ï»¿ WPS6430
Policy Research Working Paper 6430
Energy Intensive Infrastructure Investments
with Retrofits in Continuous Time
Effects of Uncertainty on Energy Use
and Carbon Emissions
Nils Christian Framstad
Jon Strand
The World Bank
Development Research Group
Environment and Energy Team
April 2013
Policy Research Working Paper 6430
Abstract
Energy-intensive infrastructure may tie up fossil value of waiting to invest. Higher energy intensity is also
energy use and carbon emissions for a long time after chosen for the initial infrastructure when uncertainty
investments, making the structure of such investments is greater. These decisions are efficient given that
crucial for society. Much or most of the resulting carbon energy and carbon prices facing the decision maker are
emissions can often be eliminated later, through a costly (globally) correct, but inefficient when they are lower,
retrofit. This paper studies the simultaneous decision which is more typical. Greater uncertainty about future
to invest in such infrastructure, and retrofit it later, in climate costs will then further increase lifetime carbon
a model where future climate damages are uncertain emissions from the infrastructure, related both to initial
and follow a geometric Brownian motion process with investments, and to too infrequent retrofits when this
positive drift. It shows that greater uncertainty about emissions level is already too high. An initially excessive
climate cost (for given unconditional expected costs) climate gas emissions level is then likely to be worsened
then delays the retrofit decision by increasing the option when volatility increases.
This paper is a product of the Environment and Energy Team, Development Research Group. It is part of a larger effort by
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
Energy Intensive Infrastructure Investments with Retrofits in Continuous
Time: Effects of Uncertainty on Energy Use and Carbon Emissions
Nils Christian Framstad and Jon Strand 1
Key words: Greenhouse gas emissions; long-term investments; retrofits; uncertainty; option
value of waiting.
JEL classification: C61; Q54; R42.
Sector Boards: Energy and Mining; Environment
1
Framstad: Department of Economics, University of Oslo, Box 1095 Blindern, 0317 Oslo, Norway. Strand:
Development Research Group, Environment and Energy Team, the World Bank, Washington DC 20433. This
paper is part of the research activities at the Development Research Group, Environment and Energy Team at the
World Bank, Oslo Centre for Research on Environmentally friendly Energy (CREE) ,and the Centre for the
Study of Equality, Social Organization (ESOP), at the Department of Economics, University of Oslo. This
research has been supported by a grant from the World Bankâ€™s Research Support Budget. Viewpoints expressed
in this paper are those of the authors and not necessarily of the World Bank, its management or member
countries.
1
1. Introduction
Large and energy-intensive infrastructure poses a serious concern for climate policy. Such
infrastructure is found on both the supply side (such as power plants and other energy supply
infrastructure) and demand side of the energy-intensive sectors of the economy (such as urban
structure and transport systems including the balance between public and private transport). It
is the source of more than half of total carbon emissions from fossil fuels in high-income
countries, as well as a significant and growing share in emerging economies. Proper concern
for such systems requires considering their impacts over long future periods (in some cases
100 years or more). Infrastructure with less permanent albeit still persistent effects on
emissions includes motor vehicles (fossil-fuel versus electric or renewable-powered),
household appliances, and home heating and cooling systems. Decisions for such long-lasting
investments will always be subject to great uncertainty. Making â€œwrongâ€? investments could
tie up inefficiently high emissions levels for long future periods, and make it difficult to reach
ambitious climate policy targets later. Many emerging economies today have currently, and
are planning over the next 20â€“30 years, a high rate of such investments.
â€œRetrofitsâ€? can help to alleviate this problem. For example, coal-fired power plants
potentially (perhaps soon) could be retrofitted with carbon capture and sequestration
technologies; or to instead utilize renewable, non-fossil, fuels. Urban areas which depend
mainly on private transport, can likewise be â€œretrofittedâ€? by adding public transport systems.
Motor vehicle fleets can be â€œretrofittedâ€? by replacing gasoline- or diesel-driven vehicles with
electric vehicles (with renewable-based power production).
In this paper we consider an initial infrastructure investment carried out at a point of time
that can be more or less energy intensive. Its emission intensity stays constant until it is
retrofitted, which we define to mean that it is purged of all of its emissions, while the utility of
its basic services is fully retained after the retrofit.
Two important issues related to energy use and GHG emissions are here 1) the optimal
energy and emissions intensity for the initial infrastructure, and 2) the optimal retrofit policy
(when, if ever, should the infrastructure be retrofitted). We first (in section 2) consider these
issues under certainty, where combined energy and environmental costs follow a deterministic,
increasing, path over time. In sections 3â€“5 we assume that energy (including climate and
2
other environmental) costs are uncertain and follow a geometric Brownian motion process
with constant positive drift. The future retrofit cost is still considered as certain, and constant.
An important issue is efficiency. If policy makers face â€œglobally correctâ€? price signals
(prices where all external effects are taken into consideration), the resulting (dynamic)
allocation will tend to be socially optimal. 2 Most of our formal presentation focuses on cases
of efficiency, focusing on questions (I) â€“ (VI) below. It thus represents a starting point for a
fuller analysis of cases, more plausible in emerging economies today, where energy prices are
too low as decision makers are not, in their investment and retrofit decisions, properly
accounting for global climate costs. Our model as presented however still has rather direct
implications for cases of inefficiency. This is discussed in the final section below.
Some key questions to be discussed in this paper are:
(I) When (if ever) will the infrastructure be â€œretrofittedâ€?; and what principles govern a
retrofit decision?
(II) What is the probability distribution of the time to a â€œretrofitâ€??
(III) What is the probability distribution of accumulated carbon emissions from the
infrastructure?
(IV) What is the optimal level of carbon emissions related to the initial infrastructure
investment?
(V) What are expected damages, and the distribution of damages, both in present
discounted value terms, due to carbon emissions over the lifetime of the infrastructure
investment?
(VI) Is such infrastructure investment likely to be optimal in the real world; and if not,
what types of distortions lead to non-optimality and in what way, and how can optimality, or
an improved solution, be achieved?
Of particular concern is the impact of â€œvolatilityâ€?, by which we here mean the
instantaneous relative variance on a continuous stochastic process by which the marginal
GHG emission cost for society changes over time. We first (in section 2) study a â€œbenchmarkâ€?
case where the climate cost trajectory is assumed to be certain. Our main concern, however
(in sections 3-4), is with the case of uncertain damages.
2
The main assumptions are that certain convexity conditions on choice and production sets are fulfilled; and that
all economic actors behave competitively (are price takers).
3
Our paper is technically similar to Pindyck (2000, 2002); and most directly builds on
Pindyck (2000). 3 Our solution for optimal timing of a technology retrofit, for given initially
invested infrastructure (questions (I) and (II) above), reproduces Pindyckâ€™s result (albeit with
a modification; see Framstad (2011)). Our more important contribution is to extend Pindyckâ€™s
analysis by deriving the initial infrastructure investment decision (question (IV)),
simultaneously with the future retrofit decision. We are also the first to directly answer
questions (III) and (V), on effects for accumulated carbon emissions. While we have less
formally to say about point (VI), we discuss this issue and its relation to our model in the final
section 6.
The combined infrastructure establishment and retrofit issue have been studied by Strand
(2011a), Strand and Miller (2010) and Strand, Miller and Siddiqui (2011), albeit in simpler
discrete (two-period) models. Shalizi and Lecocq (2009) provide a more descriptive (largely
non-technical) framing. Anas and Timilsina (2009), simulating infrastructure investments in
roads for Beijing, find that a higher level of road investments makes the chosen residential
pattern more dispersed, and also makes later investments in mass transport (â€œretrofitsâ€? in this
case) less valuable or more expensive. Vogt-Schilb, Meunier and Hallegatte (2012) discuss
timing of sectoral abatement policies within a model of overall optimal climate policy, given
that long-lasting effects of particular abatement investments can vary between sectors. They
show that marginal sectoral abatement costs should differ by sector, with more â€œrigidâ€? sectors
investing relatively more in early abatement.
2. Overview of the paper
Some of our analysis is analytically complex, but the main results are simple. Given
socially optimal pricing, increased volatility of climate damages from given emissions, when
future retrofit is an option, implies that a) retrofits will be executed later, and when marginal
climate damage has reached a higher level; and b) the initial infrastructure will be chosen with
a higher energy intensity. These two factors together imply that both current emissions while
3
See also a follow-up paper by Pindyck (2002), which generalizes the analysis in some other directions than
those pursued here (to simultaneously uncertain climate impact, and uncertain damage due to given climate
impact. See also the review by Pindyck (2007). Balikcioglu, Fackler and Pindyck (2011) and Framstad (2011)
rectify errors in the original Pindyck (2000, 2002) presentations.
4
the infrastructure operates, and its aggregate lifetime emissions, will be greater when
volatility is higher.
These results may be surprising to some, but have a simple intuitive explanation. A retrofit
will be executed later because increased volatility leads to an increased option value of
waiting. The result was shown in similar model context by Pindyck (2000, 2002). The
principle is however long well known, as exposed e g in Dixit and Pindyckâ€™s (1994)
celebrated and far more general analysis of investment decisions under uncertainty. It is due
to the asymmetric effect of volatility when retrofit is an option. Since one can guard against
(and thus avoid) â€œparticularly badâ€? outcomes by retrofitting, the higher frequency of
â€œparticularly goodâ€? outcomes that also follows from higher volatility then takes on a greater
importance, and leads to an overall benefit from â€œnot yetâ€? having carried out the retrofit.
Higher volatility also increases the expected net utility of the infrastructure investment (not
only the option value component), by reducing the expected value of actually realized
damages, when (as here) â€œvery badâ€? outcomes are avoided by either a retrofit, or (in a worst
case) full closedown of the infrastructure system. For a wide class of parameters â€“ we argue,
those for which the model is otherwise reasonable â€“ we show that the energy intensity of the
initially chosen investment is higher with higher volatility, a result which in view of the above
results is not surprising. In the same way as for the retrofit decision, higher volatility with
respect to future environmental costs has an asymmetric effect on â€œgoodâ€? versus â€œbadâ€?
outcomes: it gives the â€œgoodâ€? outcomes (with low climate damage) relatively greater weight
as the â€œworstâ€? outcomes can always be avoided by retrofitting later. This draws in the
direction of a more energy intensive infrastructure choice.
Such an outcome could clearly be problematic in a climate policy context: Greater
uncertainty about the welfare consequences of GHG emissions leads to less and not more
climate action in early problem stages. Greater unconditional uncertainty then makes it
optimal to postpone mitigation. But such climate action is optimal only given that decision
makers face correct global costs. More typically, decision makers do not face correct but
instead too low GHG emissions costs, so that more and not less climate action is desirable. In
such circumstances, greater uncertainty facing decision makers, together with â€œwrongâ€?
emissions prices, could together easily lead to a greater social loss, due to a more inadequate
climate action.
5
Several policies can deal with GHG emissions â€œex postâ€?, related to an already established
infrastructure. We consider the following 4:
(1) Energy use eliminated upon â€œretrofittingâ€?. This may represent a case where the initial
fossil energy is replaced by renewable energy sources with very low ex post marginal
production cost (which could include hydro, solar or wind); or some new energy source that is
supplied in unlimited amount. In this case we need only be concerned with one set of prices or
costs, namely the combined energy and emissions cost, from the start of the project and until a
retrofit takes place.
(2) Energy is not eliminated, but emissions are eliminated upon retrofitting. This could be the
case where CCS technology is adopted to existing power plants; or fossil fuels are replaced by
renewable energy with no net emissions load. Energy costs could here be taken to follow a
deterministic process independent of emissions costs.
(3) The infrastructure is closed down or abandoned. In this case all (energy and
environmental) costs are removed, and the infrastructure provides no utility from then on.
Should there be both a retrofit and a closedown option with given constant costs, we can
disregard the more expensive of the two.
We shall below consider a set of exogenously given constraints or background
assumptions put on the problem, A through D, assumed to hold in the main part of the
analysis, but partly relaxed in Appendix 1:
A. The infrastructure is laid down initially, with no timing optimization;
B. The infrastructure is operated forever, so that the only policy choices later would be when
and how to retrofit (change the emission rates);
C. It is possible to retrofit only once (if ever), and
D. A retrofit eliminates emissions completely. 5
In section 2 below we first study a â€œbenchmarkâ€? case with full certainty, where the process
for the marginal environmental cost of climate emissions, Î˜ , is non-stochastic and has a
constant relative growth (â€œdriftâ€?) rate Î± . This leads to a deterministic stopping rule in this
4
A further type of policy to deal with the impacts of GHG emissions, not discussed further here, is to lower
impacts directly (through â€œadaptationâ€?); see Strand (2011b) for a related analysis.
5
Note that removal of energy use is not a necessary assumption given that the cost q below only refers to
environmental costs; see footnote 2.
6
case, as the exact time of â€œstoppingâ€? can be decided already at the initial investment stage.
We assume in section 2 that the retrofit cost is proportional to the reduction of emission
level. In sections 3â€“5, we generalize this analysis by assuming that Î˜ follows a geometric
Brownian motion process, and by assuming a more general retrofit cost structure. Net
welfare to be maximized, over initial technology and retrofit implementation time, is (under
assumptions A-D) assumed to be given by
V âˆ’ Ï† ( E0 ) âˆ’ E ï£® âˆ« e âˆ’ rt Î˜t M t dt + e âˆ’ rÏ„ K ( E0 ) ï£¹
âˆž
ï£¯ 0
ï£° ï£º
ï£» (1)
where V is the discounted gross utility from the services of the infrastructure, assumed
constant; the cost of the initial infrastructure investment, Ï† (which yields a given utility flow
to the public), depends on the level of energy consumption chosen, E0 ; Ï„ is the time of
retrofit, assumed chosen optimally (where e âˆ’ rÏ„ = 0 if retrofit never occurs); K ( E0 ) is the cost
of retrofit, assumed to eliminate all emissions; and M t is the stock of greenhouse gases in the
atmosphere. E[ Â· ] is the expectation operator.
3. Deterministic climate damage
This section considers a simple â€œbenchmarkâ€? case where the future path for climate costs
is deterministic and known. This serves as a precursor to the later uncertainty analysis. It also
provides a first analysis of optimal initial infrastructure investment with a retrofit option.
Consider the simple current-value function for society:
= V âˆ’ qE
U (E) (2)
where V is (gross) current utility flow to the public associated with the infrastructure once
established, assumed given; at the initial time, the only choice variable is the initial emission
(or energy consumption) rate E of the infrastructure. q > 0 incorporates the entire (expected
discounted) damage caused by current energy use, for all future (in addition to the basic initial
fuel cost). In the following we will typically assume that damage is of this proportional form
also when a subsequent emission reduction is chosen optimally.
7
3.1 Infrastructure utility and cost
The cost of establishing the infrastructure is Ï† ( E ) , where Ï† '( E ) < 0 < Ï† ''( E ) : it is less
expensive to establish an infrastructure which requires a high ex post energy consumption
level, but marginal investment cost savings are reduced as energy consumption increases (or
opposite, when energy consumption is reduced, investment costs increase more steeply). 6 One
simple example is the power function Ï† ( E ) = AE âˆ’ a , where A , a > 0 ; then Ï† ' is increasing
from âˆ’âˆž to 0 , so we have no corner solution; under (2), the optimal choice E* for E would
be given by the first-order condition aAÂ·E*âˆ’ a âˆ’1 = q , equivalent to E* = (q / Aa ) âˆ’1/( a +1) . Net
present value of the installation is then
1 a
V âˆ’ (1 + a ) A a +1
Â·(a / q ) a +1
. (3)
Assume that V is sufficiently large so that this expression is always positive; otherwise, the
basic investment would not be worthwhile in the first place.
3.2 The retrofit decision
A â€œretrofitâ€? will remove emissions permanently from the infrastructure thereafter. This
cannot be done at the establishment time, but â€œlaterâ€?. Assumptions A-D are assumed to hold.
The cost of permanently removing emissions E is K ( E ) . Assume that it is optimal to remove
all emissions once a retrofit is carried out; this is always optimal when (as in Pindyckâ€™s (2000)
main case) retrofit costs are proportional to removed emissions. The only choice variable once
the investment is sunk is then the retrofit time, stopping emissions completely and
permanently. In the following we suppress the energy cost dimension (considering basic
energy costs as given and netted into the utility V and retrofit cost K ), and focus on the
(climate-related) environmental impact of carbon emissions. Following Pindyck (2000),
current environmental costs are given by
6
The infrastructure cost function will need to take this form: considering the set of cost-minimizing
infrastructure projects that all yield the same utility, it must be the case that economically relevant projects, that
are less energy intensive (and thus has lower current energy cost E), must have a higher establishment cost Ï† .
8
Î˜t M t (4)
M is the stock of GHGs in the atmosphere, assumed to obey the following law of motion:
dM t / dt Et âˆ’ Î´ Â·M t
= (5)
where Î´ (â‰¥ 0) denotes a constant rate of decay of GHGs, assuming 0 < M t < max t Et / Î´ .
Discounted damage caused by one unit of emissions at time t1 equals D(t1), which in the
non-stochastic case grows exponentially at rate Î± starting at D(t0), and given by
âˆž eÎ± t1
Î˜0 eÎ± t1 âˆ« e âˆ’ ( r +Î´ âˆ’Î± )t dt =
D(t0 )eÎ± t1 =
D(t1 ) = Î˜0 (6)
0 r + Î´ âˆ’Î±
The discounted value of damages due to one unit of emissions per time unit emitted forever,
=
starting at t = 0 , is D0 Î˜0 / (r âˆ’ Î± )(r + Î´ âˆ’ Î± ) . Charging for emissions costs at a socially
optimal rate along the optimal emissions path requires that the charge q(t) equal D(t) at all
times (where q is interpreted as emissions cost alone).
Suppose E is emitted per time unit from time 0 on. Solving the differential equation for
M t yields
M t M 0 e âˆ’Î´ t + (1 âˆ’ e âˆ’Î´ t ) E / Î´
=
Using that Î˜t =Î˜0 eÎ± t , we can calculate accumulated emissions as
âˆž E /Î´ M âˆ’ E /Î´ Î˜0 M 0 Î˜0 E
Î˜ 0 âˆ« e (Î± âˆ’ r ) t [ Î´
E
+ (M 0 âˆ’ Î´
E
)e âˆ’Î´ t ] dt =
Î˜0 Â·[ + 0 ]= +
0 (r âˆ’ Î± ) r + Î´ âˆ’ Î± r + Î´ âˆ’ Î± (r âˆ’ Î± )(r + Î´ âˆ’ Î± )
Here the first term is a â€œfixed costâ€? unrelated to the project. Suppose that we retrofit the
infrastructure at time t * , eliminating emissions from this project from that point on. We can
then again insert for M t (the dynamics of which are now split at t * with
M t M t* exp(âˆ’Î´ (t âˆ’ t*)) for t > t * ), and write the net social value as âˆ’Î˜0 M 0 / (r + Î´ âˆ’ Î± )
=
(which is independent of chosen infrastructure), plus terms amounting to W(E), that depend
on infrastructure type:
9
âˆž Î˜0 t*
âˆ’Ï† ( E ) + v âˆ« e âˆ’ rt dt âˆ’
W (E) = âˆ« e âˆ’ ( r âˆ’Î± ) t E dt âˆ’ e âˆ’ rt K ( E )
*
0 r +Î´ âˆ’Î± 0
(7)
Î˜0
=âˆ’Ï† ( E ) + V âˆ’ (1 âˆ’ eâˆ’ ( r âˆ’Î± )t* ) E âˆ’ eâˆ’ rt* K ( E )
( r âˆ’ Î± )( r + Î´ âˆ’ Î± )
In (7), the first two terms comprise the ordinary net present value of the infrastructure. The
third term constitutes present discounted emissions costs due to infrastructure operation; and
the last term is the present value of the retrofit cost when done at time t * ; both the two last
terms are discounted from the time of infrastructure investment.
A retrofit as noted removes Et from all periods beyond the retrofit time. (7) is concave
with respect to t * . Taking the derivative yields the first-order condition 7
dW ( E ) Î˜0 E
= âˆ’ e âˆ’ ( r âˆ’Î± )t + re âˆ’ rt K ( E )
* *
(8)
dt *
(r + Î´ âˆ’ Î± )
The solution for t* is
K (E)
Î˜0 Â·eÎ± t =Î˜(t * ) =r (r + Î´ âˆ’ Î± )
*
(9)
E
given a general retrofit function K(E). Bearing in mind that t* â‰¥ 0 , we need to have
K (E) Î˜0
â‰¥ (10)
E r (r + Î´ âˆ’ Î± )
If (10) did not hold, the â€œretrofit costâ€? K would be so low that a â€œretrofitâ€? would be chosen
already at t = 0 (the time of initial investment); and our model would not apply.
7
An alternative and more intuitive way to derive this condition in discrete time requires considering the
discounted value of climate damages caused by one (infinitesimal) unit of installed energy capacity for one
(small) discrete time unit at time t * , found from (4). The current interest cost of this instalment is simply
rK '( E ) . The optimal solution is found at the point where these two quantities are equal, which yields (9).
10
3.3 The choice of energy intensity of infrastructure
Choosing the energy intensity of the initial infrastructure investment amounts to
maximizing (7) with respect to E. We assume in this section K ( E ) = kE : the retrofit cost is
proportional to the emissions removed from the infrastructure. This is a particularly simple
case since then the optimal retrofit time, from (9), is independent of E* , the optimal E. A
more general case is discussed in section 3. E* is here found from:
âˆ’Ï† â€²( E* ) âˆ’
Î˜0
(r âˆ’ Î± )(r + Î´ âˆ’ Î± )
( )
1 âˆ’ e âˆ’ ( r âˆ’Î± )t âˆ’ e âˆ’ rt K â€²( E* ) =
* *
0
(11)
where we assume t* > 0 . With K â€²( E ) = k = constant we then have
Î³0
Î˜0 k ï£® Î˜0 ï£¹
Ï† â€²( E* ) =
âˆ’ + ï£¯ ï£º (11a
(r âˆ’ Î± )(r + Î´ âˆ’ Î± ) Î³ 0 âˆ’ 1 ï£° kr (r + Î´ âˆ’ Î± ) ï£»
)
Î³ 0 r / Î± > 1.
where=
The first main term on the right-hand side of (11) can be interpreted as the discounted cost
associated with one unit of energy consumed forever. The last term modifies this calculation:
it increases the value of energy use at the time of investment, due to the added option of a
retrofit possibility, which is here always exercised.
We are here most interested how energy intensity and overall energy consumption over the
infrastructure's lifetime, depend on the retrofit cost. Given K ( E ) = kE , with constant k (>0),
the effect of a small change in k on E* is found, from (11a), as:
dE* âˆ’1 Î˜0
= Â· k âˆ’Î³ 0 (12)
dk Ï† â€²â€²( E* ) r (r + Î´ âˆ’ Î± )
A higher retrofit cost leads, reasonably, to the choice of a lower energy intensity. This effect
is weaker when k is larger. Intuitively, a retrofit is then done in the more distant future, and is
less significant at the time of investment. Also, since E* is bounded, by (12), also the
elasticity Eï?¬ k E* = (k / E* )Â·(dE* / dk) decreases in k, in the same way as âˆ’k 1âˆ’Î³ 0 .
11
3.4 Implications of changes in cost variables
The discounted value of damages caused by energy consumption resulting from the initial
investment at t = 0 is given from (7) by
=D
Î˜0
(r âˆ’ Î± )(r + Î´ âˆ’ Î± )
( )
1 âˆ’ e âˆ’ ( r âˆ’Î± )t E*
*
(13)
Consider here first an upward shift in the path of damages, represented by a shift in Î˜0.
Such a shift will shift D up for two reasons: first, directly as seen from (13); and secondly, as
t* is now reduced from (8).
Consider next an increase in the unit retrofit cost, k. This affects the discounted damage
from (13) in two separate, opposite, ways. First, the initial energy intensity of the
infrastructure, E* , is reduced, from (12). Secondly, t * is increased, so that a retrofit occurs
later (and energy consumption continues for longer), from (8). Using the latter expression and
âˆ’( r âˆ’ Î± )
=t* Î± t *(1 âˆ’ Î³ 0 ) , discounted climate damages can be rewritten as a function of k:
Î˜0 Î˜0
=D Â·(1 âˆ’ [ ]Î³ 0 âˆ’1 )Â·E* (14)
(r âˆ’ Î± )(r + Î´ âˆ’ Î± ) rk (r + Î´ âˆ’ Î± )
whose derivative with respect to k is a difference between positive terms:
(Î³ 0 âˆ’ 1)Î˜0 E* D dE* Î˜0
[ k Â·r (r + Î´ âˆ’ Î± )]
1âˆ’Î³ 0
Â· = + Î³0( )Î³ 0 E* + D dE* (15)
Î˜0 k (r âˆ’ Î± )(r + Î´ âˆ’ Î± ) E* dk kr (r + Î´ âˆ’ Î± ) E* dk
From the expression for dE* / dk â€“ namely, in particular, the Ï† â€²â€² in the denominator â€“ we have
the following (negative) result:
Proposition 1: There is no universal sign of the relation between the magnitude of
(proportional) retrofit cost and expected discounted environmental damage.
Cases with more general retrofit cost and uncertainty will yield the same result.
4. Infrastructure investment and retrofit when environmental damage is stochastic
We now extend the above model to the more realistic and interesting but complicated case
of stochastic environmental damages. Unless otherwise stated, we assume that conditions
12
Assumptions A-D hold. 8 Several alternatives exist for modelling energy and emissions costs
over time under uncertainty. We have so far assumed that marginal emissions cost has a
positive, constant and deterministic growth rate (â€œdriftâ€?) Î± . We now assume that the
environmental cost parameter Î˜t evolves according to a geometric Brownian motion process
1
Î˜t =Î˜0 exp{(Î± âˆ’ Ïƒ 2 )t + Ïƒ Z t } (16)
2
so that the change in Î˜ can be expressed via the stochastic differential equation
dÎ˜t = Î±Î˜t dt + ÏƒÎ˜t dZ t (17)
where Z t denotes the standard Brownian motion. Marginal climate damages from one unit of
emissions depend on time with constant growth rate Î± , and in addition on a purely random
factor. A simplification is that, with this process, there is no direct relation between
accumulated emissions and marginal climate damage. Considering the impacts of one single
project, this may be reasonable as the project is likely to be â€œsmallâ€? relative to global
emissions (implementation of this project does not substantially alter the process for Î˜ ). A
similar stochastic model has already been studied by Pindyck (2000) for already committed
infrastructure investment. We here reproduce some of his results (with a modification as set
out also in Framstad (2011)), and extend them to the initial investment stage.
4.1 The retrofit decision for given investment under uncertainty
The optimal rule for timing of a retrofit (given that the infrastructure is not closed down)
has been derived by Pindyck (2000), as:
Î³ K ( E* )
Î¸*
= (r âˆ’ Î± )(r + Î´ âˆ’ Î± )Â·
Î³ âˆ’1 E* (18)
where E* is the initially (and optimally) chosen energy intensity of the infrastructure, and
8
Part of the analysis could cover the case where assumption B is relaxed, and K ( E ) capped at V . This case
however turns out to be analytically intractable, at least with parsimonious investment cost; see Appendix 1.
13
where Î³ is given as the positive zero of 1
2 Ïƒ Î³ (Î³ âˆ’ 1) + Î±Î³ âˆ’ r
2
:
1 Î± ï£«Î±
2
1ï£¶ 2r
Î³= âˆ’ 2 + ï£¬ 2 âˆ’ ï£· + 2 (âˆˆ (1, r / Î± ])
2 Ïƒ ï£Ïƒ 2ï£¸ Ïƒ (19)
This expression characterizes the solution whenever r > Î± (otherwise, welfare is âˆ’âˆž for all
positive M 0 Â·Î˜0 ). We here assume a general retrofit cost function K(E), but note that, under
the proportionality case treated in section 2, K ( E* ) / E* = k , a constant.
The key result proved by Pindyck (2000) can be stated as follows:
Proposition 2 Consider the one-shot problem of maximizing (7) with respect to retrofit-time
Ï„ for which Et changes from E to 0 (where E = E* is given exogenously). Suppose that the
laws of motion for M and Î˜ are given by (2) and (16), respectively, where Î± < r , and with
initial conditions Î˜0 = Î¸ > 0 and M =
0 m > 0 . Then the optimal Ï„ = Ï„ * is the first time â€“ if
ever â€“ when Î˜t hits [Î¸ * , âˆž) , with Î¸ * given by (18), and the optimal net social value is the
sum of âˆ’Î¸ m /(r + Î´ âˆ’ Î± ) (which would incur even if the investment were never made), and
W ( E ) given by
ï£±K (E) if Î¸ â‰¥ Î¸ * , and otherwise, if Î¸ < Î¸* :
ï£´
V Ï† (E) âˆ’ ï£²
W ( E ) =âˆ’ Î¸ (Î³ âˆ’ 1)Î³ âˆ’1 Î¸ E Î³ âˆ’1
[
ï£´ (r âˆ’ Î± )(r + Î´ âˆ’ Î± ) âˆ’ Î³
( )Î³ Â·( ) ]E
ï£³ Î³ (r âˆ’ Î± )(r + Î´ âˆ’ Î± ) K (E)
(20).
As in the deterministic case, the value function can be decomposed as follows:
âˆ’Î¸ m /(r + Î´ âˆ’ Î± ) is the (negative-valued) climate damage incurred by inheriting the system in
(Î¸ , m) , assuming no investment and no further emissions.
state (Î˜0 , M 0 ) =
âˆ’Î¸ E / (r âˆ’ Î± )(r + Î´ âˆ’ Î± ) is the additional expected discounted social cost of the climate
damage from E units of emissions emitted for all future periods 9. This can at the discretion of
the social planner be exchanged at cost K ( E* ) , and in optimum this will occur when Î˜ hits
[Î¸ * , âˆž) ; below this, there is a value of the option to stop, equal to (Î¸ / Î¸ * )Î³ Â·K ( E* ) / (Î³ âˆ’ 1) .
9
This holds only when there are no effects of current emissions on future marginal costs Î¸ ; see
comment in the text. If there are such effects, the social cost of emissions is higher.
14
This solution converges to the deterministic case as Ïƒâ†’0. Then Î³ reduces to Î³0 = r/Î± , and Î¸ *
reduces to the right-hand side of (9). From (18), Î¸ * is greater by a factor Î³ / (Î³ âˆ’ 1) > 1 , as Î³ >
1. An increase in Ïƒ 2 implies that í µí»¾ drops (moves closer to unity), so that Î³ / (Î³ âˆ’ 1) increases,
and Î¸ * thus increases.
Greater uncertainty then raises the level of damage required for mitigation action. The
reason is that the option value of waiting with carrying out the retrofit (or, possibly, close the
infrastructure down) then increases. Intuitively, greater uncertainty leads to more states of the
world where, in the immediate future, damages will be reduced; this makes waiting more
favorable. This can however be unfortunate from the point of view of mitigation: Once the
infrastructure investment is sunk, a decision maker has an incentive to delay a costly retrofit
that removes all emissions associated with the infrastructure, perhaps for a long time.
4.2 The optimal emission intensity of infrastructure
We now derive optimal emission intensity, E0 , of infrastructure at establishment and until
it is retrofitted, assuming that the later retrofit decision is optimal. We then maximize welfare
(7) with respect to E. Differentiating W (from Proposition 1) and defining
(Î³ âˆ’ 1)Î˜0 Î˜0 bE
b= , and Ï?
= = (21)
Î³ (r âˆ’ Î± )(r + Î´ âˆ’ Î± ) Î¸*
K (E)
âˆ’Ï† â€²( E ) âˆ’ K â€²( E ) for Î˜0 > Î¸ * , while for Î˜0 â‰¤ Î¸ * we find:
we have W â€²( E ) =
Î³b 1 Î³ EÎ³ d
W '( E ) =âˆ’Ï† '( E ) âˆ’ + bÂ· Î³ âˆ’1
Â· [Î³ ln E âˆ’ (Î³ âˆ’ 1) ln K ( E )]|
Î³ âˆ’ 1 Î³ âˆ’ 1 K ( E ) dE
Î³b
âˆ’Ï† '( E ) âˆ’
= + h( E ). (22)
Î³ âˆ’1
The effect of the â€œoption elementâ€?, denoted h, can be written in terms of the elasticity of K
with respect to E, Eï?¬ K , given general K(E) function, as
bE Î³ K ( E ) Î³
h( E ) ( )Â· [ âˆ’ Eï?¬ K ] (23)
K (E) E Î³ âˆ’1
15
The first-order condition for optimal E (= E* ) is straightforward. The second-order condition
is more complicated in the general case of non-linear K function. We find
bE Î³ Î³ K
=
h '( E ) ( ) Â·[ 2 {1 âˆ’ Eï?¬ K }2 âˆ’ K â€²â€²] (24)
K (E) E
If we require W to be concave for any parsimonious convex Ï† , the appropriate condition
hâ€² â‰¤ 0 requires K to be convex (see comment below Proposition 2). In particular, a convex
power function is here sufficient as long as the exponent â€“ i.e. the elasticity â€“ is between 1
and Î³ / (Î³ âˆ’ 1) .
Optimization with respect to E can in the proportional cost case be expressed by a
sufficient first-order condition involving the function
1
g (Ï? , Î³ ) :
= [max{1, Ï?}Î³ âˆ’ Î³ Â·max{1, Ï?}] ( âˆˆ [âˆ’1, 0] )
Î³ âˆ’1 (25)
The following key result can now be stated and proven.
Proposition 3 Consider the optimal stopping problem from Proposition 2, with cost
functions Ï† ( E ) and K ( E ) = kE , where k > 0 is a constant, Ï† is strictly positive and strictly
convex, and Ï† (âˆž) â‰¥ 0 , Given that an optimal initial E* exists it depends only on parameters
through b and Î³ . Then W is strictly concave, and E* is either zero (iff Ï† â€²(0) â‰¥ kg (Î¸ / Î¸ * , Î³ ) )
or uniquely characterized by the first-order condition
Ï† â€²( E* ) = kg (Î¸ / Î¸ * , Î³ ) (26)
which can be interpreted in the subgradient sense if Ï† is not C1 .
The proposition is proven for the case of proportional retrofit cost, with k constant. With
strictly convex Ï† , as assumed, the option to retrofit will increase the initially chosen energy
consumption level E* for more general K function, as long as Eï?¬ K < Î³ / (Î³ âˆ’ 1) . With larger
elasticity the presence of the option will reduce E* , the main reason being that it is then much
cheaper (per unit of purged emissions) to retrofit a less polluting infrastructure. Pindyck
(2000), section 4, shows how convex K could lead to gradual reduction in E through
successive partial retrofits; this possibility is however ruled out here. Note that in our main
case, we have constant Îº which implies Eï?¬ K = 1 , and the above stated condition holds.
16
As in subsection 2.3 we can find an expression for the derivative of E* with respect to the
retrofit cost. Assume then that the retrofit cost equals Îº Â·K ( E ) , where Îº is a shift parameter.
Instead of h , we get hÂ·Îº 1âˆ’Î³ , and the first-order condition for E* becomes
âˆ’Ï† '( E
=*) bÎ³ / (Î³ âˆ’ 1) âˆ’ h( E* )Îº 1âˆ’Î³ . (27)
Implicit differentiation with respect to Îº yields comparative statics as below. Furthermore,
the effect of the initial state Î˜0 is found by differentiating with respect to Îº . As the following
results are straightforward, proof is omitted:
Proposition 4 (effect of retrofit cost and state on the optimal E* with proportional retrofit
costs) Suppose that Proposition 3 applies, with W â€²( E* )= 0 < W â€²â€²( E* ) . Then, at least piecewise,
dE* (Î³ âˆ’ 1)Îº âˆ’Î³ h( E* ) (Î³ âˆ’ 1)h( E* )
= = âˆ’ Î³ and (28)
dÎº W ''( E* ) Îº Ï† â€²â€²( E* ) âˆ’ Îº h â€²( E* )
(Î³ âˆ’ 1)h( E* ) E*
Eï?¬ Îº E* = âˆ’ Î³ âˆ’1
Îº Ï† â€²â€²( E* ) âˆ’ hâ€²( E* ) (29)
which have opposite signs to h( E* ) , i.e. the same signs as Eï?¬ K âˆ’ Î³ / (Î³ âˆ’ 1) . Also the
elasticity tends to 0 monotonically as Îº (and thus the denominator) increases.
As the retrofit cost increases, the value of the option to stop is reduced. Optimal emissions
intensity in infrastructure is reduced and converges to the case with no retrofit option. The
characterization, in terms of elasticity of E* with respect to Îº, yields conditions for this
convergence to be monotone and smooth, assuming that Ï† is strictly convex.
5. Distributional properties of the solution in the stochastic case
5.1 Probability distributions
We now discuss distributional properties and comparative statics with respect to volatility,
represented by the variance Ïƒ 2 on the stochastic process for climate damages. Assume that the
initial value (at time t = 0 when the infrastructure investment is made) of Î˜t equals
17
Î˜0 = Î¸ < Î¸ * . Since a retrofit intervention completely removes emissions from then on,
aggregate lifetime emissions from this infrastructure is the optimized E* multiplied by the
(stochastic optimal stopping) time Ï„ * > 0 to retrofit; it therefore suffices to give the
distribution of Ï„ for each choice of E . The solution for Î˜t can be written in log terms as
1
ln(Î˜t / Î¸ ) =(Î± âˆ’ Ïƒ 2 )t + Ïƒ Z t
2 (30)
Ï„ * is therefore the hitting time at the positive level L* = âˆ’ ln Ï? of the right-hand side, which is
a Brownian motion with drift Î±, and Ï„ * has probability density (Borodin and Salminen (2002)
p. 295)
L* 2 Ïƒ )t âˆ’ L )
((Î± âˆ’ 1 2 * 2
exp(âˆ’ )t âˆ’3/2 (31)
| Ïƒ | 2Ï€ 2Ïƒ 2
t
We distinguish between the following parametric cases:
â€¢ For Î± < Ïƒ 2 / 2 (where Î˜t tends to zero almost surely), this density does not integrate
to 1 , but to Pr[Ï„ = < âˆž] Ï? 1âˆ’ 2Î± /Ïƒ < 1 . In this case the upward drift is too low to
2
*
guarantee that the process will ever hit the trigger level for the infrastructure
investment. This is however a case on which we here do not focus.
â€¢ For Î± = Ïƒ 2 / 2 , then (25) corresponds to the LÃ©vy distribution (i.e. stable distribution
with index of stability equal to 1/2), which is finite with probability one but has
infinite moments of order 1/2 and above.
â€¢ For Î± > Ïƒ 2 / 2 , the process will hit the target value Î¸ * in finite time, with probability
one. (31) is now the density of the inverse-Gaussian distribution with mean
=
E[Ï„ * ] L* / (Î± âˆ’ 1
2
Ïƒ 2 ) < âˆž and shape ( L* / Ïƒ ) 2 , i.e. variance L*Ïƒ 2 (Î± âˆ’ 1
2
Ïƒ 2 ) âˆ’3 < âˆž .
Starting from m < E / Î´ , the pollution stock M t will keep increasing to a peak level
M Ï„ = E / Î´ + (m âˆ’ E / Î´ ) exp(âˆ’Î´Ï„ ) . For the optimal Ï„ = Ï„ * , which does not depend on the
Ë† = M * will be distributed as
initial pollution stock m , this peak level M Ï„
= Ë† (m
CDFM 1
Ë† ) CDFÏ„ * ( Î´ ln m âˆ’ E /Î´ Ë† âˆˆ (m, E / Î´ ),
Ë† âˆ’ E /Î´ )
m
on M
18
where CDFÏ„ is the distribution of Ï„ * as above â€“ notice that for Î± < Ïƒ 2 / 2 , the point mass at
*
Ï„ * = âˆž translates into an equal point mass for M
Ë† at E / Î´ (a limit not attained in finite time).
We focus on our central case where Î± > Ïƒ 2 / 2, and where Ï„ * is integrable 10. From the
moment-generating function evaluated at âˆ’Î´ , it follows that the peak M
Ë† has expected value
Ë† ] = E âˆ’ [ E âˆ’ m]exp{Ï€ Â·L [1 âˆ’ 1 + 2Î´Ïƒ 4Ï€ âˆ’2 ]} = E âˆ’ [ E âˆ’ m]Ï? Ã²
*
E[ M (32)
Î´ Î´ | Ïƒ |3 Î´ Î´
where Ã² and Ï€ (both positive, the latter by assumption) are defined as:
Ï€ 2Î´ | Ïƒ |âˆ’1 Î± 1
Ã²= [ 1 + 2Î´Ï€ âˆ’2 âˆ’ 1] = , and Ï€ = âˆ’ (33)
|Ïƒ | Ï€ + Ï€ 2 + 2Î´ Ïƒ2 2
The peak current marginal cost factor is simply the non-random Î¸ * , so that the peak
environmental current damage rate is simply Î¸ * M
Ë† . The expected environmental total damage
is the braced expression of (20) corrected for the expected discounted retrofit cost
E[e âˆ’ rÏ„ ]K ( E ) , which (Borodin and Salminen (2002) p. 295) equals (Î¸ / Î¸ * )Î³ K ( E ) . The
*
following formula for expected environmental total damage â€“ note the Î³ factor at the end due
to the correction â€“ is then (valid for non-optimized E* and Î¸ * as well):
Î³
=D { Î¸ âˆ’ ( Î¸ )Î³ }K ( E )
Î³ âˆ’1 Î¸ * Î¸* *
(34)
We can find an expression for the comparative static wrt a multiplicative Îº on retrofit cost;
however, as pointed out in subsection 2.4, we have no universal sign on this.
10
Finite mean will hold in many plausible cases (Dixit and Pindyck (1994), page 81). To exemplify, consider
â€œreasonableâ€? values for Î± and Ïƒ, say, Î± (= the mean rate of increase in climate damage) = 2 percent per year, and
Ïƒ (= the relative random change around trend, positive or negative, in impact of GHG accumulation) = 10
= Î± / 2.
Ïƒ 2 0.01
percent per year - this, we would claim, is a relatively high value). In this case, =
19
5.2 Effects of increased volatility
We now consider how the retrofit decision and initial infrastructure investment are affected
by changes in Ïƒ2. These together determine the time profile for carbon emissions, and thus
also expected aggregate emissions resulting from the infrastructure. Note that
(Î¸ / Î¸ * )Î³ Â·K ( E ) / (Î³ âˆ’ 1) is an option value which increases in Ïƒ 2 for fixed E . By the envelope
theorem, this turns out to be the only first-order effect of increased volatility on both the
expected environmental damage (decreasing in volatility) and optimal value of E (increasing
in volatility) when optimized. At first glance counterintuitive, the reduced environmental
damage will â€“ as long as Eï?¬ K ( E* ) âˆˆ [1, Î³ / (Î³ âˆ’ 1)] , as we shall see â€“ be accompanied by an
increased emissions rate, an increased (finite) expected time to retrofit, and increased
expected total emissions.
It turns out to be convenient to give comparative statics results in terms of Î³ , from (19).
From Proposition 2, for given retrofit cost function K , the optimal E* depends on the
parameters only via Î³ and b , where b depends on Î³ but not directly on volatility. The
comparative static with respect to Î³ suffices to analyze the effect of volatility Ïƒ 2 , and we
note that Î³ is strictly decreasing in volatility: since Ïƒ 2 =
2(r âˆ’ Î±Î³ ) / (Î³ 2 âˆ’ Î³ ) , we have
dÎ³ 1 Î³ 2 (Î³ âˆ’ 1) 2
= âˆ’ Â· = : âˆ’âˆ† ( < 0)
d(Ïƒ 2 ) 2 (r âˆ’ Î±Î³ )Î³ + r (Î³ âˆ’ 1) (35)
Using âˆ† as shorthand notation, we arrive at the following:
Proposition 5 (Impact on value and decision rules)
Ï? 1 . Then the derivatives with
Suppose the conditions of Proposition 3 hold, with Î˜0 / Î¸ * =<
respect to volatility Ïƒ 2 of the optimal E* , the optimal Î¸ * , and the value function W * = W ( E* ) ,
are found as, given sufficient differentiability:
dE* âˆ† K ( E* ) Î³ Eï?¬ K ( E* ) âˆ’ 1 Î³ 1
= Â· Ï? [ +( âˆ’ Eï?¬ K ( E* )) ln ] (36)
d(Ïƒ ) âˆ’W â€²â€²( E* ) E*
2
Î³ Î³ âˆ’1 Ï?
dÎ¸ * Eï?¬ K ( E* ) âˆ’ 1 dE* âˆ†
= Î¸ *Â·[ Â· + ] (37)
d(Ïƒ )
2
E* d(Ïƒ ) Î³ (Î³ âˆ’ 1)
2
20
d d K ( E* ) Î³ 1
W * = âˆ’âˆ†Â· W * = âˆ†Â· Ï? ln > 0 (38)
d(Ïƒ )
2
dÎ³ Î³ âˆ’1 Ï?
W * always increases in volatility, while E* and Î¸ * always do so given that
Eï?¬ K ( E* ) âˆˆ [1, Î³ / (Î³ âˆ’ 1)] . There is no universal sign of dD / d(Ïƒ 2 ) , i.e. no universal sign of
the relation between volatility and optimized expected environmental damage â€“ not even
under proportional retrofit cost.
The optimal values of E* and Î¸ * increase as expected in volatility under the stated conditions.
The impact on the decision rules depend on the more precise shape of the retrofit cost
function, where a proportionate retrofit cost function, K(E) = kE, is a special case (in which
Eï?¬ K ( E* ) = 1 ). We are also interested in effects of increased volatility on retrofit time and
emissions. The discussion on the time to retrofit will be simplified if we merely consider its
expectation, and assume it to be finite, i.e. Î± > Ïƒ 2 / 2 . In that case, we have the following
result:
Proposition 6 (Impact on time to retrofit, total emissions and peak pollution stock)
Assume the conditions for Proposition 5 hold, and in addition that Î± > Ïƒ 2 / 2 . Then
d E[Ï„ * ] 1 1 1 1 dÎ¸ *
= ln + Â· Â· (39)
d (Ïƒ 2 ) 2(Î± âˆ’ 1
2
Ïƒ 2 ) 2 Ï? (Î± âˆ’ 1
2
Ïƒ 2 ) Î¸ * d (Ïƒ 2 )
which is always positive given dÎ¸ * / d(Ïƒ 2 ) â‰¥ 0. Expected total emission, E[Ï„ * ]Â·E* , always
increases with respect to volatility given that Eï?¬ K ( E* ) âˆˆ [1, Î³ / (Î³ âˆ’ 1)] .There is no universally
Ë† ] = E[ M * ] and
valid sign on the relationship between expected peak pollution stock E[ M Ï„
volatility, not even when K ( E ) = Îº E .
The proof (given in Appendix 2) is straightforward, noting that Eï?¬ K ( E* ) âˆˆ [1, Î³ / (Î³ âˆ’ 1)]
grants that E* and Î¸ * , and thus also Ï„*, all increase in volatility. Expected emissions thus
increase â€œfasterâ€? in volatility than either initial energy intensity E* , or expected time to
retrofit, Î¸ * , since each is increasing.
Appendix 2 gives examples of cases with either sign of the relationship for expected peak
stock. It is unclear whether the slightly more complicated peak expected current damage rate
21
Î¸ *E[ M
Ë† ] has any universal sign of relation with volatility. It is also not clear whether expected
total emissions increase with respect to volatility given that E* does not â€“ a case where the
first-order condition for E* does not guarantee uniqueness.
6. Conclusions
In this paper we have derived solutions for the following two decisions:
1) The (fossil-fuel) energy and carbon emissions intensity embedded in an infrastructure
object, from the time of investment until the time it is (later, if ever) â€œretrofittedâ€?. As long as
operated and not retrofitted, this infrastructure gives rise to a constant energy consumption
and carbon emissions per unit of time.
2) The retrofit time for the infrastructure. Fossil energy use and carbon emissions are then
eliminated, while the infrastructure in other respects keeps operating with a given and
constant utility stream to the public.
Both decisions have implications for aggregate fossil energy consumption and carbon
emissions resulting from the infrastructureâ€™s lifetime operation. In section 3 of the paper, we
assume that marginal emission costs evolve according to a geometric Brownian motion
process with constant positive drift. We derive analytical results related to each of the two
decisions; one of these is an already known result, while the other is new. The first result,
previously shown by Pindyck (2000), is that increased variance of the stochastic process
(â€œvolatilityâ€?), facing the party making the investment, leads to postponement of the retrofit
decision, due to the increased option value of waiting when volatility increases. Our second
(and new) result deals with optimal energy and emissions intensity of infrastructure when
established, which is shown to also increase in volatility. This is not surprising: it follows
from the first-order effect of an increase in the associated value function when the â€œupsideâ€?
(or benign) risk increases. This effect is reinforced by the increased option value of waiting to
retrofit, which works in the same direction. The retrofit decision here â€œtakes care ofâ€?
(eliminates) any major increases in â€œdownsideâ€? risk; the principal effect of greater volatility is
that â€œgoodâ€? outcomes become better and more frequent, which increases the value of the
project and of having a high established energy consumption level. A related and also new
result is that the aggregate expected damage from carbon emissions, from the lifetime
operation of the infrastructure (considering also the future retrofit), also increases in volatility.
This result is less obvious: it happens despite the increased frequency of â€œgoodâ€? outcomes
22
which in isolation lowers emissions damage. The higher energy intensity in the initially
established infrastructure, and the longer time to a retrofit, is here shown to always more than
outweigh this benign effect.
For efficient investment decisions involving long-lasting and potentially energy-intensive
infrastructure, both at the initial infrastructure investment stage, and at later stages when the
infrastructure can (at least in principle) be retrofitted, decision makers need to face globally
correct energy and emissions prices. Otherwise, two serious related problems, to be read
directly out of our analysis, can result. First, infrastructure whose structure is difficult or
expensive to change later, will be too energy intensive and established at a too large scale.
Secondly, too low emissions and energy costs can lead to inadequate action also in the
process of operating the infrastructure, as necessary retrofits are not implemented. This
problem would be exacerbated if, at the same time, the cost of executing a retrofit is too high,
which it may often be in less developed economies that do not have access to the most
advanced retrofit technology.
We find that when the level of volatility or uncertainty facing policy makers increases,
energy consumption due to both the energy intensity of infrastructure, and the unwillingness
to retrofit this in the future, at the same time increase. This result indicates that great attention
should be paid to correct pricing of emissions and energy, for all decision makers in both the
short and long run. An indication is also that appropriate pricing becomes even more crucial
under uncertainty, and in the long run, with the additional pressure thereby created to raise the
fossil energy consumption related to long-lasting infrastructure.
The issues discussed in this paper are of particular concern for emerging and expanding
economies which are making, or plan to make, large volumes of infrastructure investments of
a long-lasting nature. Given that decision makers in these economies face globally correct
emissions and retrofit costs both today and in the future, our results show that they will
choose to make their infrastructure investment more energy- and emissions-intensive, and to
retrofit them later, when volatility of the process for emissions costs increases.
But policy and wider implications of the concrete results we derive are not entirely obvious
at this stage. A major political complication today is that the relevant decision makers, most
of whom are in emerging economies, in most cases do not face globally efficient emissions
prices, but instead prices that are (often far) lower. In our model as it stands, since increased
volatility of global emissions costs leads to a higher globally optimal emissions level, it might
be reasonable to conjecture that when decision makers face lower than optimal global
23
emissions costs, resulting emissions rates will be excessive. We know however less about the
relationship between these two; it requires comparisons of discretely different systems, most
likely necessitating simulations. Then the problem of excessive energy intensity and retrofits
to be made too late, associated with such infrastructure will be exacerbated by greater
volatility. A high perceived likelihood of â€œupsideâ€? (low-cost) risks, when such perceived
effects are not warranted, might be particularly harmful. Such â€œupsideâ€? risks can also be
interpreted as a low rate at which carbon emissions stemming from the infrastructure will
actually be charged; and not necessarily as a low (true) climate cost to society (be it the local,
national or global).
Note also that high â€œdownsideâ€? (high-cost) risks play no effective role in decision makersâ€™
choices in our model, as such high costs can always be purged by retrofitting later (at a
limited and â€œmoderateâ€? cost). This is of course unrealistic. If retrofits are impossible or very
costly in some cases (such as completely altering urban structure or transport systems), and
high â€œdownside riskâ€? (such as the possibility of a â€œclimate catastropheâ€?) is underrated,
emissions, in particular those associated with the initial infrastructure investment, will also
tend to be excessive. A further formal analysis of such cases must however await future
research.
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Shalizi, Zmarak and Franck Lecocq (2009), Economics of Targeted Mitigation Programs in
Sectors with Long-Lived Capital Stock. Background paper for the World Development
Report 2010.
Strand, Jon (2011a), Inertia in Infrastructure Development: Some Analytical Aspects, and
Reasons for Inefficient Infrastructure Choices. Journal of Infrastructure Development, 2,
2011, 51-70.
Strand, Jon (2011b), Implications of a Lowered Damage Trajectory for Mitigation in a
Continuous-Time Stochastic Model. Unpublished, the World Bank.
Strand, Jon and Sebastian Miller (2010), Climate Cost Uncertainty, Retrofit Cost Uncertainty,
and Infrastructure Closedown: A Framework for Analysis. Policy Research Working Paper
5208, the World Bank.
Strand, Jon, Sebastian Miller, and Sauleh Siddiqui (2011), Infrastructure Investment with the
Possibility of Retrofit: Theory and Simulations. Policy Research Working Paper 5516, World
Bank.
Vogt-Schilb, Adrien, Guy Meuner, and Stephane Hallegatte (2012), How Inertia and Limited
Potential Affect the Timing of Sectoral Abatements in Optimal Climate Policy. Policy
Research Working Paper WPS6154, Development Research Group, World Bank.
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Appendix 1: Alternative assumptions
A1. Multiple retrofit times
Thus far we have assumed that emissions are reduced at most once, and if so to zero. We shall
give conditions for the latter; if it is not optimal even in a one-shot model to remove all
emissions, then it is neither optimal when gradual emissions are allowed.
Suppose the problem is solved, with initial investment E* and retrofit trigger level Î¸ * chosen
optimally, however this time under the assumption that at the hitting time Ï„ * for the level Î¸ * ,
the emission level is reduced not necessarily to zero, but to some optimized level E * âˆˆ [0, E* ) ,
where it is kept forever; this yields a discounted environmental damage from Ï„ * on, of
Î¸ * E * / (r âˆ’ Î± )(r + Î´ âˆ’ Î± ) . At Ï„ * , the optimal E * must therefore minimize the sum of this
damage and the retrofit cost which we now write as two variables, K ï€¥ ( E ,Î· ) . Sufficient for
*
ï€¥ ' + Î¸ * / (r âˆ’ Î± )(r + Î´ âˆ’ Î± ) is
choosing Î· = 0 is then convexity, or alternatively that K 2
nonnegative for all Î· âˆˆ (0, E* ) Inserting for Î¸ * , this is ensured if
ï€¥ ( E , Â·) â‰¤ Î³
âˆ’Eï?¬ 2 K (A1)
*
Î³ âˆ’1
Arguably, a natural generalization of assumption D (all emissions eliminated by the retrofit),
where we have assumed a cost K ( E ) of reducing emissions to zero, would be to instead
,Î· ) K ( E âˆ’ Î· ) for reducing emissions from E to Î· . Then (A1)
ï€¥ ( E=
impose a functional form K
is a direct generalization of the condition Eï?¬ K ( E* ) â‰¤ Î³ / (Î³ âˆ’ 1) under which the presence of
the retrofit option leads to higher level of initial emissions under conditions A-D. The
interpretation of this is that there will not be the same incentive to adapt for lower retrofit cost,
if those lower costs may be attained nevertheless, at the expense of (linear!) cost of climate
damage.
Note also that the allowable maximum value Î³ / (Î³ âˆ’ 1) for the elasticity is due to the
constraints on the retrofit action. Likely, a model admitting more general strategies would not
share this property.
26
A2. Abandoning the infrastructure
Thus far we have taken as given that the infrastructure is operated forever. If we drop that
assumption, the cost K is capped at V , as one can at any time close down, abandoning the
services from the infrastructure. We shall see that in this case, the model will very often be
losing its validity, as the optimal choice when initial time is non-negotiable, could be to invest
at level E* = +âˆž and then immediately abandon the infrastructure. Assume that the cap
becomes effective at some E ï€¥âˆ’) =
ï€¥ + ) âˆ’ h( E
ï€¥ ; then h gets an upward jump h( E ï€¥ âˆ’ )Â·(Î¸ / Î¸ * )Î³ .
K '( E
If this is positive, then W â€² will be discontinuous unless Ï† â€² has an equal jump; assuming that
ï€¥ + ) âˆ’ W '( E
Ï† â€² continuous, we have W '( E ï€¥âˆ’) = ï€¥ âˆ’ ) min{1, Î¸ / Î¸ * }Î³ . Obviously, we do not
K '( E
ï€¥ , but we know nothing of whether
have concavity. There might be a local max to the left of E
ï€¥ , then with parsimonious convex Ï† ,
it will be optimal. Assuming that K ( E ) = V for all E â‰¥ E
W will to the right of V / k be a difference between two convex functions, and any hope for
uniqueness of any stationary point > V / k would require further conditions or specification
of Ï† . For given Î¸ , notice that for large enough E (making Î¸ * decrease), W â€²( E ) =
âˆ’Ï† â€²( E ) > 0 .
Further assumptions have to be made to ensure E* < âˆž . However, E* = âˆž would lead to total
payoff W * =âˆ’Ï† (âˆž) âˆ’ K (âˆž) < 0 , and it becomes absurd to assume non-negotiable initial time.
This leads to the next subsection: what if initial time is negotiable?
A3. Endogenizing initial time
Introducing initial time as another choice variable will arguably adds another level of
complexity to the problem, but it will resolve the objectionable properties of the previous
subsection; it guarantees nonnegative value. Although the deduction is somewhat beyond the
scope of this paper, the optimal rule will be to wait for for the first time Ï„ * for which Î¸Ï„ * â‰¤
some sufficiently low value Î¸* â€“ chosen subject to optimized choices of E* and Ï„ * .
Obviously, this will ensure Ï„ * > Ï„ * and prevent immediate retrofit/closedown action.
27
A4. Availability of retrofit technology
Thus far, we have assumed that the initial technology E is freely available instantly. It is hot
hard to accommodate a compact interval of admissible emission levels. However, what
happens if the availability of technology changes over time?
Consider a situation where the new technology will not be available until some future time T ,
which then lower bounds the intervention time Ï„ . We require the initial investment to be
committed at time zero, but the optimal stopping problem has to be restarted at T , with the
value
Î˜T
K ( E )Â·g ( )
Î¸*
K ( E ) if Î˜T â‰¥ Î¸ * , and otherwise:
(Î³ âˆ’ 1)Î³ âˆ’1 Î˜T Î˜T Â·E
( )Î³ Â·E Î³ K ( E )1âˆ’Î³ âˆ’
Î³ Î³
(r âˆ’ Î± )(r + Î´ âˆ’ Î± ) (r âˆ’ Î± )(r + Î´ âˆ’ Î± )
At time zero, one has to discount this by exp(âˆ’rT ) and take the expectation; this applies even
if T is a stochastic time. The positive probability that Î˜T > Î¸ * together with the split
definition makes this analytically intractable, however, we still have the form (cf. (20) and the
subsequent discussion)
Î˜T
V Ï† ( E ) + E[e âˆ’ rT g (
W ( E ) =âˆ’ )]Â·K ( E )
Î¸*
The following result can now be shown:
Proposition 7 Suppose that (39) has been maximized over all Ï„ â‰¥ T , where T is a
nonnegative finite random variable with distribution not depending on E nor the cost
function Ï† . Then E ï?¡ E[e âˆ’ rT g (Î˜t / Î¸ * )]Â·K ( E ) is concave if K is convex, and affine if K is
affine.
Remark: This result shows that the form of the value function, and of the optimization wrt
emission level, are maintained, at least to a certain degree. However, there is not much hope
28
that the conclusion of Proposition 2 (namely, dependence merely upon k , Î³ and Î¸ / Î¸ * ) will
carry over, as g is nonlinear and with a split definition.
A different approach could be to model the retrofit unit cost as a stochastic process
(reasonably, a supermartingale after discounting). This is work in progress.
Appendix 2: Proofs
Only propositions 4-6 require several lines of calculation. Proposition 1 follows Pindyck
(2000). Proposition 2 is straightforward. For the first part, the problem only depends on the
parameters in question. For the second part, consider the first-order condition or the derivative
at zero. Proposition 3 is straightforward calculation, and the proof is omitted.
Proof of Proposition 5
We calculate dE* / dÎ³ by differentiating the first-order condition, writing
= Ï? Î¸ / Î¸ * < 1:
(Î³ âˆ’ 1)Î¸ E*
âˆ’W â€²â€²( E* )
dE* âˆ‚
[( )Î³ Â·K ( E* ) [ Î³ âˆ’ Eï?¬ K ( E* )]]
dÎ³ âˆ‚Î³ Î³ (r âˆ’ Î± )(r + Î´ âˆ’ Î± ) K ( E* ) E* Î³ âˆ’ 1
(Î³ âˆ’ 1)Î¸ E*
K ( E* )
= [[ Î³ âˆ’ Eï?¬ K ( E* )]Ï? Î³ âˆ‚ [Î³ ln ]âˆ’
1
Ï?Î³ ]
E* Î³ âˆ’1 âˆ‚Î³ Î³ (r âˆ’ Î± )(r + Î´ âˆ’ Î± ) K ( E* ) (Î³ âˆ’ 1) 2
K ( E* ) Î³ Î³ 1 1 1
= Ï? [[ âˆ’ Eï?¬ K ( E* )]( âˆ’ ln ) âˆ’ ]
E* Î³ âˆ’1 Î³ âˆ’1 Ï? (Î³ âˆ’ 1) 2
and the result follows by gathering terms and multiplying by âˆ’âˆ† . Then for Î¸ * :
dÎ¸ * âˆ‚ Î³ dE* âˆ‚ K ( E* )
= Î¸ * Â·[âˆ’âˆ† ln + Â· ln ]
d(Ïƒ )
2
âˆ‚Î³ Î³ âˆ’ 1 d(Ïƒ ) âˆ‚E*
2
E*
and the rest is straightforward. Now by the envelope theorem, dW * / dÎ³ can be calculated by
partially differentiating W from (20):
29
dW* Ï?Î³ âˆ‚ (Î³ âˆ’ 1)Î¸ E*
K ( E* ) Â· [Î³ ln + ln(Î³ âˆ’ 1)]
dÎ³ Î³ âˆ’ 1 âˆ‚Î³ Î³ (r âˆ’ Î± )(r + Î´ âˆ’ Î± ) K ( E* )
Ï?Î³ 1 1 1
= K ( E* ) Â·[ln Ï? + Î³ [ âˆ’ ]âˆ’ ]
Î³ âˆ’1 Î³ âˆ’1 Î³ Î³ âˆ’1
and everything but the ln term vanishes from the bracket.
Finally, consider the environmental damage, which by the form of E[e âˆ’ rÏ„ ] and assuming
*
proportional retrofit costs ( K ( E ) = kE ), turns out to be
Î¸ E* (Î³ âˆ’ 1)Î³ âˆ’1 Î¸
=D âˆ’ Î³
( )Î³ Â·kE*Î³
(r âˆ’ Î± )(r + Î´ âˆ’ Î± ) Î³ (r âˆ’ Î± )(r + Î´ âˆ’ Î± )k
The effect of volatility enters by way of E* and Î³ :
dD D d E* âˆ‚ (Î³ âˆ’ 1)Î³ âˆ’1 Î¸
= + âˆ†kE [ Î³ âˆ’1 ( )Î³ ]
d(Ïƒ ) E* d(Ïƒ ) âˆ‚Î³ Î³ (r âˆ’ Î± )(r + Î´ âˆ’ Î± )k
2 2 *
D âˆ†k Ï? Î³ Î³ 1 âˆ†Î¸ E* âˆ‚ Î¸ (Î³ âˆ’ 1)
= Â·( âˆ’ 1) ln + [( )Î³ âˆ’1 ]
E* âˆ’W ''( E* ) Î³ âˆ’ 1 Ï? (r âˆ’ Î± )(r + Î´ âˆ’ Î± ) âˆ‚Î³ Î³ (r âˆ’ Î± )(r + Î´ âˆ’ Î± )k
D âˆ†k Ï? Î³ 1 âˆ†Î¸ E* Ï? Î³ âˆ’1 1
= Â·ln + [ + ln Ï? ]
E* Ï† â€²â€²( E* )(Î³ âˆ’ 1) Ï? (r âˆ’ Î± )(r + Î´ âˆ’ Î± ) Î³
âˆ†k Ï? Î³
= { D 1 1
Â·ln + [1 âˆ’ Î³ ln ]E*}
Î³ âˆ’ 1 E*Ï† â€²â€²( E* ) Ï? Ï?
This expression is positive for Ï? near 1 . Choose therefore Ï? so small that the bracket is
negative. Fix an E* and construct a sequence of strictly convex investment cost functions Ï†
which all lead to E* being the optimal choice, i.e. with the same Ï† â€²( E* ) ; but with the
sequence of Ï† â€²â€²( E* ) values tending to +âˆž .
Proof of Proposition 6
Differentiation is straightforward, and whenever Eï?¬ K ( E* ) âˆˆ [1, Î³ / (Î³ âˆ’ 1)] , E[Ï„ * ]Â·E* is a
product of two nonnegative increasing functions. It remains to show that the derivative of
expected peak pollution stock can take either sign, assuming proportional retrofit cost Îº E .
We first manipulate differentials:
30
1 E
Ë†]=
dE[ M (1 âˆ’ Ï? Ã² ) dE âˆ’ [ âˆ’ m] d( Ï? Ã² )
Î´ Î´
where, by slight abuse of notation, we interpret â€œ dE â€? to be zero if the initial investment is
already made, and dE * if it can still be optimized. By manipulating logarithmic derivatives,
we get
1âˆ’ Ï?Ã² E
= dE + [ âˆ’ m]Ï? Ã²Â·Ã²Â·{dL* + L*Â·d ln Ã²}.
Î´ Î´
Observe that when taking derivatives wrt Ïƒ 2 , all terms are nonnegative except possibly d ln Ã² ;
this does not depend on Ï? , so in the limit Ï? ï?š 1 , we would get a positive derivative wrt
volatility. This is due to Î¸ * increasing.
1
(Ïƒ 2 ) Î± / Ïƒ 2 âˆ’ , so that Ï€ â€² = âˆ’Î±Ïƒ âˆ’4 , and write also ln Ã²in
Consider d ln Ã² / d(Ïƒ 2 ) . Write Ï€=
2
terms of volatility:
1
ln Ã² = ln 2 + ln Î´ âˆ’ ln(Ïƒ 2 ) âˆ’ ln[ Ï€ (Ïƒ 2 ) 2 + 2Î´ + Ï€ (Ïƒ 2 )]
2
so that
d ln Ã² 1 Î±Ïƒ âˆ’4 1 2Î±
=âˆ’ + = 2 [âˆ’1 + ]
d(Ïƒ )
2
2Ïƒ 2
Ï€ + 2Î´
2 2Ïƒ 1
(Î± âˆ’ Ïƒ 2 ) 2 + 2Î´Ïƒ 4
2
which is nonnegative if (and only if)
1 Î± 1 Î± 1
Î´ â‰¤ ( 2 + )(3 2 âˆ’ )
2 Ïƒ 2 Ïƒ 2
Ë† ] / d(Ïƒ 2 ) also takes negative values, let Î´ grow, but adjust Î¸ to keep
To establish that dE[ M
Ï? constant (and fairly close to 0 ). For given level of volatility, Î³ and k , this will fix E* ,
and therefore dE* / d(Ïƒ 2 ) ; choose a cost function Ï† so that Ï† â€²â€²( E* ) = +âˆž and dE* / d(Ïƒ 2 )
31
Ë† ] / d(Ïƒ 2 ) depends only on the sign of the braced expression
vanishes. Then the sign of dE[ M
of (39) that is, the sign of
2(Ïƒ 2 ) dL* 2Î±
âˆ’1+ .
L* d(Ïƒ 2 ) 1
(Î± âˆ’ (Ïƒ 2 )) 2 + 2Î´ (Ïƒ 2 ) 2
2
Only the last term depends on Î´ , and vanishes as Î´ â†’ âˆž when everything else is fixed.
Choosing Ï? sufficiently small, will keep the first term < 1 .
Proof of Proposition 7
It suffices to prove that K ( E )Â·g (Î˜T / Î¸ * ) is concave for every Î˜T > 0 . For Î˜T â‰¥ Î¸ * we have
Î˜T / Î¸ * < 1 , we differentiate wrt E . Using that
concavity if K is convex. For Ï?T =
âˆ’ Ï?T (Eï?¬ K âˆ’ 1) / E , we obtain
dÏ?T / dE =
K Ï?T
K â€²( E ) g ( Ï?T ) âˆ’ g â€²( Ï?T )Â·(Eï?¬ K âˆ’ 1)
E
Now insert for the function g and write K Ï?T / E as (Î³ âˆ’ 1)Î˜T / Î³ (r âˆ’ Î± )(r + Î´ âˆ’ Î± ) to get,
after some simplifications:
Î³ Î˜T
âˆ’ K â€²( E ) Ï?T âˆ’ ( Ï?T Î³ âˆ’1 âˆ’ 1)
(r âˆ’ Î± )(r + Î´ âˆ’ Î± )
The pasting of this with âˆ’ K â€²( E ) across Ï?T = 1 , is continuous. For concavity, it therefore
suffices to prove that this expression decreases in E ; for affinity, we merely need to observe
that this is constant in E , since Ï?T is constant. Differentiating once more, we get:
Î³ Ï?T Î³ âˆ’1 (Î³ âˆ’ 1)Î˜T
âˆ’ K â€²â€²( E ) Ï?T âˆ’ (Eï?¬ K âˆ’ 1)Â·[ K â€²( E )Î³Ï?T âˆ’ Ï?T Î³ âˆ’ 2 ]
E (r âˆ’ Î± )(r + Î´ âˆ’ Î± )
Ï?T Î³
= âˆ’ K â€²â€²( E ) Ï? Î³
T âˆ’ (Eï?¬ K âˆ’ 1)Â·[Î³ K â€²( E ) âˆ’ Î³Ï?T
K (E)
]
E E
Ï?T Î³ K ( E )
= âˆ’ K â€²â€²( E ) Ï?TÎ³ âˆ’ Î³ 2
(Eï?¬ K âˆ’ 1) 2
E
which is negative if K â€²â€² â‰¥ 0 .
32