higher, leading to energy use (retrofit) being the chosen application.
The last (bottom) tables in each of Figures 4-6 show overall effects of changes in closedown
costs; or rather effects of the possibility to exercise the closedown option. We assume that the
cut-off value for costs (beyond with the closedown option will be exercised) is non-stochastic
but is varied parametrically in the table. As can be seen from the figures, for "reasonable"
values of the cut-off level y* (beyond 5; a quite low level), 29 the closedown option has
consequence for overall costs only in the high variance alternative (Figure 6); and seemingly
not at all in the two other alternatives (this is seen by the "total cost" curves not being
sensitive to y*). The reason is, obviously, that in the (low- and moderate-variance)
alternatives, the closedown option will be exercised in few cases; and, when exercised, on
average lead to only moderate cost savings. This is different in the high-variance alternatives
(Figure 6), where the "total cost" curves are obvious (increasing) functions of y*. The
simulations here indicate an overall (expected energy plus retrofit) cost saving of about 10
percent when the cut-off level y* is lowered from 15 to 5) (and a similar relative cost saving
for energy costs alone).
Figure 4: Expected energy, retrofit and total costs,
and the expected values of q, y and y*.
3 2
1.5
2
Costs
Costs
1
1
0.5
0 0
1 2 3 4 1 2 3 4
Expected q Expected y
2
1.5 Energy Costs
Costs
Retrofit Costs
1 Total Costs
0.5
5 10 15
Y*
29
Note that, for y* = 5, the overall net utility value of the infrastructure after infrastructure costs have been sunk
is 2.5 times ex ante expected energy costs; which seems quite moderate.
22
Figure 5: Expected energy, retrofit and total costs, and the expected values of q,
y and y*. (Low variance of q and y; V(q)=0.07; V(y)=0.09)
3 2
1.5
2
Costs
Costs
1
1
0.5
0 0
1 2 3 4 1 2 3 4
Expected q Expected y
2
Energy Costs
Retrofit Costs
1.5
Total Costs
Costs
1
0.5
0
5 10 15
Y*
Figure 6: Expected energy, retrofit and total costs, and the expected values of q,
y and y* (High variance of q and y; V(q)=7; V(y)=15)
2 1.5
1.5
1
Costs
Costs
1
0.5
0.5
0 0
1 2 3 4 1 2 3 4
Expected q Expected y
1.4
1.2
1 Energy Costs
Costs
Retrofit Costs
0.8 Total Costs
0.6
0.4
5 10 15
Y*
23
Some striking results follow from these simulations. While the expectation of unconditional
energy cost E(q(2)) equals 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 due to the decision maker optimally exercising either of our
two other options ex post, retrofitting or closing down. Energy costs are avoided in states of
the world where they are particularly high, and also in states when they are low but retrofit
costs are even lower. These effects are stronger, the more variable energy and retrofit costs
are (for given unconditional expectations).
Concentrating on energy costs alone, (conditional) expected energy costs can be reduced
when costs become more variable (for given unconditional expectations), for three separate
reasons. It is then 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.
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.
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 neither energy consumption nor 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.
Another reservation is that the distributions of energy and retrofit costs are assumed to be
independent. 30 Perhaps more likely, these costs are in practice positively correlated (when one
is high, it is more likely than otherwise that the other is high, etc.). Positive correlations lead,
in general, to less scope for cost savings.
Note also that these simulations do not directly address a key issue for the overall analysis,
namely, the effect on energy intensity of the initial infrastructure investment. The general
conclusion, from section 2 above, is that having options (through retrofit and closedown
possibilities) to reduce energy use later increases the energy intensity of the infrastructure. We
have no general conclusions on the strength of this effect, but it could be substantial, in
particular when infrastructure investment choice (unlike in our model) implies a choice
between two discretely different infrastructure systems (such as between a transport system
"largely" dependent on private, or public transport), and where the policy maker is initially
"almost indifferent" between the two.
8. Summary of Sources of Market Inefficiency, and Final Comments
A main aim of this paper has been to study decisions to invest in infrastructure that commits
society to potentially high levels of energy use, and carbon emissions, for a long future
30
The log-normal distribution assumption could, potentially, also be attacked. Log-normality is however a rather
robust assumption in this context; see e g Schuster (1984).
24
period. We have studied factors behind inefficiency of such investments, and implications for
GHG emissions (which would be excessive in the long run), and considered how any
inefficiencies can be avoided or counteracted. Another aim of the paper has been to study the
impacts of two types of policy interventions that may be applied after infrastructure
investment have been sunk: namely first, "retrofitting" the infrastructure (by making an
additional, later, investment that removes the energy demand and/or emissions due to the
infrastructure, while retaining its utility value to the public); and secondly, closing it down (a
more drastic alternative, as the utility value of the infrastructure is then is removed together
with energy use and emissions). Most of the focus here has been on the retrofit alternative, on
its ability to reduce subsequent (energy and environmental) costs, and its effect back on
energy intensity of the initial infrastructure.
Considering the first of these objectives, inefficient infrastructure choice can result from all
the types of market failure discussed in section 2 above. The five types discussed were A)
anticipated energy and emissions costs below actual prices; B) too low actual energy
(including emissions) costs facing private agents, set by the respective governments; C)
incorrectly anticipated (and lower than actually realized) retrofit costs; D) too high realized
retrofit costs; and E) excessive discounting.
Roughly, these explanations can be classified into two groups: one related to insufficient or
faulty general climate-related or energy policies (including insufficient emissions pricing and
technology support); and another related to inefficiencies and incompleteness in the execution
of policy. Points B and D fall largely into the former category, while points A and C mainly
into the latter. Point E might conceivably fall into either category.
One easily understands the main reason for faulty or inadequate policies of the individual
governments hosting infrastructure projects: 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 for so doing. The second group of explanations has a more diverse set
of explanations, which are however all related to either policy incompleteness or to various
forms of "irrationality" (or "behaviorism") in the policy process.
Policy incompleteness arises when policies are based on ad hoc rules that are outcomes not of
a deliberate and complete optimisation process, but instead of a much simplified process that
may lead to systematic biases in a climate context. One such is case is when discounting of
public projects with climate impacts is determined administratively by a common rule 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 based on no explicit cost-benefit calculation whatsoever.
Considering implications for initial infrastructure design of problems related to categories A-
E above, all categories except D are likely to make energy consumption excessive. 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 here too infrequent (as the initially excessive energy intensity makes it
excessively likely that the infrastructure is later retrofitted or abandoned). On balance
expected energy use is still typically excessive. The fourth category (D) is different in that it
25
tends to make energy intensity of the initial investment too low. An efficient policy would in
this case lead to lower total infrastructure cost associated for given energy intensity (as the
optimal R&D investment would reduce retrofit costs), and this cost reduction would make an
initial energy intensity increase attractive. The infrastructure will however be retrofitted in
more cases; this constitutes the main effect for energy consumption, which is reduced in
response. In the fifth case, E, the inefficiency could go either way. In one case, excessive
optimism over future retrofit possibilities (as reflected in sentiments such as, "future
technology will solve everything") could here trigger excessive fossil-fuel intensity in period
1, and overall excessive fuel consumption and emissions.
The second main objective of the paper was to study impacts of the option for later retrofit of
an established infrastructure on total expected costs, expected energy (including
environmental) costs, and initial energy intensity of infrastructure. These are topics of
sections 4 and 6, and of the simulations in section 7. These simulations 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. The simulations also indicate that this fraction
might be large, under arguably plausible assumptions. 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. These simulations however have limitations. First, they
are based on ex ante distributions of energy and retrofit costs that are both log-normal and
known, and these two distributions are assumed to be independent. When costs are instead
positively correlated, a smaller fraction of the overall expected costs can be avoided by ex
post exploitation of low-cost retrofit options. Positive correlation of energy and retrofit costs
is here perhaps as likely as cost independence (or more so), as the different cost items that
apply to the energy sector may easily co-vary (possibly, with the costs of the whole range of
alternative energy technologies, including both renewable energies and pure retrofit
technologies co-varying, and fossil-fuel energy costs co-varying with renewables costs). 31
This issue however awaits further analysis, and will be pursued in extensions of this work.
Note that with a long expected time from infrastructure investment to the availability of a
relevant alternative option (retrofit, or closedown), and the decision maker discounts 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.
Some limitations of our analysis must be pointed out. One is our assumption of only two
periods, "the present", and "the future", allowing 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 far less tractable, but should still be pursued in follow-up work. Relevant costs (of
31
Positive correlation between renewables costs and fossil-fuel costs may in turn follow from a variety of
factors. These include downward pressure on fossil fuel market prices from higher levels of renewables
production; that less fossil fuels will actually be extracted by any given time frame thus moving less up along the
marginal extraction cost curve for such fuels; and that efficiency in fossil fuel extraction may be positively
correlated with renewable energy production costs.
26
energy, emissions, and related to the retrofit technology), as well as benefits (the current
utility value of the infrastructure technology), all in reality evolve 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 in fact considered in a
companion paper to the current one, Framstad and Strand (2009), where energy and
environmental costs evolve continuously; in other respects however assumptions are here
much simpler, in particular, a fixed retrofit cost is assumed. 32
Another extension is to consider different types of (partial) retrofit, where neither fossil
energy use nor carbon emissions are reduced to zero. Such alternatives are typically less
costly than full retrofit, and may be more efficient in some cases. Another feature that may
need amendment is our assumption that, upon a possible infrastructure closedown, "nothing
happens". A more satisfactory analysis would involve a new 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.
32
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.
27
References
Arthur, Brian (1983), Competing Technologies, Increasing Returns, and Lock-In by Historical
Events. Economic Journal, 99, 116-131.
Barrett, Scott (2006), Climate Treaties and Breakthrough Technologies. American Economic
Review, 96 (2), 22-25.
Barrett, Scott (2009), The Coming Global Climate-Technology Revolution. Journal of
Economic Perspectives, 23 (2), 53-73.
Brueckner, Jan K. (2000), Economics of Cities Theoretical Perspectives. Cambridge, UK:
Cambridge University Press.
Buonanno, P., Carraro, C. and Galeotti, M. (2003), Endogenous Induced Technical Change
and the Costs of Kyoto. Resource and Energy Economics, 25, 11-34.
David, Paul A. (1992), Path Dependence and Predictability in Dynamic Systems With Local
Network Externalities: A Paradigm for Historical Economics. In D. Foray and C. Freeman
(eds.): Technology and the Wealth of Nations, pp 208-231. London: Pinter.
Framstad, Nils Christian and Strand, Jon (2009), Inertia in Infrastructure Investment With
"Retrofit": A Continuous-Time Option Value Approach. In process, DEC-EE, World Bank.
Geweke, John (2001), A Note on Some Limitations of CRRA Utility. Economics Letters, 71,
341-345.
Glaeser, Edward L. and Kahn, Matthew E. (2008), The Greenness of Cities: Carbon Dioxide
Emissions and Urban Development. NBER Working Paper, no 14238.
Golombek, Rolf and Hoel, Michael O. (2005), Climate Policy Under Technology Spillover.
Environmental and Resource Economics, 31, 201-227.
Golombek, Rolf and Hoel, Michael O. (2006) "Second-Best Climate Agreements and
Technology Policy", Advances in Economic Analysis & Policy, 6, No. 1, Article 1.
Goulder, L. H. and Mathai, K. (2000), Optimal CO2 Abatement in the Presence of Induced
Technical Change. Journal of Environmental Economics and Management, 39, 1-38.
Goulder, L. H. and Schneider, S. H. (1999), Induced Technological Change and the
Attractiveness of CO2 Abatement Policies. Resource and Energy Economics, 21, 211-253.
Greaker, Mads and Pade, Lise-Lotte (2008), Optimal CO2 Abatement and Technological
Change. Discussion Paper no 548, Research Department, Statistics Norway.
28
Gusdorf, François and Hallegatte, Stéphane (2007a), Behaviors and Housing Inertia are Key
Factors in Determining the Consequences of a Shock in Transportation Costs. Energy Policy,
35, 3483-3495.
Gusdorf, François and Hallegatte, Stéphane (2007b), Compact or Spread-Out Cities: Urban
Planning, Taxation and the Vulnerability to Transportation Shocks. Energy Policy, 35, 4826-
4838.
Ha-Duong, M., Grubb, Michael, and Hourcade, Jean-Charles (1997), Influence of
Socioeconomic Inertia and Uncertainty on Optimal CO2-emission Abatement. Nature, 390,
270-273.
Ha-Duong, M. (1998), Quasi-Option Value and Climate Policy Choices. Energy Economics,
20, 599-6210.
Hourcade, Jean-Charles and Crassous, Renaud (2008), Low-Carbon Societies: A Challenging
Transition for an Attractive Future. Climate Policy, 8, 6-0-612.
Jaccard, M. (1997), Heterogeneous Capital Stock and Decarbonating the Atmosphere. Does
Delay make Cents? Working Paper, Simon Fraser University.
Jaccard, M. and Rivers, N. (2007), Heterogeneous Captal Stocks and the Optimal Timing for
CO2 Abatement. Resource and Energy Economics, 29, 1-16.
Lecocq, Franck, Hourcade, Jean-Charles and Ha-Duong, M. (1998), Decision Making Under
Uncertainty and Inertia Constraints: Sectoral Implications of the When Flexibility. Energy
Economics, 20, 539-555.
Leibowitz, S. and Margulis, S. (1995), Path Dependency, Lock-In and History. Journal of
Law, Economics and Organization, 11, 205-226.
Lovins, Amory B. (1977), Soft Energy Paths: Toward a Durable Peace. Harmondsworth,
UK: Penguin Books.
Shalizi, Zmarak and Lecocq, Franck (2009), Economics of Targeted Mitigation Programs in
Sectors with Long-Lived Capital Stock. World Bank Working Paper, WPS 5063. Background
paper for the World Development Report 2010.
Schuster, Eugene F. (1984), Classification of Probability Laws by Tail Behavior. Journal of
the American Statistical Association, 79 no 388, 936-939.
Strachan, N., Foxon, T., and Fujino, J. (2008), Modelling Long-Term Scenarios for Low-
Carbon Societies. Climate Policy, 8 (supplement).
Strachan, N., Pye, S., and Hughes, N. (2008b), The Role of International Drivers on UK
Scenarios of a Low-Carbon Soceity. Climate Policy, 8 (supplement), S125-S139.
Strand, Jon (2009), Inertia in Infrastructure Development: Some Analytical Aspects, and
Reasons for Inefficient Infrastructure Choices. Unpublished, DECRG, World Bank.
29
Weitzman, Martin L. (2009), On Modeling and Interpreting the Economics of Catastrophic
Climate Change. Review of Economics and Statistics, 91, 1-19.
Wigley, T. M. L., Richels, Richard and Edmonds, Jae (1996), Economic and Environmental
Choices in the Stabilization of Atmospheric CO2 Concentrations. Nature, 379, 240-243.
World Bank (2003), World Development Report 2003: Sustainable Development in a
Dynamic World. Transforming Institutions, Growth, and Quality of Life. Washington DC:
The World Bank.
World Bank (2009), World Development Report 2010: Development and Climate Change.
Washington DC: The World Bank.
30