Modeling Adaptation as a Flow and Stock Decision with
Mitigation: Electronic Supplementary Material
Tyler Felgenhauer (corresponding)
Energy and Climate Assessment Team, Environmental Protection Agency,
Office of Research and Development, Research Triangle Park, NC 27711, USA
tel: (919)541-5150
fax: (919) 541-7885
e-mail: felgenhauer.tyler@epa.gov
Mort Webster
Engineering Systems Division, Massachusetts Institute of Technology,
Cambridge, MA 02139-4307, USA.
Disclaimer: This research was conducted while Tyler Felgenhauer was a doctoral student in
the Department of Public Policy at the University of North Carolina at Chapel Hill. Any
opinions expressed are those of the authors and not necessarily of the U.S. Environmental
Protection Agency.
1
1. Analytical Integration of Mitigation and Adaptation
Several integrated assessment models (IAMs) of the climate and economy have been
applied with a focus on determining optimal mitigation levels both in cost-benefit
optimizations and in cost effectiveness analyses, as surveyed in Mastrandrea (2010).
Examples of those based on DICE or its approach include Dumas and Ha-Duong
(2008), de Bruin et al. (2009b), and Felgenhauer and de Bruin (2009). Versions of
RICE with adaptation as a policy variable allow for analysis of a geographically
heterogeneous representation of mitigation and adaptation tradeoffs (Bosello 2008; de
Bruin et al. 2009a; de Bruin et al. 2010; Dellink et al. 2010). Other dynamic
optimizations with both policies have been conducted using PAGE (Hope et al. 1993;
Hope 2006), AD-WITCH (Bosello et al. 2009, 2010), FUND for coastal adaptation to
sea-level rise (Tol 2007), and Settle et al. (2007) using Stella system dynamics
software. Independent analyses of optimal adaptation levels (apart from mitigation)
have been performed as well (Fankhauser et al. 1999; Mendelsohn 2000). In
conjunction with mitigation, policy-driven adaptation, especially when anticipatory or
ex ante to the incidence of damages, may reduce costs more than would reactive
autonomous adaptation (under a business as usual (BAU) trend) or ex post
remediative adaptation.
1.1. Static representations of mitigation and aggregated adaptation
Initial static exploration of climate policy portfolios recognized that optimal
approaches will include a mix of both mitigation and adaptation responses.
Fankhauser (1996) lays out the basic policy allocation problem in a simple global
model of tradeoffs between mitigation and adaptation, in which the policymaker
chooses policy levels to minimize the societal welfare losses of mitigation costs,
adaptation costs, and residual climate damages. Total policy costs rise with the
increased use of either strategy while damage costs fall with both. Kane and Shogren
(2000) look at the same policy allocation choice within an endogenous risk
framework. National-level policymakers seek to maximize national wealth
commensurate with society’s chosen level of risk avoidance, and damages depend on
these policies, the unknown state of the climate system, and the level of mitigation by
other countries. The authors find that more risk can either increase or decrease the
2
marginal effectiveness of mitigation, which leads to chosen mitigation levels that are
either higher or lower, respectively. More adaptation results if the increased risk does
not decrease its marginal productivity; otherwise the outcome is ambiguous. Ingham
et al. (2005) use a simplified version of the Kane and Shogren (2000) approach to
show how mitigation and adaptation are usually policy substitutes.
1.2. Dynamic analyses of mitigation and flow adaptation
Recall from the paper that flow adaptation is a relatively cheap and quickly
implemented response to climate change, providing short-term benefits that must be
repeated over time. In its purest sense, flow adaptation is a continuous variable, but
as applied it comes in discrete bursts of spending or application. The spending is
relatively reversible, not because the money once spent is recoverable but because
such sunk costs are low enough that the cost of revising the decision is also low.
Such flow adaptation can be seen pejoratively as a “coping” (Kelly and Adger 2000)
or “triage” mechanism (Craig 2010) that alleviates some damages but comes with no
expectation that these benefits will continue after the spending stops. Viewed
favorably, flow spending allows us to fill in the gaps of our adaptation needs at will
and as needed, supplementing existing infrastructure or policy mechanisms. It occurs
under relative certainty on the level of damages faced and the amount of spending
that is needed to alleviate these damages.
Ingham et al. (2007) take two approaches in dynamically modeling mitigation with
such flow adaptation. In the first, adaptation is characterized as additional mitigation,
while in the other adaptation acts to reduce total damage costs. Either representation
of adaptation within a model of mitigation optimization under uncertainty and
learning reduces the irreversibility effect. In other words, with future learning the
existence of an option to adapt reduces near-term mitigation levels.
De Bruin et al. (2009b) developed the AD-DICE model by modifying DICE
(Nordhaus and Boyer 2000) to incorporate flow adaptation, where the benefits of the
adaptation are felt immediately but have no lasting effect beyond that time period.
Using AD-DICE, de Bruin et al. (2009b) confirm the results of Ingham et al. (2005)
that an optimal response to climate change will be a mix of both mitigation and
3
adaptation. Adaptation is found to reduce damage costs primarily in early years
while mitigation reduces damage costs in later years. The authors also find that in
model runs employing only mitigation or adaptation, adding the alternative policy
response decreases the optimal level of the initial policy, and optimal mitigation
levels are more sensitive to the addition of adaptation than vice versa.
Felgenhauer and de Bruin (2009) extend AD-DICE to explore tradeoffs between
climate change mitigation and flow adaptation policies under both certainty and
uncertainty with learning. Using a two-stage stochastic programming version,
uncertainty was represented by a discrete probability distribution of five different
values for climate sensitivity CS, a measure of the temperature increase that would
result from a doubling of atmospheric CO2 concentrations. Compared to the case
where the CS is known with certainty, the addition of uncertainty (with the same
expected value) leads to three insights: 1) before learning occurs, optimal levels of
both mitigation and adaptation are lower under uncertainty than under certainty; 2) in
this same early period, the levels of optimal mitigation and adaptation are most
sensitive to their own respective policy costs (rather than gross climate damages or
the costs of the alternative policy), with the mitigation level more dependent on
adaptation costs than vice versa; and 3) variance in CS – a parameter with long-term
effects – affects optimal mitigation levels more than it affects optimal adaptation
levels.
1.3. Dynamic analyses of mitigation and stock adaptation
As with flow adaptation, mitigation and stock adaptation have been modeled in
simultaneous implementation, where long-lived stock adaptation is comprised of
large-scale and expensive infrastructure investments, with a high adaptive capacity
that declines over time due to capital depreciation and rising climate damages.
Compared with flow adaptation, such investments are likely to be episodic and
politically-triggered by extreme events that elevate climate change in the public
consciousness and give legitimacy to large-scale governmental intervention by
opening a temporary “policy window” for significant policy changes to be made
(Kingdon 1995; Adger et al. 2005).
4
Dumas and Ha-Duong (2008) stipulate an adaptation stock with efficiency ranges for
adapting to climate change, which rise with temperature but fall as temperatures
begin to exceed the stock’s designed efficiency range. Even with over-adaptation in
early periods, this proactive adaptation is found to be optimal, as the point of the
investment is to adapt to future rather than present damages. They also find that
transitional adaptation has a higher but one-time cost while absolute adaptation has
lower relative costs.
Bosello (2008) models a policy choice as one among mitigation, adaptation stock,
and research and development (R&D), using a revised version of RICE (the regional
version of DICE), FEEM-RICE. Stock adaptation and R&D investments are found to
penalize capital accumulation in the long-term, while mitigation only affects present
output. Therefore, in contrast to de Bruin et al. (2009b), the optimal mix of responses
in this model is dominated in the first several periods by mitigation, until damages
become more substantial (in 2040) and adaptation begins to dominate the policy
portfolio.
1.4. Dynamic analyses of mitigation, flow adaptation, and stock
adaptation
An emerging modeling literature is including mitigation along with adaptation with
both flow- and stock-type attributes. Lecocq and Shalizi (2008) elaborate on Kane
and Shogren (2000) to develop a dynamic partial equilibrium model with mitigation
along with temporally-distinguished adaptive responses – ex ante proactive
adaptation (like a stock), and ex post reactive adaptation (like a flow). The authors
find that the policies are jointly determined, uncertainty in the location of damages
reduces the value of proactive relative to reactive adaptation, and rainy day funds
would allow for a future correction of current investment misallocation due to climate
damage uncertainty. The adaptation tradeoff is between spending resources to cope
with damages at the time they occur (reactive ~ flow adaptation) and spending
resources now to address future damages (proactive ~ stock adaptation), where this
proactive adaptation does not lower any damages that occur now. The stock and flow
adaptations here, in contrast, are delineated by the longevity of the investments, with
stock adaptation working over its entire lifetime rather than just in the future.
5
Bosello et al (2009, 2010) use the AD-WITCH optimization model to examine the
choice among mitigation, anticipatory adaptation (stock), reactive adaptation (flow),
and R&D (a stock of knowledge that improves the effectiveness of reactive
adaptation), over different global regions. As with other model results, they find
strategic complementarity between mitigation and adaptation, in which outcomes are
improved with the implementation of both strategies, and preference is given to
adaptation if there is no possibility of catastrophic damages. While the policy mix of
anticipatory and reactive adaptation is found to be relatively equal over time in nonOECD nations, the mix in the relatively richer OECD nations is heavily weighted
toward anticipatory adaptation. The authors attribute this to the fact that relatively
richer nations are able to afford expensive investments that only become productive
in the future, and non-OECD nations are more climatically vulnerable, with damages
better addressed by reactive adaptation and the reverse being the case for OECD
nations.
Yohe et al. (2011) examine two types of adaptation to sea-level rise, with rough
equivalence to the flow and stock options, respectively: 1) environmentally benign
individualized home protection measures with low ongoing costs, and 2) an
engineering response with large up-front costs and ongoing maintenance costs.
Increasing risk aversion is found to raise the value of the stock-type engineering
adaptation, moving its date of economically efficient implementation closer to the
present, while actuarially fair insurance improves overall efficiency. The engineering
stock is designed specifically for the damages of sea-level rise, with a focus on the
effect that insurance as a public policy has on both adaptation strategies. The focus
of this paper is on the pathways of flow and stock adaptation over time and across all
sectors. Furthermore, we do not incorporate ongoing operations and maintenance
costs as in Yohe et al. (2011); rather the costs associated with adaptation stock are
ongoing investments that add to its volume.
2. Key Equations of the DICE Model, with Revisions
The Dynamic Integrated Climate Economy (DICE) model, developed originally by
Nordhaus (1994), is a simple, forward-looking integrated assessment model of the
6
world climate and economy. In the first part of this section we present the key
equations of the DICE-99 model. Equations denoted with a “B” are taken from the
appendix of Nordhaus and Boyer (2000). Explanatory notes are summarized from
chapter 2 (ibid.) and more details are available from the same source. Our model is
based on DICE-98, and we make note of the small differences with DICE-99. We
also summarize the additional equations that were modified or added for this model,
with the numbering from the main paper, and refer the reader there for a more
detailed explanation.
The objective in DICE is to maximize aggregate utility (social welfare), calculated as
the discounted natural logarithm of consumption,1
𝑈[𝐶(𝑡), 𝐿(𝑡)] = 𝐿(𝑡){log[𝐶(𝑡)]}.
(1)
In each time period, consumption and savings/investment are chosen endogenously,
subject to available income that is reduced by the costs of climate change (residual
damages and mitigation costs). As noted in the paper, the formal statement of the
problem is
max 𝑊 = ∑ 𝑈[𝐶(𝑡), 𝐿(𝑡)]𝑅(𝑡)
𝐼(𝑡),𝜇(𝑡)
𝑡
(1)
subject to 𝑆(𝑡) = 𝐺{𝑆(𝑡 − 1), 𝐶(𝑡 − 1)}
where t is the time period index, 𝑈(∙) is the utility function from equation (1), 𝐶(𝑡) is
consumption, 𝐿(𝑡) is population, and 𝑅(𝑡) is the discount factor. The model is
constrained such that 𝑆(𝑡) is the vector of states (capital stocks), 𝐺(∙) is the state
transition function, and 𝑃(𝑡) is the vector of policy controls (emissions control rate
𝜇(𝑡) and investment in physical capital stocks 𝐼(𝑡)). The constraint equations are
defined in Nordhaus and Boyer (2000), and for convenience are also given in the
1
Alternatively, more recent versions of DICE use an iso-elastic utility function (Nordhaus 2007),
𝑈[𝐶(𝑡), 𝐿(𝑡)] = 𝐿(𝑡) [
𝐶(𝑡)1−𝛼
].
(1−𝛼)
7
Online Resources. Below we only explicitly list the additional or modified variables
and equations that represent flow and stock adaptation.
The pure time preference discount factor is a function of the social time preference
discount rate 𝑆𝑅𝑇𝑃(𝑡), which is denoted as 𝜌(0) in Nordhaus and Boyer (2000).
This declines at the decadal rate of 𝑔𝜌 because of declining impatience,
𝑡
𝑅(𝑡) = ∏[1 + 𝑆𝑅𝑇𝑃(𝑣)]−10
(B.2)
𝑣=0
𝑆𝑇𝑅𝑃(𝑡) = 𝑆𝑅𝑇𝑃(0)exp(−𝑔𝜌 𝑡).
World population grows at the rate of 𝑔𝑝𝑜𝑝 (𝑡) from the initial time period 𝑔𝑝𝑜𝑝 (0),
but at a declining rate of growth δ𝑝𝑜𝑝 ,
𝑔𝑝𝑜𝑝 (𝑡) = 𝑔𝑝𝑜𝑝 (0)exp(−δ𝑝𝑜𝑝 𝑡)
t
𝐿(𝑡) = 𝐿(0)exp (∫ 𝑔𝑝𝑜𝑝 (𝑡))
(B.4)
0
Global output 𝑄(𝑡) comes from the production function,
𝑄(𝑡) = 𝛺(𝑡)(1 − 𝑏1 (𝑡)𝜇(𝑡 𝑏2 ))𝐴(𝑡)𝐾(𝑡)𝛾 𝐿(𝑡)1−𝛾 ,
(B.5)
where 𝐴(𝑡) is technological change (total factor productivity), 𝐾(𝑡) is capital, and
𝐿(𝑡) is population (labor). Capital and population interact via the elasticity of output
with respect to population, 𝛾. Output is tempered by the damages of climate change,
as represented by the coefficent 𝛺(𝑡).
Output in the production function above (B.5) is also tempered by the costs of
mitigation 𝜇(𝑡), which has the cost parameters 𝑏1 (𝑡) and 𝑏2. Mitigation is on a [0-1]
scale, representing the fractional reduction of greenhouse gas emissions relative to the
baseline for that period. Mitigation reduces gross damages from the level that would
occur under a no-policy BAU scenario. As baseline emissions are always rising,
increasing mitigation will not necessarily result in decreasing emissions over time.
Mitigation costs decline over time, as described here:
8
𝑔𝑏 (𝑡) = 𝑔𝑏 (0)exp(−𝛿 𝑏 𝑡)
𝑏1 (𝑡) =
𝑏1 (𝑡 − 1)
(1 + 𝑔𝑏 (𝑡))
(B.9)
𝑏1 (0) = 𝑏1 ∗ ,
where 𝑔𝑏 (𝑡) is the rate of decline of 𝑏1 (𝑡), and 𝛿 𝑏 (t) is the rate of decline of 𝑔𝑏 (𝑡).
Though its costs decline, the benefits of mitigation lag far into the future due to the
long time scales of the carbon cycle.
Capital stock evolves as a function of investment and the depreciation rate of capital,
𝛿𝐾
𝐾(𝑡) = 𝐾(𝑡 − 1)(1 − 𝛿𝐾 )10 + 10 ∗ 𝐼(𝑡 − 1)
𝐾(0) = 𝐾 ∗
(B.15)
As a measure of productivity, technological growth occurs at the rate 𝑔 𝐴 , but this
growth slows at the constant rate of 𝛿 𝐴 :
𝑔 𝐴 (𝑡) = 𝑔 𝐴 (0)exp(−𝛿 𝐴 𝑡)
(B.6)
t
𝐴(𝑡) = 𝐴(0)exp (∫ 𝑔
𝐴 (𝑡)
)
0
The climate change damage coefficient is a function of the climate damages 𝐷(𝑡),
𝛺(𝑡) =
1
[1 + 𝐷(𝑡)]
(B.7)
These damages rise exponentially with temperature. We update from the Nordhaus
and Boyer (2000) version,
𝐷(𝑡) = 𝜃1 𝑇(𝑡) + 𝜃2 𝑇(𝑡)2 ,
(B.8)
to use a three parameter damage function with the new term gross climate damages,
𝐺𝐷(𝑡),
𝐺𝐷(𝑡) = 𝛼1 𝑇(𝑡) + 𝛼2 𝑇(𝑡)𝛼3 .
(6)
9
In our model, gross damages are those that can be lowered by mitigation, or which
are unaffected if there is no mitigation.
World industrial CO2 emissions are a function of mitigation, the base case emissions
intensity of the economy 𝜎𝑒 (𝑡), and the previously listed factors of production
(technology, capital, and population):
𝐸(𝑡) = (1 − 𝜇(𝑡))𝜎𝑒 (𝑡)𝐴(𝑡)𝐾(𝑡)𝛾 𝐿(𝑡)1−𝛾
(B.10)
Nordhaus and Boyer (2000) denote the base case emissions intensity as simply 𝜎(𝑡);
we rename it here as 𝜎𝑒 (𝑡) in order to avoid confusion with our 𝜎, which is the
elasticity of substitution between flow and stock adaptation types. The emissions
intensity of the economy falls in a manner similar to the declining costs of mitigation
above,
𝑔𝜎 (𝑡) = 𝑔𝜎 (0)exp(−𝛿 𝜎 1 𝑡 − 𝛿 𝜎 2 𝑡 2 )
𝜎𝑒 (𝑡) =
𝜎𝑒 (𝑡 − 1)
(1 + 𝑔𝜎 (𝑡))
(B.11)
𝜎𝑒 (0) = 𝜎 ∗
The 1999 version of DICE has an additional source of greenhouse gas emissions from
land use 𝐿𝑈(𝑡) that is not in the our version. This combines with industrial emissions
to form total emissions 𝐸𝑇(𝑡):
𝐿𝑈(𝑡) = 𝐿𝑈(0)(1 − 𝛿1 )𝑡
(B.16)
𝐸𝑇(𝑡) = 𝐸(𝑡) + 𝐿𝑈(𝑡)
Thus, in our model, 𝐸(𝑡) = 𝐸𝑇(𝑡).
Global CO2 accumulation and transportation is represented in a three reservoir model.
𝑀𝐴𝑇 (𝑡), 𝑀𝑈𝑃 (𝑡), and 𝑀𝐿𝑂 (𝑡) are the end of period mass of carbon in the atmosphere,
biosphere and upper oceans, and lower oceans, respectively; 𝜑() are the parameters
of the carbon transition matrix.
10
𝑀𝐴𝑇 (𝑡) = 10 ∗ 𝐸𝑇(𝑡 − 1) + 𝜑11 𝑀𝐴𝑇 (𝑡 − 1) − 𝜑12 𝑀𝐴𝑇 (𝑡 − 1)
+ 𝜑21 𝑀𝑈𝑃 (𝑡 − 1)
(B.17a)
𝑀𝐴𝑇 (0) = 𝑀𝐴𝑇 ∗
𝑀𝑈𝑃 (𝑡) = 𝜑22 𝑀𝑈𝑃 (𝑡 − 1) + 𝜑12 𝑀𝐴𝑇 (𝑡 − 1) + 𝜑32 𝑀𝐿𝑂 (𝑡 − 1)
(B.17b)
𝑀𝑈𝑃 (0) = 𝑀𝑈𝑃 ∗
𝑀𝐿𝑂 (𝑡) = 𝜑33 𝑀𝐿𝑂 (𝑡 − 1) + 𝜑23 𝑀𝑈𝑃 (𝑡 − 1)
(B.17c)
𝑀𝐿𝑂 (0) = 𝑀𝐿𝑂 ∗
In DICE, radiative forcing of the atmosphere 𝐹(𝑡) increases with increasing
emissions in combination with other exogenous forcings, 𝑂(𝑡):
𝐹(𝑡) = 𝜂 {log [
𝑀𝐴𝑇 (𝑡)
𝑀𝐴𝑇 𝑃𝐼
] /log(2)} + 𝑂(𝑡)
𝑂(𝑡) = −0.1965 + 0.13465𝑡
= 1.15
𝑡 < 11
(B.18)
𝑡 < 10
The increased in globally averaged temperature in the atmosphere and upper oceans
since 1900 is 𝑇(𝑡), which comes from the radiative forcing of the atmosphere. The
warming of the lower ocean 𝑇𝐿𝑂 (𝑡) lags due to thermal inertia. The coefficients 𝜎𝑖
reflect thermal flow rates and capacities of the three sinks, and 𝜆 is a feedback
parameter:
𝑇(𝑡) = 𝑇(𝑡 − 1)
+𝜎1 {𝐹(𝑡) − 𝜆𝑇(𝑡 − 1) − 𝜎2 [𝑇(𝑡 − 1) − 𝑇𝐿𝑂 (𝑡 − 1)]}
(B.19)
𝑇(0) = 𝑇 ∗
𝑇𝐿𝑂 (𝑡) = 𝑇𝐿𝑂 (𝑡 − 1) + 𝜎3 [𝑇(𝑡 − 1) − 𝑇𝐿𝑂 (𝑡 − 1)]
𝑇𝐿𝑂 = 𝑇𝐿𝑂 ∗
(B.20)
11
In addition to mitigation, the present model incorporates two types of adaptation as
additional policy responses to reduce climate change damages. First, short-lived flow
adaptation 𝐴𝐹(𝑡) reduces damages in the current time period 𝑡 but has no lasting
effects. As in AD-DICE De Bruin et al. (2009b), adaptation flow costs 𝐴𝐹𝐶(𝑡) rise
as a multiple of implementation:
𝐴𝐹𝐶(𝑡) = 𝛾1 𝐴𝐹(𝑡)𝛾2 .
(10)
Second, a supply of long-lived adaptation stock 𝐻(𝑡) can provide additional
adaptation services 𝐴𝑆(𝑡) through the adaptation stock production function,
𝐴𝑆(𝑡) = 𝜃1 + 𝜃2 𝑙𝑛 (
𝐻(𝑡)
),
𝐾(𝑡)
(9)
where 𝐾(𝑡) is the total amount of non-adaptive capital stock in the economy, and
𝜃1 and 𝜃2 are parameters used for calibration. Adaptation stock is replenished every
period by the policymaker’s chosen amount of adaptation stock investments 𝐴𝑆𝐼(𝑡),
and depreciates at the rate 𝛿𝑠 such that
𝐻(𝑡 + 1) = [(1 − 𝛿𝑠 )10 ]𝐻(𝑡) + 10𝐴𝑆𝐼(𝑡).
(4)
The replenishment and depreciation behavior of adaptation stock is the same as for
physical capital stock in DICE, for adaptation stock in other studies (Bosello 2008;
Agrawala et al. 2010, 2011; de Bruin 2011), and for knowledge stock in economic
models of R&D (e.g., Popp (2004)).
Both the short-lived flow adaptation and the long-lived stock adaptation are on a [01] scale, with zero representing no adaptation and one being the hypothetical upper
bound representing the elimination of all climate damages in that time period. As in
Agrawala et al. (2010, 2011) and de Bruin (2011), flow and stock adaptation
responses are combined through a constant elasticity of substitution (CES) production
function to provide a total level of adaptation services 𝐴𝐷(𝑡),
𝛼
𝐴𝐷(𝑡) = [𝛽𝐴𝐹(𝑡)𝜌 + (1 − 𝛽)𝐴𝑆(𝑡)𝜌 ]𝜌 ,
(8)
where 𝜌 is the substitution parameter between the two adaptation inputs (the elasticity
of substitution is 𝜎 = 1/(1 + 𝜌)), and 𝛼 is the returns to scale for adaptation. The
12
relative share of flow adaptation as an input factor in the amount of total adaptation is
𝛽, and (1 − 𝛽) is the share of stock adaptation in the production of adaptation.
As an aside, two other total adaptation production functions were explored and
rejected in favor of the CES form in this model. A linear (additive) combination,
where AD(t) = c1 (c2 AF(t) + c3 AS(t)), requires values for the scaling parameters c(.)
that have no support in the literature. Alternatively, a Cobb-Douglas production
function, where AD(t) = A(AF(t)a ∗ AS(t)b ), with A as the scaling parameter and a
and b as the input share parameters, is a special case of the CES function with ρ = 0
and σ = 1. This form restricts the ability to conduct sensitivity analysis and explore
model assumptions. Further discussion of production functions is found in Nicholson
(1995), Chiang (1984), and Friedman (2002).
The stock and flow adaptation types act jointly to reduce “post-mitigation” gross
damages (of equation 6) to a further lower level of net damages 𝑁𝐷(𝑡) such that
𝑁𝐷(𝑡) = 𝐺𝐷(𝑡)(1 − 𝐴𝐷(𝑡)).
(7)
In DICE, output 𝑄(𝑡) equates to the sum of total consumption 𝐶(𝑡) and investment
𝐼(𝑡). In the no policy case, the number of carbon emissions permits 𝛱(𝑡) equals total
emissions. The parameter 𝜏(𝑡) is a carbon tax, though it can also be interpretted as
the price of emissions permits, such that
𝑄(𝑡) + 𝜏(𝑡)[𝛱(𝑡) − 𝐸(𝑡)] = 𝐶(𝑡) + 𝐼(𝑡)
(B.12)
𝛱(𝑡) = 𝐸(𝑡).
(B.13)
Per capita consumption 𝑐(𝑡) comes from total consumption over total population:
𝑐(𝑡) =
𝐶(𝑡)
𝐿(𝑡)
(B.14)
In this model, income 𝑌(𝑡) is the total economic output 𝑄(𝑡), which is reduced by net
damages 𝑁𝐷(𝑡), adaptation flow costs 𝐴𝐹𝐶(𝑡), and mitigation costs 𝑀𝐶(𝑡)
13
𝑌(𝑡) = 𝑄(𝑡)
1 − 𝑀𝐶(𝑡) − 𝐴𝐹𝐶(𝑡)
1 + 𝑁𝐷(𝑡)
(3)
where 𝑀𝐶(𝑡) = 𝑏1 (𝑡)𝜇(𝑡 𝑏2 ) from equation (B.5) above. As with investments in
normal productive capital 𝐼(𝑡), investments in adaptation stock 𝐴𝑆𝐼(𝑡) reduce
available consumption:
𝐶(𝑡) = 𝑌(𝑡) − 𝐼(𝑡) − 𝐴𝑆𝐼(𝑡).
(11)
Total adaptation costs are the sum of spending on adaptation flow efforts and
investments in adaptation stock.
4. Model Calibration and Additional Reference
Results
4.1. Calibration
Here we present results from the calibration of our model to Agrawala et al. (2011),
as described in Section 3 of the paper, in three figures. In Fig. 1 we compare the
relative ratios of stock to flow adaptation investment from the models over time. Fig.
2 depicts the net present values (NPV) of adaptation spending as a percentage of
global GDP. And in Fig. 3 we show the policy cost pathways for total adaptation,
mitigation, and climate damages.
Our results reproduce qualitatively those from Agrawala et al. (2011), particularly the
relative shares of flow vs. stock adaptation and the NPV of stock and flow adaptation
spending levels in the mitigation and adaptation scenario. The numerical differences
are primarily a result of our assumption of diminishing marginal returns in the stock
adaptation production function. This assumption has the effect of making stock and
aggregate adaptation more costly (equivalently, less productive for the same levels of
expenditures). As a result, our model solution has lower aggregate adaptation,
slightly higher mitigation, and slightly higher net damages. However, the patterns
over time and the relative ratios of stock adaptation, flow adaptation, and mitigation
are the essentially the same.
14
Fig. 1 Comparison of flow and stock adaptation spending ratios.
50%
Flow percentage of total
adaptation spending
Stock percentage of total
adaptation spending
100%
40%
80%
30%
60%
Agrawala et al. (2011)
20%
40%
this paper
10%
20%
0%
2005 2025 2045 2065 2085 2105
0%
2005 2025 2045 2065 2085 2105
year
year
Fig. 2 Comparisons of the NPV of climate policy costs over three scenarios, with 3% discounting: 1)
no climate policy, 2) optimal adaptation (A*) only, and 3) optimal adaptation with optimal mitigation
(M*).
no policy
(2010-2100),
Agrawala et al.
(2011)
no policy
(2015-2115),
this paper
A* only
(2010-2100),
Agrawala et al.
(2011)
A* only
(2015-2115),
this paper
1.5%
A-costs
residual damages
1.0%
gross damages
0.5%
0.0%
stock
flow
adaptation costs, % of world GDP
percentage of world GDP
2.0%
0.3%
0.2%
0.1%
0.0%
A* only
(2010-2100),
Agrawala et al.
(2011)
A* only
(2015-2115),
this paper
A* and M*
(2010-2100),
Agrawala et al.
(2011)
A* and M*
(2015-2115),
this paper
15
Fig. 3 Comparisons of the policy costs of optimal adaptation and mitigation over time, as a
percentage of world GDP.
percent of world GDP
3%
Total Adaptation Costs
Mitigation Costs
2%
1%
0%
percent of world GDP
3%
Residual (Net) Damages
Total Policy Costs
2%
1%
0%
2005
2025
2045 2065
year
2085
2105
2005
2025
Agrawala et al. (2011), Fig. 9
2045 2065
year
2085
2105
this paper
4.2. Additional Reference Case Results
In Table 1 we list the reference and calibrated values for the model runs, and in Fig. 4
we present a set of model reference case runs for four scenarios over 2015-2115,
compared on the alternative metric of total policy costs.
16
Table 1 Parameters in the model – reference and chosen for sensitivity analysis.
Reference
Value
Reference
Source
gross climate damages, GD
𝛼1
first, second, and third damage coefficients
𝛼2
𝛼3
0.0075
0.0025
3.1958
calibrated to
de Bruin (2011)
adaptation flow cost function, AFC
𝛾1
first and second parameters
𝛾2
0.115
3.60
de Bruin
et al. (2009b)
1.062
calibrated to
Agrawala et
al. (2011)
author choicea
Agrawala et al.
(2011)b
Function and Parameters
Sensitivity
Values
adaptation stock production function, ASPROD
𝜃1
𝜃2
first and second parameters
0.15
𝐻0
adaptation stock initial value
0.25
𝛿𝑠 
depreciation rate of adaptation stock
5%
total adaptation production function, ADPROD
relative distribution of flow adaptation as an input
0.49

factor share
α
returns to scale for all adaptation types in totalc
0.8
elasticity of substitution between flow and stock
σ
2
adaptation
substitution parameter between the two
ρ
-0.11111
adaptation inputs
other parameters
capacity limits on flow adaptation
𝛬
CS
climate sensitivity
SRTP(0)
initial social time preference discount rated
0
2.9078
3%
Agrawala et al.
(2011)
0.2, 0.4,
0.6, 0.8
0.5, 0.999,
3, 4
derives from σ
model default
Nordhaus and
Boyer (2000)
0.025
1.5, 4, 5, 6
0.1%, 1.5%,
4%, 5%
a
This is chosen for model tractability. A large amount of capital stock already exists that actively
provides adaptation services, even if the vast majority of it was built without explicitly planning for
climate change. Bosello (2008) chooses an initial adaptation stock level of zero.
b
De Bruin (2011) chooses a reference of 10%, and conduct sensitivity over 5, 10, and 50%.
c
The returns are such that α < 1 is decreasing returns to scale, α = 1 is constant returns to scale, and
α > 1 is increasing returns to scale.
d
This parameter is denoted as 𝜌(0) in Nordhaus and Boyer (2000). Agrawala et al. (2010, 2011) conduct
sensitivity analysis over two discount rates, while de Bruin (2011) conducts sensitivity analysis over
three discount rates and two levels of climate damages. In Section 4 of the Online Resources we
present sensitivity analyses for both discount rates as well as climate sensitivity.
17
Fig. 4 Total policy costs as a percent of world GDP, under four policy scenarios, 2015-2115.
7.5%
flow adaptation
percnet of world GDP
adaptation stock investment
mitigation
5.0%
net damages
2.5%
0.0%
no climate policy (BAU)
mitigation only
adaptation only
optimal mitigation
and adaptation
four policy scenarios, 2015-2115
Total policy costs are the sum of flow adaptation spending, adaptation stock
investments, mitigation costs, and residual climate damages after policy
implementation. In descending order from left to right in Fig. 4, total policy costs of
the no climate policy/BAU scenario are higher than those of a mitigation-only
approach, an adaptation-only response, and the optimal response that uses both
mitigation and adaptation, which has the lowest net damages and overall costs of the
four scenarios. An adaptation-only approach brings slightly lower total costs than
does a mitigation-only strategy, as well as slightly lower net (post-policy) damages
for the first 20 periods. The same ordering of scenarios by policy costs is found in
other work, e.g., Felgenhauer and de Bruin (2009) and Bosello et al. (2009). It is
clear that the policy costs of all responses to climate change (both types of adaptation
plus mitigation) remain small when compared to both the damages that they eliminate
as well as the damages that they cannot eliminate.
5. Additional Sensitivity Analyses
5.1. Climate sensitivity
Climate sensitivity (CS) is the change in the global mean surface
temperature that comes with a doubling of CO2 concentrations from pre18
industrial levels, providing a measure of how sensitive the climate is to
GHG forcing. In 5.2. Discount Rates
In the model, the social rate of time preference is a measure of how much value is put
on outcomes that happen in the long-term future versus those that occur in the near
term. As expected, sensitivity analysis of the model over five discount rates (Fig. 6,
p. 21) shows a strong relationship between a very low discount rate and a willingness
to devote high levels of resources to mitigation, in order to lower damages. With a
high discount rate, this willingness evaporates, replaced with a tolerance of higher
climate damages and an allocation of policy emphasis towards adaptation, but at
much lower spending levels. With a higher emphasis on adaptation relative to
mitigation under the higher discount rate, spending on flow adaptation rises relative
to that devoted to stock adaptation as well, which reflects a preference for short-term
over long-term climate policy responses.
Fig. 5 (p. 20) we show results for policy pathways and portfolio compositions of the
three responses (flow adaptation stock adaptation, and mitigation) under five climate
sensitivities. As expected, higher CS shifts the levels of all policies higher, with
mitigation increasing the most relative to its levels under a lower CS. This is similar
to results from Felgenhauer and de Bruin (2009) where mitigation, with its lagged
climate benefits, is most affected by a long-term parameter such as CS. As
adaptation derived from stock is longer-lived compared to flow adaptation, but
shorter-lived compared to mitigation, its response to changing CS is in the middle.
Measuring policy costs in terms of their NPVs – both as a percentage of world GDP
and in terms of shares of the policy portfolio – leads to two insights. First, higher
levels of all policies due to higher CS lead to higher costs, with mitigation and higher
damages forming the largest shares of all costs, but with mitigation rising the most
relative to other cost categories. Second, as mitigation spending rises with higher CS
levels, it substitutes primarily for stock adaptation rather than for flow adaptation.
5.2. Discount Rates
In the model, the social rate of time preference is a measure of how much value is put
on outcomes that happen in the long-term future versus those that occur in the near
term. As expected, sensitivity analysis of the model over five discount rates (Fig. 6,
19
p. 21) shows a strong relationship between a very low discount rate and a willingness
to devote high levels of resources to mitigation, in order to lower damages. With a
high discount rate, this willingness evaporates, replaced with a tolerance of higher
climate damages and an allocation of policy emphasis towards adaptation, but at
much lower spending levels. With a higher emphasis on adaptation relative to
mitigation under the higher discount rate, spending on flow adaptation rises relative
to that devoted to stock adaptation as well, which reflects a preference for short-term
over long-term climate policy responses.
Fig. 5 Sensitivity of adaptation and total policy levels to climate sensitivity.
flow adaptation
0.3
0.3
0.2
0.2
stock adaptation
mitigation
1
0.5
0.1
0.1
0.0
0
year
CS=1.5
0
year
CS=2.9078
year
CS=4
CS=5
CS=6
20
4%
3%
2%
1%
NPV (2015-2115)
as a Percentage
of World GDP
flow adaptation
stock adaptation
mitigation
0%
net damages
100%
75%
50%
25%
NPV (2015-2115)
of Policy Costs as
a Share of Climate
Policy Spending
flow adaptation
stock adaptation
mitigation
0%
21
Fig. 6 Sensitivity of adaptation and total policy levels to discount rates.
flow adaptation
mitigation
stock adaptation
0.3
0.3
0.2
0.2
1
0.5
0.1
0.1
0
0
year
0
year
year
SRTP = 0.001 (Stern)
SRTP = 0.015 (UK Treasury)
SRTP = 0.03 (Nordhaus)
SRTP = 0.04
SRTP = 0.05
4%
NPV (2015-2115)
as a Percentage
of World GDP
3%
2%
flow adaptation
stock adaptation
mitigation
net damages
1%
0%
SRTP = 0.001
(Stern)
SRTP = 0.015
(UK Treasury)
SRTP = 0.03
(Nordhaus)
SRTP = 0.04
SRTP = 0.05
100%
NPV (2015-2115)
of Policy Costs as
a Share of Climate
Policy Spending
75%
flow adaptation
stock adaptation
mitigation
50%
25%
0%
SRTP = 0.001
(Stern)
SRTP = 0.015
(UK Treasury)
SRTP = 0.03
(Nordhaus)
SRTP = 0.04
SRTP = 0.05
Acknowledgements
The authors would like to thank Tim Johnson, Richard Andrews, Doug Crawford-Brown, Jonathan
Wiener, Gary Yohe, Shardul Agrawala, Rob Dellink, and Kelly de Bruin, as well as three anonymous
reviewers, for comments on earlier versions of this paper. Tyler Felgenhauer gratefully acknowledges
financial research support from the Joseph L. Fisher Doctoral Dissertation Fellowship from Resources
22
for the Future, as well as the Royster Society of Fellows at the University of North Carolina at Chapel
Hill.
References
Adger, W. N., N. W. Arnell and E. L. Tompkins (2005). "Successful Adaptation to Climate Change
Across Scales." Global Environmental Change 15: 77-86.
Agrawala, S., F. Bosello, C. Carraro, K. d. Bruin, E. D. Cian, R. Dellink and E. Lanzi (2010). Plan or
React? Analysis of Adaptation Costs and Benefits Using Integrated Assessment Models.
OECD Environmental Working Papers. No. 23 (10 August), Organization for Economic
Cooperation and Development.
--- (2011). "Plan or React? Analysis of Adaptation Costs and Benefits Using Integrated Assessment
Models." Climate Change Economics 2(3): 175-208.
Bosello, F. (2008). Adaptation, Mitigation, and "Green" R&D to Combat Global Climate Change:
Insights From an Empirical Integrated Assessment Exercise. Research Paper 0048 / Working
Paper 20 (October), Centro Euro-Mediterraneo per i Cambiamenti Climatici (CMCC),
Climate Impacts and Policy Division, Milan, Italy.
Bosello, F., C. Carraro and E. D. Cian (2009). An Analysis of Adaptation as a Response to Climate
Change. Copenhagen Consensus on Climate, Copenhagen Consensus Center, Copenhagen,
Denmark.
--- (2010). Climate Policy and the Optimal Balance Between Mitigation, Adaptation, and Unvoided
Damage. Working Papers, Department of Economics, Ca' Foscari University of Venice. no.
10 (February), Dipartimento Scienze Economiche (DSE), Venice, Italy.
Chiang, A. C. (1984). Fundamental Methods of Mathematical Economics. New York, NY, McGrawHill, Inc.
Craig, R. K. (2010). ""Stationarity is Dead" – Long Live Transformation: Five Principles for Climate
Change Adaptation Law." Harvard University Environmental Law Review 34: 9-73.
de Bruin, K. C. (2011). Distinguishing Between Proactive (Stock) and Reactive (Flow) Adaptation.
CERE Working Paper. 2011:8 (August), Centre for Environmental and Resource Economics
(CERE).
de Bruin, K. C., R. Dellink and S. Agrawala (2009a). Economic Aspects of Adaptation to Climate
Change: Integrated Assessment Modelling of Adaptation Costs and Benefits. Environment
Working Paper No. 6. ENV/WKP(2009)1 (24 March), Environment Directorate, Organization
for Economic Cooperation and Development (OECD), Paris, France.
de Bruin, K. C., R. B. Dellink and R. S. J. Tol (2009b). "AD-DICE: An Implementation of Adaptation
in the DICE Mode." Climatic Change 95: 63-81.
--- (2010). International Cooperation on Climate Change Adaptation from an Economic Perspective.
Sustainable Development Series. 63C. Carraro, Fondazione Eni Enrico Mattei (FEEM),
Milan, Italy.
Dellink, R., K. C. de Bruin and E. v. Ierland (2010). Incentives for International Cooperation on
Adaptation and Mitigation. The Social and Behavioural Aspects of Climate Change. Linking
Vulnerability, Adaptation and Mitigation. P. Martens and C. Chang Ting. Sheffield UK,
Greenleaf Publishing.
Dumas, P. and M. Ha-Duong (2008). Optimal Growth with Adaptation to Climate Change. 16th Annual
Conference of the European Association of Environmental and Resource Economists
(EAERE). Gothenburg, Sweden.
Fankhauser, S. (1996). The Potential Costs of Climate Change Adaptation. Adapting to Climate
Change: An International Perspective. J. B. Smith, N. Bhatti, G. V. Menzhulinet al. New
York, NY, Springer-Verlag, Inc.: 80-96.
Fankhauser, S., J. B. Smith and R. S. J. Tol (1999). "Weathering Climate Change: Some Simple Rules
to Guide Adaptation Decisions." Ecological Economics 30: 67-78.
Felgenhauer, T. and K. C. de Bruin (2009). "The Optimal Paths of Climate Change Mitigation and
Adaptation Under Certainty and Uncertainty." The International Journal of Global Warming
1(1/2/3): 66-88.
Friedman, L. S. (2002). The Microeconomics of Public Policy Analysis. Princeton, NJ, Princeton
University Press.
23
Hope, C. (2006). "The Marginal Impact of CO2 from PAGE2002: An Integrated Assessment Model
Incorporating the IPCC's Five Reasons for Concern." The Integrated Assessment Journal
6(1): 19-56.
Hope, C., J. Anderson and P. Wenman (1993). "Policy Analysis of the Greenhouse Effect: An
Application of the PAGE Model." Energy Policy 15: 327-338.
Ingham, A., J. Ma and A. Ulph (2007). "Climate Change, Mitigation, and Adaptation with Uncertainty
and Learning." Energy Policy 35: 5354-5369.
Ingham, A., J. Ma and A. M. Ulph (2005). Can Adaptation and Mitigation be Complements? Tyndall
Centre Working Papers. 79 (August), Tyndall Centre for Climate Change Research, Norwich,
UK.
Kane, S. and J. F. Shogren (2000). "Linking Adaptation and Mitigation in Climate Change Policy."
Climatic Change 45: 75-102.
Kelly, P. M. and W. N. Adger (2000). "Theory and Practice in Assessing Vulnerability to Climate
Change and Facilitating Adaptation." Climatic Change 47: 325-352.
Kingdon, J. (1995). Agendas, Alternatives, and Public Policies, Adison-Wesley Educational
Publishers, Inc.
Lecocq, F. and Z. Shalizi (2008). Balancing Expenditures on Mitigation of and Adaptation to Climate
Change: An Exploration of Issues Relevant to Developing Countries. 16th Annual Conference
of the European Association of Environmental and Resource Economists (EAERE).
Gothenburg, Sweden.
Mastrandrea, M. D. (2010). Representation of Climate Impacts in Integrated Assessment Models.
Assessing the Benefits of Avoided Climate Change: Cost-Benefit Analysis and Beyond
(March 16-17, 2009). J. Gulledge, L. J. Richardson, L. Adkins and S. Seidel. Arlington, VA,
Pew Center on Global Climate Change: 85-99.
Mendelsohn, R. (2000). "Efficient Adaptation to Climate Change." Climatic Change 45: 583-600.
Nicholson, W. (1995). Microeconomic Theory: Basic Principles and Extensions. Fort Worth, TX, The
Dryden Press.
Nordhaus, W. D. (1994). Managing the Global Commons: The Economics of Climate Change.
Cambridge, MA, MIT Press.
--- (2007). A Question of Balance: Weighing the Options on Global Warming Policies. New Haven,
CT, Yale University Press.
Nordhaus, W. D. and J. Boyer (2000). Warming the World: Economic Models of Global Warming.
Cambridge, MA, MIT Press.
Popp, D. (2004). "ENTICE: Endogenous Technological Change in the DICE Model of Global
Warming." Journal of Environmental Economics and Management 48: 742-768.
Settle, C., J. F. Shogren and S. Kane (2007). "Assessing Mitigation-Adaptation Scenarios for Reducing
Catastrophic Climate Risk." Climatic Change 83: 443-456.
Tol, R. S. J. (2007). "The Double Trade-Off Between Adaptation and Mitigation for Sea Level Rise:
An Application of FUND." Mitigation and Adaptation Strategies for Global Change 12: 741753.
Yohe, G., P. Kirshen and K. Knee (2011). "On the Economics of Coastal Adaptation Solutions in an
Uncertain World." Climatic Change 106: 71-92.
24