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Energy Intensive Infrastructure Investments with Retrofits in Continuous Time
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© 2015 The World Bank
Energy Intensive Infrastructure Investments with Retrofits in Continuous
Time: Effects of Uncertainty on Energy Use and Carbon Emissions1
Nils Chr. Framstad2 and Jon Strand3
Abstract
Energy-intensive infrastructure may tie up fossil energy use and carbon emissions for a long
time after investment, and thus be crucial for the ability to control long-run emissions. Much or
most of the resulting carbon emissions can often be eliminated later, through a retrofit that may
however be costly. This paper studies the joint decision to invest in such infrastructure, and
retrofit it later, given that future climate damages are uncertain and follow a geometric
Brownian motion process with positive drift. We find that higher climate cost volatility (for
given unconditional expected costs) then delays the retrofit decision by increasing the option
value of waiting to invest. The initial infrastructure is also chosen with higher energy intensity,
further increasing total emissions, when volatility is higher. We provide conditions under
which higher climate cost volatility increases total expected discounted climate damage from
the infrastructure, which happens in a wide set of circumstances.
Key words: Greenhouse gas emissions; long-term investments; retrofits; uncertainty; option
value of waiting.
JEL classification: C61; Q54; R42.
1 This paper is part of the research activity at the Development Research Group, Environment and Energy Team at
the World Bank, the 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.
The research has been supported by a grant from the World Bank’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.
2 Department of Economics, University of Oslo, Box 1095 Blindern, 0317 Oslo, Norway.
3 Development Research Group, Environment and Energy Team, the World Bank, Washington DC 20433.
1
1. Introduction
Large and energy-intensive infrastructure poses a serious concern for climate policy. Some of it,
both supply-side (power plants) and demand-side based (urban structure and transport
systems), is very long-lasting (up to 100 years or more). Other shorter-lasting infrastructure,
with still persistent effects on emissions, includes motor vehicles (fossil-fuel versus electric or
renewable-powered), household appliances, and home heating and cooling systems. Such
capital gives rise to more than half of total greenhouse gas (GHG) emissions from fossil fuels in
high-income countries. 4 Importantly also, in many emerging economies the rates of such
investments, and planned rates over the next 20–30 years, are very high. A basic dilemma is that
“wrong” decisions about such investments could tie up inefficiently high greenhouse gas (GHG)
emissions levels for long future periods. One consequence is to make it difficult to later reach
ambitious climate policy targets.
Retrofits can in some cases help to alleviate this problem. Coal-fired power plants might
(perhaps soon) be retrofitted with carbon capture and sequestration technologies; or possibly be
modified to instead run on renewable, non-fossil, fuels. Urban areas which depend mainly on
private transport might be retrofitted by adding major public transport systems.
This paper considers an abstract or generic infrastructure investment that can, at investment
time, be made more or less energy intensive, and where emissions are assumed to stay constant
until it is retrofitted. Upon a retrofit, the infrastructure is purged of all of its emissions (and
possibly, energy use), while its basic services remain unaltered.
The two key issues explored in this paper, are 1) the optimal energy and emissions intensity
of the initial infrastructure, and 2) the optimal retrofit policy (when, if ever, should the
infrastructure be retrofitted). We assume that energy (including climate and other
environmental) costs are uncertain and follow a geometric Brownian motion process with
constant positive drift. The retrofit cost is known, and constant. Our solution for optimal timing
of a retrofit for given initial infrastructure reproduces results from Pindyck (2000).5 The main
novel result in our paper is to derive the initial infrastructure investment decision,
4 See e.g. Davis et al (2010); Ramaswami (2013)
5 In a follow-up paper Pindyck (2002) generalizes the analysis in some other directions (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.
2
simultaneously with the future retrofit decision. We are also the first to directly derive
implications for accumulated carbon emissions over the infrastructure’s lifetime.
Volatility – the variance on the random component of the process for marginal GHG
emission cost – here plays a major role. Our main messages are simple: Increased volatility of
climate damages from given emissions, when future retrofit is an option, implies – under certain
plausible parametric assumptions specified below – that (a) retrofits will be carried out when
marginal climate damage has reached a higher level and thus, in expectation, later (strictly later
except in the event when it never happens); and (b) the initial infrastructure will be chosen with
higher energy and emissions intensity.
These results have a simple intuitive explanation. First, retrofits occur later when volatility
is higher since higher volatility increases the option value of waiting to retrofit. This principle is well
known from e.g. Dixit and Pindyck’s (1994) analysis of investment decisions under uncertainty,
and from Pindyck’s (2000, 2002) analysis of similar climate-related investments. It is due to the
asymmetric effect of increased volatility when retrofit is an option, which is exercised only when
the climate cost is high. Since one can avoid “bad” outcomes by retrofitting, the better prospects
for “good” outcomes that follow from higher volatility increases the benefit from “not yet”
having carried out the retrofit. Secondly, 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) “bad” outcomes are avoided by
either retrofitting or (in a worst case, when the retrofit cost is also high) abandoning the
infrastructure. For a wide, and plausible, parametric class we show that higher volatility leads
to a higher energy intensity of the initially chosen investment, which, in view of above, is not
surprising. In the same way as for retrofits, the effect of higher volatility on future environmental
cost is asymmetric by giving “good” outcomes (with low climate damage) relatively greater
weight as “bad” outcomes can be avoided by retrofitting later. This makes a more energy-
intensive infrastructure optimal.
Our solution is socially optimal given that decision makers face globally correct energy, emissions
and retrofit costs. 6 More typically, however, decision makers face too low energy and GHG
emissions costs, usually due to either general energy subsidies, or insufficient environmental
taxes. They often also face too high retrofit costs, as least-cost technologies are not available in
6Certain other assumptions are required for this result, in particular, that certain convexity conditions on choice
and production sets are fulfilled; and that all economic actors behave competitively (are price takers).
3
many countries. More climate action than that initiated by private agents and planners is then
desirable. These issues are elaborated in the final section 3.
Combined infrastructure investment and retrofit decisions have been studied by Strand
(2011), Strand and Miller (2010) and Strand, Miller and Siddiqui (2014), albeit in simpler, discrete
(two-period), models. Lecocq and Shalizi (2014) provide a more descriptive (less technical)
presentation. Anas and Timilsina (2009) simulate infrastructure investments in roads for Beijing,
finding that more road investments make the chosen residential pattern more dispersed, and
later investments in mass transport (retrofit in our terminology) less valuable or more expensive.
Vogt-Schilb, Meunier and Hallegatte (2012) and Vogt-Schilb, Hallegatte and Gouvello (2014)
discuss timing of sectoral abatement policies within similar models, and where long-lasting
effects of particular abatement investments can vary between sectors. They show that marginal
sectoral abatement costs should differ by sector, with more “rigid” sectors investing relatively
more in early abatement; and that uncertainty can lead to option values whereby energy-saving
investment is scaled down initially. Our analysis supports the basic conclusions from most of
this literature; but goes farther by providing more precise arguments and analytical results.
Results from other contributions, including Meunier and Finon (2013), and in Ha Duong, Grubb
and Hourcade (1997), however differ from ours; here inertia and learning spillover effects (not
considered in our framework) are shown to overturn the option value induced incentive to wait,
so that early action is instead spurred.
Let us outline the rest of the paper. In subsection 2.1 we introduce the model and its basic
assumptions; alternative assumptions are discussed in Appendix A. Subsection 2.2 considers
the retrofit decision for given initial infrastructure investment, in Proposition 1; for details, see
Appendix B.1. Subsection 2.3 derives the initial infrastructure choice (in Proposition 2;
Proposition A in Appendix A.3 considers the same optimization problem under an alternative
assumption), and effects of increased retrofit cost (in Proposition 3). Subsection 2.4 considers the
effect of increasing volatility on optimal decision rules, optimized project value, and climate
damage (Proposition 4). Section 3 concludes. Appendix B1 derives the main results under a more
general stochastic process; while Appendix B.2 reviews distributional properties of our basic
stochastic process. Results concerning impacts on the probability distributions of time to retrofit
and stock and running damage rate at retrofit time, are found in Appendix B.3 (Proposition B).
Appendix C provides additional proofs.
4
1. Climate Damage in a Stochastic Dynamic Model
1.1. Basics
Assumptions (A)-(C) are assumed to hold for most of the analysis:
(A) The infrastructure is established at a given point of time t 0.
(B) The infrastructure is operated forever. After infrastructure establishment, the only
policy choice is a later retrofit.
(C) A retrofit eliminates emissions forever from the time it is implemented. A retrofit can
be carried out at any point of time after establishment of the infrastructure, but only
once.
Several policies can be applied to affect ex post fossil energy consumption, and GHG emissions,
related to an already established infrastructure:7
(I) Fossil energy use of the infrastructure is eliminated upon “retrofitting”. An example of
such a case is where the initial fossil energy is replaced by renewable energy sources
with very low ex post marginal production cost (which might include hydro, solar or
wind); or some new energy source supplied in unlimited amount (such as nuclear
fusion). We then need only to 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.
(II) Fossil energy use is not eliminated, but GHG emissions are eliminated upon retrofitting.
This might happen when CCS technology is applied to existing power plants; or fossil
fuels replaced by renewable energy with no net emissions load.
(III) The infrastructure can be closed down or abandoned. All (energy and environmental)
costs are then eliminated, and the infrastructure provides no further utility.
In the following we will ignore case (III); cf. assumption (B). When invoking case (II), we simplify
by assuming that future energy costs are deterministic and given (and independent of whether
a retrofit occurs or not, thus not affecting the retrofit decision).
The climate damage rate is given by t Mt , where the environmental cost parameter
follows an exogenous stochastic process, and M is the GHG stock level at time t , affected by
the emissions rate from the infrastructure in question here, Et ; which from assumption (C) will
7A further type of policy to deal with the impacts of GHG emissions, not discussed further here, is to lower impacts
directly (through “adaptation”); see Strand (2014a) for a related analysis.
5
be constant E 0 until time of retrofit, and zero from then on. The cost of retrofit is a
nonnegative, increasing function K of E 0 . The infrastructure has a potentially infinite lifetime.
The gross discounted utility V from the services it provides, from investment time until infinity,
is assumed given. Net social value to be maximized, over initial technology and time of retrofit
implementation, is then given by
rt r
V (E0 ) E e t Mt dt e K (E 0 ) (1)
0
where E[ ] – upright, sans serif font – is the expectation operator and the discount rate r is
constant and 0 . The infrastructure investment cost is assumed to satisfy 0 :
investment costs are lower for infrastructure requiring a high ex post energy consumption level,
whilemarginal investment cost savings are reduced when energy consumption rises.8
We shall assume that the pollutant stock M evolves linearly, as
dMt / dt Et Mt , starting at M 0 0 (2)
where ( 0) denotes a constant rate of decay of GHGs, assuming 0 Mt maxt Et / , where
Et is the emission rate at time t ; we shall assume that E 0 is chosen initially, while by
assumption (C), Et E0 up to retrofit time , and zero from then on. Under the linear
dynamics, M could be interpreted as this project’s contribution to the GHG stock as well as the
overall stock itself; we shall exploit the fact that the dependence upon initial stock splits out
linearly, and refer to this as “everyone else’s contribution”. While linearity is a crucial feature,
the assumption that there is no uncertainty in the dynamics of M apart from through the Et
process is made for simplicity and admits wide generalizations (see Framstad (2014)).
1.2. Optimal Retrofit for Given Investment
We shall throughout the analysis assume that the process is a geometric Brownian motion
2
with drift (0, r ) and volatility 0:
1 2
( )t Zt
t 0 e 2 , (3)
8 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), must have a higher establishment cost .
6
as e.g. in Pindyck (2000, 2002). Z is a standard Brownian motion. The evolution of can also
be expressed via the stochastic differential equation d t t ( dt dZt ) . The marginal climate
damage is independent of this project’s emissions, as t Mt , is linear in M with exogenous ,
and the differential equation for M is linear.9 It turns out that the optimal retrofit rule is to wait
for a sufficiently high state *
that does not depend on M ; this result is not specific to the
geometric Brownian model, but follows much more generally, see Appendix B.1 or Framstad
(2014).
We will need the following properties of the geometric Brownian motion (see Appendix B).
t
Its expected value is 0e , and since 0 , the process will with positive probability hit any
given positive level ˆ . Assume ˆ 0 and denote the first hitting time by ˆ . The power form
E[e rˆ
] ( 0 / ˆ) is valid for any r 0 , where the exponent is the positive zero of
2
1
( 1) r , namely
2
2
1 1 2r
2 2 2
(4)
2 2
2
Given r 0 , we have 1 0 : r/ ; is strictly decreasing wrt. , where 0 and 1
2
are the limiting values as 0 , resp. .
A similar model has already been studied by Pindyck (2000), who shows that the optimal
retrofit strategy for given investment is characterized by the threshold value
* K (E )
(r )(r ) (5)
1 E
which optimally trades off the damage reduction against the discounting of the retrofit cost. This
form is derived in Appendix B.1 as a first-order condition for this trade-off. We find it convenient
to work with the quantities , G and Q defined by
0
*
, G( , ) and Q(E ) K (E ) (6)
1 1
1
( 1) 0 E 1
so that Q(E ) ( ) ( ) E (7)
(r )(r ) K (E )
9Linearity of climate damage wrt. M – jointly with the assumption of exogenous – also follows as an
approximation if the project is small relative to global emissions.
7
is the ratio (the convenient metric for a geometric process) of 0 to the threshold value. The
following key result from Pindyck (2000) exhibits Q(E0 ) as – given 0
*
– the value of the
option to retrofit and G K (E0 ) Q(E0 ) as the climate-related cost incurred when operating
forever without retrofitting:
Proposition 1: Consider maximizing (1) with respect to a single retrofit time, , at which Et changes
from E0 0 to 0, with laws of motion for M and given by (2) and (3), respectively, and r . Then
the optimal retrofit-time *
is the first time – if ever – when t hits *
given by (5); if 0 exceeds *
,
retrofit immediately. The maximal value of (1) is the sum of 0M 0 / (r ) (which would incur
even if the investment were never made), and W (E0 ) given by
W (E 0 ) V (E 0 ) G (min{1, }, ) K (E 0 )
0 *
E0 Q(E 0 ) if 0 (8)
V (E 0 ) (r )(r )
K (E 0 ) otherwise
Furthermore, the expected discounted climate damage, D , turns out as (see Appendix B):
0E 0
D ( )K (E0 ) (1 1
) (9)
1 (r )(r )
Given r 0 , the solution in Proposition 1 converges to the deterministic case as 0,
2
with 0 r/ and /( 1) r / (r ) . A higher results in a lower (closer to
*
unity), so that /( 1) increases, and increases.
Greater uncertainty raises the current damage rate required for mitigation action, as the
option value of waiting to retrofit (or closedown) then increases. Intuitively, greater uncertainty
leads to more states of the world where damages are reduced (and/or reduced by more), which
makes more advantageous to wait. However, as we shall see in section 2.4, greater uncertainty
does not necessarily increase expected environmental damage (9) for given E 0 .
1.3. The Optimal Initial Emission Intensity
We can now derive the optimal initial emission intensity of infrastructure, E* , assuming that
the retrofit decision is optimal (from subsection 2.2). Let us use subscript asterisk to denote a
8
quantity optimized at initial time, e.g. * denotes with the optimal E* inserted. Denote by
the elasticity of K with respect to E , i.e.:
E K (E )
(E ) : K (E ) (10)
K (E )
We maximize welfare (8) with respect to E 0 , which then affects the trigger *
except in the
proportional cost case. Differentiating W in (8) yields the first-order condition
Q (E* ) (E* ) 0
(11)
(r )(r )
*
(valid as long as 0 ). Using
d (E ) 1
(12)
dE E
we can write Q (E ) in terms of as
K (E )
Q (E ) 1 0
K (E ) [ (E )] (13)
(r )(r ) E 1
Notice that the elasticity could get so high that ( K so convex that) the value of the retrofit
option decreases with emissions level. The threshold /( 1) is precisely when, at the margin,
the benefit of eliminating emissions is balanced by the increasing cost. We have the following
two forms of (11) exhibiting the effects of the two particular elasticity values 1 vs
/( 1) :
K (E* ) K (E* )
(E* ) 0
[ (E* )] * (G( *, ) [ (E* ) 1] * )
(r )(r ) 1 E* E*
(14)
where G is given by (6). The second-order condition is more complicated in the general case of
*
non-linear K function. We find for 0 that:
K (E* )
Q (E* ) * ( 2
[ (E* ) 1]2 K (E* )) (15)
E*
which should be negative to ensure a concave relationship between E and W (without having
to impose stronger conditions on than convexity). We summarize:
Proposition 2: Consider the optimal stopping problem from Proposition 1, with continuously
differentiable cost functions K (strictly increasing) and (strictly decreasing, strictly convex and with
9
( ) 0 ). Given that an optimal initial E * exists, it depends only on parameters through and
0 / (r )(r ) . Suppose furthermore that Q (E* ) (from (15)) is nonpositive. Then W is strictly
concave, and E* is either zero, or uniquely characterized by (14), or such that immediate retrofit is
optimal.
For Q 0 it is necessary, but not sufficient, that K 0 . The following example illustrates
why the condition
1 (E* ) (16)
1
will show up in many of our results. Consider power functions K (E ) kE for various constant
, so that (15) has same sign as ( 1)( /( 1)) . If (1, / ( 1)) , then from the first-
order condition, the presence of the retrofit option will increase the chosen E* , and the second-
order condition holds under the assumption of strictly convex . If takes one of the endpoint
values of this interval, then Q is constant in E and we can solve for E* by inverting . At
larger elasticity we face the following properties: from the first-order condition, the presence of
the option reduces E* , as it is then much cheaper (per unit of purged emissions) to retrofit a less
polluting infrastructure10; however, there is no guarantee that the second-order condition holds,
as Q is positive and tends to as E* 0 . However, at low enough E* we will have *
0 ,
i.e. immediate retrofit, in which case the model does arguably lose validity. To find how E*
changes with the level of retrofit cost, defined by K (E ) k J (E ) , and consider changes in the
scaling k . Since K and J have the same elasticity , the first-order condition for E* then takes
the following form, cf. (14):
1 1
(E* ) 0
{1 * [ (E* )]} (17)
(r )(r ) 1
where E* depends on k only through * ; at the particular elasticity /( 1) , the
dependence degenerates completely. Straightforward implicit differentiation with respect to k
yields the following comparative statics, and proof is omitted:
10Pindyck (2000) shows how convex K could lead to gradual reduction in E through successive partial retrofits, a
possibility which is however here ruled out.
10
Proposition 3 (effect of retrofit cost on the optimized E* and D* ): Suppose that Proposition 2
applies, the retrofit cost takes the form K (E ) k J (E ) , and W (E* ) 0 W (E* ) . Then, assuming
sufficient differentiability,
dE* ( 1) J (E* )
* [ (E* )] (18)
dk W (E* ) E* 1
and
( 1) K (E* )
k E* * [ (E* )] (19)
W (E* ) 1
both of which are negative iff /( 1) , in which case also (19) tends to 0 as k increases, provided
that W (0 ) 0 . For the effect of k on D* , we find
(r )(r ) dD* dE* 1
d ln(1/ * ) 1
(1 * ) * ( 1)E*
dk dk dk (20)
dE* E*
{1 *
1
( 1) *
1
[ (E* ) 1]} *
1
( 1)
dk k
and
(r )(r )
k D* {1 *
1
( 1) *
1
[ (E* ) 1]} k E* ( 1) *
1
(21)
A higher retrofit cost reduces the value of the option to stop (cf. formula (6)) for given
emissions level; proposition 3 gives a condition that the emission level is reduced with the
retrofit cost level, approaching the case with no retrofit option as the retrofit cost grows.
However, the effect on the damage need not follow the effect on the optimized value. To
understand this result, observe that higher retrofit cost both postpones the retrofit and reduces
the optimal E* (as long as /( 1) ). The indirect effect – through E * – could take any
magnitude, depending on (E* ) . When is very large, the initial investment is insensitive to
changes in a future retrofit cost, which will then only postpone the retrofit, increasing the
environmental damages. When is very small, the dominating impact is on the initial decision.
Example: Let us give an example where dD* / dk could indeed take either sign depending on
. Fix an initial state 0 with optimal choices – without loss of generality by scaling units –
*
E* 1 and (r )(r )K (1) / ( 1) assumed strictly greater than 0 . Those choices
are then optimal for all strictly convex decreasing positive cost functions for which
11
(1) (1) – in the limiting cases even with (1) being (with inelastic E* , (21) reduces to
( 1) * ) or zero. In the latter case, consider proportional K , such that W (1) (1) and the
elasticity tends to . We leave to the reader to verify that power functions do yield tractable
expressions also when (1, / ( 1)) .
1.4. Effects of Increased Volatility
Consider now effects on the retrofit decision and initial infrastructure investment due to changes
2
in , which measures per time-unit variance of the log of climate cost. These two decisions
together determine the time profile for carbon emissions, and thus expected aggregate emissions
resulting from the infrastructure. From Proposition 2, for given retrofit cost function K , the
optimal E* depends on the parameters only via and 0 / (r )(r ) . Since
2
d 1 ( 1)2
: (>0) (22)
d( 2 ) 2 (r ) r( 1)
2
we can instead calculate derivatives wrt. in place of (the negative to keep signs
consistent). Using as shorthand notation, we find (proof in Appendix C.1):
Proposition 4 (Impact on decision rules, welfare and environmental damage): Suppose the
*
conditions of Proposition 3 hold, with 0 / * 1 . Then the value function W* W (E* ) increases
2
wrt. , with derivative
d d K (E* ) 1
W*
2
W* * ln 0 (23)
d( ) d 1 *
*
The optimal E* and both increase wrt. volatility provided that (E* ) [1, / ( 1)] , with
derivatives, given sufficient differentiability:
dE* dE* K (E* ) (E* ) 1 1
2 * { [ (E* )]ln } (24)
d( ) d W (E* ) E* 1 1 *
*
d d * * 1 (E* ) 1 dE*
2
[ ( )] (25)
d( ) d ( 1) E* d
For the optimized D* we find
12
dD* 0 E* dE*
2
[{ 1 ln
1
} *
1
(1 *
1
( 1) *
1
[ (E* ) 1])
1
]
d( ) (r )(r ) * E* d
(26)
1/ *
Provided dE* / d 0 (as when [1, / ( 1)] ), (26) is nonnegative given 0 e . For
lower values of 0 , the sign of (26) is ambiguous and depends on (E* ) .
From Proposition 4, E* and *
both increase in volatility – this provided that the elasticity
is within the given interval11. The impacts depend on the shape of the retrofit cost function,
and with proportionality, K (E ) kE (i.e. 1 ), as a special (borderline) case that admits
unique sign of (24). Note that when the initial technology E 0 is exogenously fixed, dD / d( 2 )
1/ *
changes sign precisely at 0 e , cf. the curled difference in (26); with E* optimized, its
derivative contributes positively to (26), and this may or may not be enough to ensure a positive
1/ *
sign when 0 e . For examples with either sign possible, copy the argument of the
example following Proposition 3: on one hand we can have (E* ) and thus W (E* ) arbitrarily
large, yielding a zero in (26) for ln(1/ * ) arbitrarily close to 1/ ; on the other hand, with
proportional retrofit cost we can have (E* ) and thus W (E* ) arbitrarily close to zero, yielding
a positive ln(1/ * ) coefficient in (26) and thus a sum of positive terms.
Our final results, presented as Proposition B in Appendix B.3, concern impacts of increased
volatility on retrofit time and emissions. As volatility increases, so does the expected time to
retrofit, if finite (i.e. if 2 2
), while if the probability that retrofit never occurs is positive, then
*
that probability will increase. When (E* ) [1, / ( 1)] , then E * , (from Proposition 4) and
*
(in expected value or point mass at , from Proposition B) all increase in volatility.
Moreover, expected emissions increase “faster” in volatility than either initial energy intensity,
*
E * , or expected time to retrofit, , since both increase. Proposition B also gives examples of
cases with either sign of the relationship between volatility and expected peak stock.
11 Otherwise, (24) takes both signs; it has a zero when ln *
[ 1] / [( 1) ] , and becomes negative for
*
0 closer to resp. closer to 0 for (E* ) 1 resp. / 1.
13
2. Discussion
In this paper we have studied two combined decision problems:
1. The choice of initial fossil-fuel and carbon emissions intensity of an infrastructure
object at the time of investment and until the object is retrofitted.
2. The chosen time of retrofit (if ever), at which time the carbon emissions, and possibly
also the fossil-fuel consumption, are eliminated from the infrastructure forever
thereafter.
Both decisions affect the aggregate fossil-fuel consumption and carbon emissions resulting from
the infrastructure’s lifetime operation. It is assumed that the marginal cost of carbon emissions
follows a geometric Brownian motion process with positive drift.
Our solution to problem 2 above (in Proposition 1) restates (but slightly modifies) a well-
known result from Pindyck (2000): Greater volatility of the stochastic process for climate costs
leads to postponement of the retrofit decision, due to an increased option value of waiting when
volatility increases.
Our solution to problem 1 (Proposition 2) is new, and gives conditions for the chosen optimal
energy and emissions intensity of the infrastructure, also to increase in volatility. Intuitively, the
value of such a project increases in volatility, as the resulting increase in benign risk (when
climate costs turn out to be small) is given greater weight relative to adverse risk (when future
climate costs turn out high): the latter risk, associated with this project, can and will be avoided
by retrofitting. This effect is reinforced by the increased option value of waiting to retrofit when
volatility increases: both effects work to make higher volatility more attractive. In consequence,
the dominating effect of a more uncertain future climate cost (for given expected cost) is that
benign, low-cost, outcomes become more frequent relative to high-cost ones. This increases both
the project value, and the expected return to a high energy consumption level. As volatility
increases, so does the expected time to retrofit, if finite (i.e. if 2 2
); while with positive
probability that retrofit never occurs, this probability increases in volatility. We then also find
that when the initial emissions level is endogeneous, increased volatility typically leads to
increased climate damage in more cases than when initial infrastructure is exogeneously given.
In order for investment decisions involving long-lasting and potentially energy-intensive
infrastructure to be socially efficient, both at the investment stage and the later retrofit stage,
decision makers need to face both globally correct energy and emissions prices, and correct
retrofit costs. When decision makers instead face too low energy and emissions prices, two
14
problems can result. First, the infrastructure will be established as overly energy intensive. This
is particularly problematic (e g for long-run climate policy) when the infrastructure is difficult
or expensive to retrofit or replace later. Secondly, necessary retrofits might be unduly postponed,
or never implemented. Decision makers in such countries would then tend to choose more
emissions-intensive infrastructure, and retrofit it later, when future emissions costs become
more volatile. A similar problem with retrofit postponement can arise when the retrofit cost is
excessive. We argue that this is likely to often occur in less developed economies with limited
access to advanced and low-cost retrofit technologies. 12 Proposition 3 gives conditions that
higher retrofit cost also reduces the chosen energy intensity of infrastructure (a benign effect
when this intensity is otherwise excessive), and conditions that the total effect is either a
reduction or an increase in expected climate damage.
Higher energy and emissions intensities chosen by decision makers when emissions prices
are more uncertain, are in our model simply a feature of the optimal policy choice for rational
economic actors facing such uncertainty. In an ideal world where decision makers face (globally)
correct energy and emissions prices, this should not be particularly worrisome as a global
concern. It is more likely to be a concern when decision makers instead face too low energy
prices, and/or too high retrofit costs. Our model points to (although does not explicitly analyze)
the possibility of adverse outcomes, with corresponding social losses, when decision makers do
not face the full global emissions costs, or excessive costs of retrofits or in establishing low-
carbon infrastructure. In emerging economies, with large planned infrastructure investments
over coming years, decision makers are (now and for the foreseeable future) likely to face
emissions prices below optimal levels. Such decision makers could also conceivably take
advantage of (possibly large) uncertainties about these cost variables, emphasizing benign risk
effects while ignoring adverse risks, on the presumption (or with the hope) that they can avoid
the consequences of more adverse risks. 13 A reasonable conjecture could then be that the
combined problem of excessive energy intensity and postponed retrofits will be exacerbated by
increased climate cost volatility. A high perceived likelihood of benign (low-cost) risks, when
such risks are not warranted, might then be particularly harmful. Such benign or favorable risks
12 A similar issue is that the cost of implementing low-carbon infrastructure technologies could be excessive in
many low-income countries; also leading to socially excessive carbon intensity of the established infrastructure.
13 Another related problem, focused by Strand (2014a), is that defining the “baseline” for any future climate action
may be a strategic choice; and that this baseline is affected by the infrastructure policy. For a country, which today
is not committed to a specific climate policy, the established baseline or norm for emissions may help to define the
degree of climate action required by this country under a future agreement. It may then have incentives to commit
to a high emissions baseline in order to convince other parties that reducing future emissions is expensive, thus
affecting the required commitments under the future agreement. High climate cost volatility could then make it
politically easier to choose an infrastructure policy that leads to high emissions.
15
can also be interpreted as a low rate at which carbon emissions from the infrastructure will
actually be charged; and not necessarily as a low (true) climate cost to society (be it local, national
or global). While such issues are not part of our model or analysis, we conclude that they are
particularly relevant as topics for future research; and that our analysis represents a good
starting point for such research.
An unrealistic feature of our model is that the “tail risk” of very high future climate costs
plays no effective role in decision makers’ choices, since very high costs are assumed to always
be avoided by retrofitting. But if retrofits are impossible or very costly in some cases (as could
apply to alternatives such as completely altering urban structure or transport systems), and such
downside risk is underrated, emissions will also tend to be excessive. Lock-in of energy-
intensive infrastructure could in such cases make certain climate policy goals infeasible.14
A complicating factor, also not discussed formally, is that low energy prices could imply that
their variance (volatility) is also low. This could affect investment and retrofit decisions in the
opposite direction, and lead to retrofits being executed too early.15 A more complete analysis of
these issues must await future research.
14 This adverse feature of lock-in of long-lasting, energy-intensive infrastructure is stressed also by Lecocq and
Shalizi (2014), and Vogt-Schilb et al. (2012; 2014).
15 See Strand (2014b) for recent analysis of retrofit decisions in such cases. It is there shown that when reduced
climate damage and volatility are reduced proportionately, the retrofit decision may in some cases be executed
earlier when the climate damage variable, facing decision makers, is reduced.
16
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18
Alternative Assumptions
This section outlines consequences of relaxing assumptions (A) through (C) in section 2.
Although the expressions become less tractable – and in section A.2 we lose concavity –the
qualitative properties will to some extent carry over.
A.1 The Possibility of Multiple Retrofits: Relaxing Assumption (C)
Thus far we have assumed that emissions are reduced through a retrofit 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 E* and retrofit trigger level chosen
optimally, however this time under the assumption that at the hitting time *
for the level *
,
the emission level is reduced not necessarily to zero, but to some optimized level E * [0, E* ) , at
which it is kept forever; this yields a discounted environmental damage from *
on, of
*
E * / (r )(r ) . At *
, the optimal E * must therefore minimize the sum of this damage
and the retrofit cost which we now write as two variables, K (E*, ) . Sufficient for choosing 0
*
is then convexity, or alternatively that K2 / (r )(r ) is nonnegative for all
*
(0, E* ) . Inserting for , this is ensured if
2 K (E* , ) (27)
1
Arguably, a natural generalization of the assumption that all emissions are eliminated by the
retrofit at cost K (E ) , would be to instead impose a functional form K (E, ) K (E ) for
reducing emissions from E to . Then (27) is a direct generalization of the condition
(E* ) /( 1) under which the presence of the retrofit option leads to higher level of initial
emissions under conditions (A)–(C). 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 /( 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.
19
A.2 Abandoning the Infrastructure: Relaxing Assumption (B)
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* and then immediately abandon the infrastructure. Assume that the cap becomes
effective at some E ; then Q gets an upward jump Q (E ) Q (E ) = K (E ) * . Assuming
*
continuous, we have W (E ) W (E ) K (E )min{1, 0 / } . Obviously, we do not have
concavity. There might be a local max to the left of E , but we know nothing of whether it will
be optimal. Assuming that K (E ) V for all E E , then with merely assumed convex, 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
*
. For given 0 , notice that for large enough E (making decrease), W (E ) (E ) 0.
Further assumptions have to be made to ensure E* . However, E* would lead to total
payoff W* ( ) K( ) 0 , and it becomes absurd to assume non-negotiable initial time.
This leads to the next subsection: what if initial time is subject to choice?
A.3 Endogenizing the Initial Investment Time: Relaxing Assumption (A)
Introducing an endogenous initial time for the investment as another choice variable will
arguably add a further level of complexity to the problem, but it will resolve certain
objectionable properties of the previous subsection; it guarantees a nonnegative value. A full
analysis of this case is beyond the scope of the present paper, but certain properties can be found
in Framstad (2014); The optimal rule is, however, to wait for the first time * for which *
*
some sufficiently low value * – chosen subject to optimized choices of E* and , and not
depending on M . Obviously, this will ensure *
* and prevent immediate
retrofit/closedown action.
A.4 Availability of Retrofit Technology: Relaxing Assumption (C)
Thus far, we have assumed that any initial technology E is freely and instantly available. A
question is what happens in the model if the availability of technology changes over time.
20
Consider a situation where the new technology will not be available until some future time
T , which then lower bounds the intervention time . Once at T , we will stop the subsequent
* *
first time we hit (immediately if T ). Rather than the post-investment value
V (E0 ) G(min{ ,1}, ) K (E0 ) in (8) valid for T 0 , we get
V (E0 ) E[e rT
G(min{ T
,1}, )] K (E 0 ) (28)
*
– the discount factor is kept inside the expectation, as the formula is valid also if T is a non-
*
deterministic stopping time. However, the positive probability that T together with the
split definition makes this somewhat analytically intractable, but the following result can now
be shown; the proof consists of copying the arguments leading to the concavity part of
Proposition 2 for each possible value of T , and is omitted:
Proposition A: Suppose that (1) has been maximized over all T , where T is a nonnegative finite
stopping time independent of everything else, and with distribution not depending on 0 , E 0 nor the cost
function . Then
E0 E[e rT
G(min{ T
*
,1}, )] K (E0 ) (29)
is convex if K is also convex, and is affine if K is proportional.
Remark: This result shows that the form of the value function, and of the optimization wrt.
emission level, will be maintained, at least to a certain degree. However, we cannot count on the
dependence upon merely k , and 0 / (r )(r ) from Proposition 2 to carry over, due
* *
to the split definition and the probability that T even when 0 .
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.
Probabilistic Properties of the Model
In subsection B.1 we demonstrate that the form of the optimal retrofit rule does not depend on
the specific geometric Brownian model, but is a consequence of the strong Markov property and
the linear dynamics of the M process. Subsection B.2 displays some basic properties of
geometric Brownian motion that we invoke in the analysis. These results can also be found in
21
Borodin and Salminen (2002), see in particular pp. 295 and 622. Subsection B.3 gives the last
result of this paper, involving certain additional effects of volatility.
B.1 The Linearity of the Pollutant Stock, the Strong
Markov Property and the Retrofit Optimization
t
t t
Solving for Mt M 0e e e s Es ds , we see that the dependence upon M 0 splits out
0
additively in the objective (1). Also, splitting M additively into this project’s contribution and
everyone else’s, it is easy to see that the latter splits out additively in (1), and the term we can
actually affect through E . Therefore the optimization of emissions – be it initial level or timing
of retrofit – does not depend upon M .
In our case, with Et E0 up to the retrofit time and zero from then on, solving out M t
yields
max{0,t } t t max{0,t }
t e e t 1 e 1 e
Mt M 0e E0 M0e E0 E0 (30)
t
This allows us to decompose as follows: the first term M 0 e can be interpreted as everyone else’s
contribution, independent of the project; adding the second term yields the project’s contribution
if operated forever without retrofit (i.e. like putting in the left-hand side); the last term is
the reduction of the emissions stock from retrofitting at time . For a given strategy – i.e. a given pair
(E0 , ) , total discounted climate damage can thus be decomposed into
(r )t
M0 e E[ t ] dt D, where (31)
0
t max{0,t }
1 e 1 e
D D( 0 ; E0 , ) E[ e rt
t dt e rt
t dt ] E 0 (32)
0
The positive contribution to D in the last line is D( 0 ; E0, ) as the last term vanishes for .
In the rightmost term, we integrate only from due to zero contribution for t . Only this
last term – a damage reduction, hence a benefit – is affected by the retrofit decision. This needs
r
to be traded off against the expected discounted cost K (E0 )E[e ] of performing the retrofit.
Assuming that is a time-homogeneous strong Markov process, we can make a time-change
shift by in the reduction term, which turns into
22
t
1 e
E[ e rt
t dt e r
] E0 (33)
0
where evolves as an independent copy of except inserted as initial value. Thus, the -
conditional expectation of the integral in (33) is D( ; E0, ) , and by the double expectation law,
r
(33) becomes E[D( ; E0, ) e ] . If furthermore is continuous, then we can restrict ourselves
to hitting times ˆ , being the first time hits some interval [ ˆ, ) (the interval extending to the
right because higher is a “bad”). If we then start no higher than ˆ we know what state we
stop at (if ever), namely ˆ . This holds for arbitrary ˆ , including for the optimal *
that
ˆ
solves the following problem that yields the optimized option value:
Q max ˆ {(D( ˆ; E0, ) K (E0 )) E[e rˆ
]} (34)
*
Notice that, because D is proportional to E 0 , the optimal level at which a retrofit is
executed will depend on E 0 only through the average K (E0 ) / E0 – in particular, if the retrofit
cost K is proportional, the dependence vanishes. Only apparently does the maximization in (34)
depend on what state 0 we are in; the first-order condition for stopping “now”, i.e. for the
initial state to be precisely the optimal intervention threshold, is
d d
0 {E[e rˆ
] D( ˆ; E0, ) (D( ˆ; E0, ) K (E0 )) E[e rˆ
]}|ˆ * (35)
dˆ dˆ 0
rˆ
where E[e ] becomes 1 when inserting, and where (as increased threshold postpones retrofit)
the first term is positive and the second negative. For the particular case of geometric Brownian
motion, we can now insert and evaluate, using well-known properties reviewed in the next
section:
t
1 e
D( ˆ; E 0 , ) e rt
e t
dt ˆE 0
0
ˆE ˆE (36)
1 1 0 0
( )
r r (r )(r )
while (cf. (39) below) E[e rˆ
] ( 0/ˆ) for ˆ , with from (4) as used throughout the paper.
0
So the first-order condition (35) reduces to
E0 ˆE
0 0
0 1 ( K (E 0 )) |ˆ *
(r )(r ) (r )(r ) ˆ 1 0
(37)
E0 1
(1 ) K (E 0 )
(r )(r ) *
23
*
confirming (5). Now with *
being the optimal choice, we can evaluate D D( 0 ; E0, )
(retrofit-optimized, but for given E 0 ) as follows: In (32), the first term is D( 0 ; E0, ) from which
*
we subtract D( * ; E0 , ) (cf. the argument following (33)), where – as usual – 0 / .
Using, (36) the difference becomes
*
0E 0 E0 1 0E 0
(1 ) (38)
(r )(r ) (r )(r ) (r )(r )
confirming (9).
B.2 Distributional Properties of Geometric Brownian Motion
2 t
The parameterization t 0 exp{( / 2)t Zt } yields expectation E[ t ] 0 e , as
Brownian motion is a martingale and applying the expectation to the (Itô) stochastic differential
equation d t t dt t dZt yields dE[ t ] E[ t ]dt 0 . Assume in the following that 0
and that 0 ˆ where ˆ is an arbitrary candidate for trigger level with corresponding first
0
hitting time ˆ 0 ; for optimum values, put ˆ *
and ˆ *
. As the log of the gBm is some
Brownian motion with drift, the hitting times are those for the latter. has positive probability
of hitting any given positive value (this in contrast to the deterministic case). We have
distributional properties, which can be found in e.g. Borodin and Salminen (2002 p. 295): putting
ˆ ln( ˆ / ˆ has the probability density
L 0) , the first hitting time ˆ for
ˆ ( 1 2
)t ˆ
L
L 1
PDFˆ (t ) t 3/2
exp{ [ 2
]}
2
(39)
2 2 2t
however with the reservation that there might be a point mass at infinity, see below. Regardless
of finiteness of , the following expression holds true for each given exponent R 0 and each
0 /ˆ 1:
2
Rˆ (R) 1 1 2R
E[e ] ( 0) with (R) (40)
ˆ 2 2 2 2 2
In particular, for the discount rate we recover (R) as in (4).
For the distribution itself, we need however distinguish between the following parametric
cases:
24
For 2 2
, the drift is too low to guarantee that the process will hit ˆ 0, and in
fact t tends to zero almost surely. The density (39) does not integrate to 1 , but – as
it should – to
2
1 2 /
Pr[ ˆ ] ( 0) (41)
ˆ
For 2 2
, ln is a Brownian motion without drift, and which hits every level; thus
hits every positive level in finite – but, in fact, infinite-mean – time; (39)
corresponds to the so-called Lévy distribution (i.e. the totally skewed stable
1/2
distribution with index of stability equal to one half, so that even E[ ] is infinite).
This is a borderline case which we omit from the exposition.
For 2 2
, the hitting time has finite moments of all orders16, and (39) is now the
inverse-Gaussian distribution with
2
1 ˆ, ˆ
mean: E[ ˆ] 2
L and variance: E[ ˆ2 ] E[ ˆ]2
2 3
L (42)
1 ( 1 )
2 2
B.3 Impact of Volatility on Some Functionals of the Probability Distribution
The following gives the impact in optimum of increased volatility on the expected value of the
*
following quantities: time to retrofit (provided finite; otherwise the probability of it being
*
infinite); total emissions E* , peak pollution stock M * and on the running environmental
damage rate at retrofit time, *
M * . Provided (E* ) [1, / ( 1)] , they all increase with
volatility except in some cases the latter, for which the relationship could take either sign.
Bearing in mind that the agent’s optimized choices do not depend on M , we might still be
interested in information on the following: at what state of the atmosphere – and at what current
damage rate – would we expect the retrofit to be implemented? Contrary to the previous
propositions, the initial M 0 would now matter, and we therefore allow it to be nonzero.
Expected peak pollution stock is
16Finite mean will hold in many plausible cases (Dixit and Pindyck (1994), page 81). To exemplify, consider
“reasonable” values for and , say, (= the mean rate of increase in climate damage) = 2 percent per year, and
(= the relative random change around trend, positive or negative, in impact of GHG accumulation) = 10 percent
per year - this, we would claim, is a relatively high value). In this case, 2 0.01 .
25
*
E[M * ] E* / (M 0 E* / ) E e E* / (M 0 E* / ) * (43)
by (40), and where we – here and in the rest of the paper – write for ( ) ; analogous to (22)
we have
2
d 1 ( 1)2 1
:
2 2 2
(44)
d( ) 2 ( ) ( 1) 2 /
which has the same sign as . This explains why the following result splits between
and :
Proposition B (Further effects of increased volatility): Assume that the conditions for Proposition
4 hold.
We have in optimum
d
2
E[M * ] [(E* M0 ) *
(E* ) 1 1 *
] dE*
2
E
( * M0 ) * { ln
1
} (45)
d( ) E* d( ) ( 1) *
which is nonnegative provided that both (i) dE* / d( 2 ) 0 (which in particular holds if
(E* ) [1, / ( 1)] ) and (ii) the curled difference term in (45) is 0 . Sufficient for (ii) is that
or if , that
0
exp{ }. (46)
* ( 1)
However, if , then for each retrofit cost function K there is a such that (45) does attain negative
values for small 0 .
Furthermore, the expected current damage rate at peak stock, E[M * ] (in optimum), has the
*
2
following derivative wrt. :
dE*
0
2
[1 * {1 ( 1)(1 M 0 / E* ) * } ( (E* ) 1)]
d( )
(47)
0E * ( 1)
{ ( 1)(1 M 0 / E* ) * [1 2 2
]ln(1 / * )}
( 1) 2 /
*
which is positive for all 0 (0, ) if dE* / d( 2 ) 0, (E* ) 1 and 0 all hold. If ,
then for all large enough r there exists some and some 0 such that (31) is negative.
2
Assume now that / 2 . Then
26
* *
dE[ ] 1 1 1 1 d
2 2 2
ln
2 * 2
(48)
d( ) 2( 1 ) ( 1 ) d( )
2 * 2
*
which is always positive given d / d( 2 ) 0 . The impact on expected total emission E[ *
] E* is
*
d 1 1 1 1 d
2
(E[ *
] E* ) ( 1 2 2
ln
1 2
·
* 2
)E*
d( ) 2( ) ( ) d( )
2 * 2 (49)
K (E* ) (E* ) 1 1
· * [ [ (E* )]ln ] E[ *
]
W (E* ) E* 1 1 *
which is the sum of positive terms provided that (E* ) [1, / ( 1)] .
2 *
Assume instead / 2 . Then is infinite with positive probability, and from (41),
*
d Pr[ ] 2 2 1 2 d ln *
1 2 /
{ ln (1 ) } (50)
d( 2 ) 4
*
2
d( 2 )
which from Proposition 4 is positive if (E* ) [1, / ( 1)] .
Proofs
Proposition 1 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.
Propositions 2 and 3 are also straightforward calculations, Proposition A follows by copying the
argument of Proposition 2. Propositions 4, and B, require several lines of calculations, which are
given in the following.
C.1 Proof of Proposition 4
Differentiating the first-order condition wrt. , we find that W (E* ) dE* / d equals the -
derivative of the rightmost expression of (13) for Q (E ) (at E* ) – where we keep E fixed (i.e. we
regard as a function of , not the E* , and only after differentiation we evaluate at E* ). We
shall differentiate the powers for {0,1} to get
d 0E 1 d
( ) [ln ] (51)
d (r )(r )K (E ) d
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and so the -derivative of 1 0
K (E ) becomes
(r )(r )
1 1 d d
[ln ] [ln ] K (E )
d (r )(r ) d
(52)
K (E ) 1 d
{[ (E )] ln [ (E ) 1] }
E 1 d
and the result follows by inserting for the elasticity ( / ) d / d 1/( 1) , evaluating at E *
and then multiplying by / W (E* ) (which is negative, by the second-order condition). Then
*
for :
* dE* K (E* )
d * d d
[ ln ln ] (53)
d( 2 ) d 1 d( 2 ) dE* E*
and the rest is straightforward. Now by the envelope theorem, dW* / d can be calculated by
partially differentiating the expression for W (E ) from (8):
dW* ( 1) 0E*
K (E* ) [ ln ln( 1)]
d 1 (r )(r )K (E* )
(54)
1 1 1
K (E* ) [ln ( ) ] K (E* ) ln
1 1 1 1
C.2 Proof of Proposition B
The differentiations (48), (49) and (50) are straightforward; note that whenever
*
(E* ) [1, / ( 1)] , the expression for E[ ] E* is a product of two nonnegative increasing
*
functions; for the infinite-mean case, observe that from Proposition 4, ln is increasing in
volatility as long as (E* ) [1, / ( 1)] .
Consider now E[M * ] * M0 (1 * )E* / ; differentiation is also fairly straightforward,
taking into account that volatility enters by way of E* directly and through *, by way of
through * and by way of . In (45), note that the derivative of E* could take any magnitude,
depending on (E* ) ; in particular, we can choose it as close as we want to zero by merely
modifying the second derivative at E* . Thus by choosing a sequence of functions for which
the first derivative in optimum is fixed, we can get (45) negative if we can get the second term
negative, and we can if 0 (i.e. iff ) by choosing * close enough to 0 . Finally, consider
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the current damage rate at peak stock, namely *
E[M * ] . For convenience, multiply by / 0 and
1 1 1
differentiate instead [ * * ] E* * M 0 to get:
[ *
1
*
1
] dE2* [ *
2
E* ( 1)(E* M0 ) *
2
] d *
2
(E* M0 ) *
1 d
2
ln *
d( ) d( ) d( )
1 dE * 1 1 d *
(1 * ) [E* ( 1)(E* M0 ) * ] (E* M0 ) *
1
ln(1/ * )
d( 2 ) *
d( 2
)
* *
(55)
1 dE *
2
[1 * {1 ( 1)(1 M 0 / E* ) * }·( (E* ) 1)]
* d( )
E* ( 1)(E* M0 ) * 1
(E* M0 ) * ln(1/ * )
* ( 1)
The dE* / d( 2 ) coefficient is positive: 1 * 0 , if 1 , then 1 ,1 M 0 / E* and * are
all in the unit interval. Consider now the last line, multiplied by ( 1) * / E* ( 0) :
1 ( 1)(1 M 0 / E* ) * [1 ln * ] (56)
2 2
where for short we have put ( 1) / ( 2 ) (positive). Formula (56) could attain
negative values for certain parameters, but never if (which 1 ), in which case the
minimum wrt. * is attained for * 1 . Suppose therefore that . Then
1 1/
argmin [0,1] (1 ln ) e so there is some 0 for which (56) equals
2 2 2
( 1 / 2) 2 2
1 ( 1)[ 1](1 M 0 / E* )exp{ } (57)
2 2 2
2 / ( 1/2) 2
2
where inside the bracket we have inserted for and for r 1
2
( 1) . Now r does not
enter directly, but only through , and nothing else than E* depends on or * . Letting r
grow – so that and with it the bracketed term – the exponential converges to something
strictly positive. We can also keep E* constant by choosing a corresponding to each r .
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