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Simplified-IAM-for-teaching.pdf
1
A Simple Climate-Solow Model for Introducing the Economics of Climate Change to
Undergraduate Students
Panagiotis Tsigaris1
Professor
Department of Economics
and
Joel Wood
Assistant Professor
Department of Economics
Thompson Rivers University
900 McGill Road
Kamloops, B.C.,
CANADA, V2C 0C8
May 24, 2016
Abstract
In this paper the simplest integrated assessment model is developed in order to illustrate to
undergraduate students the economic issues associated with climate change. The growth model
developed in this paper is an extension of the basic Solow model and includes a simple climate
model. Even though the model is very simple it is very powerful in its predictions. Students use
the model to explore various scenarios illustrating how economic activity today will inflict
damages from higher temperatures on future generations. But students also observe that future
generations will be richer than today’s generation due to productivity growth and population
stabilization. Hence, the richer future generations will not be as rich as they would be without
climate change. Since the cost of action is absorbed by the current generation and the benefits of
action accrue to future generations students can conduct a cost-benefit analysis and explore the
importance of the discount rate. The appendix provides step-by-step instructions for students to
setup the model in MS Excel and to conduct simulations.
Keywords: Integrated Assessment Models, Climate Change, Solow Growth Model, Teaching
Economics.
JEL: A22, O44, Q54.
1 Authors email addresses: [email protected] and [email protected]
2
Introduction
“Greenhouse gas (GHG) emissions are externalities and represent the biggest market failure
the world has seen” –Sir Nicholas Stern (2007)
Climate change caused by the Greenhouse Gas (GHG) emissions released by the burning of
fossil fuels and land use changes imposes damages to future generations.2 GHG emissions trap
heat and affect the future climate resulting in damages from increased temperatures. For
example, increased temperatures are expected to cause sea level rise, increased floods, increased
droughts and heat waves, and possibly even increased human conflict. The current generation
benefits from using fossil fuels, but does not internalize these external costs. As a result, climate
change is what economists call a negative externality. Covert et al., (2016) examine historical
data on fossil fuel production and consumption and conclude that neither supply (e.g., Peak Oil)
nor demand (e.g., development of low-carbon technologies) factors will sufficiently reduce GHG
emissions. Without government intervention, humans will overproduce GHG emissions.
The climate change problem is further complicated as being a global externality rather
than a local one. Even though each nation emits a different amount of greenhouse gases (GHG),
the marginal impact of a tonne of GHG is independent of where it is emitted (Stern, 2007);
whereas, the effects of smog in a city are local and heterogeneous depending on the geography
and demographics of the city. Furthermore, GHGs accumulate in the atmosphere and stay a long
time, i.e., carbon dioxide has an average atmospheric life of over a century (Archer et al., 2013).
The impact is persistent and long term, whereas the effects of smog in a city are relatively
immediate following exposure.
2 For scientific consensus on the issue see Oreskes, (2004).
3
Due to the persistence of GHGs in the atmosphere, the climate change problem is
characterized by the issue of inter-generational equity: The current generation is imposing
external costs on future generations and would have to forego some economic growth to limit
those costs. How the costs of action of the current generation varies relative to the benefits, in
terms of reduced damages, to future generations depends heavily on the discount rate used. The
discount rate in turn depends on the social rate of time preference, risk aversion and, per capita
economic growth. Discounting at normal discount rates does not put too much value on what
happens 100 or 200 years from now; however, a very low discount rate, such as that used in the
Stern Review (Stern, 2007), places much more weight on future damages. Discounting plays a
significant role as to whether it is optimal, from an inter-generational perspective, to undertake
strong emission reduction action immediately or to start reducing emissions more slowly and to
follow an increasingly stringent climate policy (a ramp up climate policy). 3
In addition to the inter-generational equity, climate change is also characterized by issues
of intra-generational equity. For example, rich nations which are relatively GHG intensive are
located in temperate climates and have the funds and strong institutions to more easily adapt to
climate change; whereas, poorer nations, say in sub-Saharan Africa, are expected to be hit
relatively harder by the negative impacts of higher temperatures.4
Complicating the problem is the fact that uncertainty and risk are significant. Damages
from climate change could be potentially large and irreversible (Weitzman, 2009, 2011).
Furthermore, the continuous disposal of carbon into the atmosphere, oceans, and land could thus
result in the tragedy of the commons (Broome, 2012). Finally, reducing GHG emissions can be
characterized as a public good in that the benefits of mitigation are non-rival and non-
3 This issue will be explored in more detail in section 4. 4 For a critical review of inter-generational and intra-generational climate justice see Forsyth (2013).
4
exclusionary resulting in a free-rider problem and the under-provision of mitigation policy. 5 This
free-rider problem can provide insights into the failure of the Kyoto Protocol and subsequent
annual meetings. It is no wonder that Sir Nicholas Stern considers this issue the biggest market
failure the world has ever seen.
One of the most common approaches to evaluate the impact of climate change is to use
an Integrated Assessment Model (IAM). These models integrate a model of the world economy
with a representation of the global climate system. The models assess different scenarios from
these complex systems and are used by governments when evaluating the impact of climate polices
(e.g., estimating the Social Cost of Carbon) and informing the general public (Schwanitz,
(2013)).6
In spite of these significant issues and all the research being undertaken to study the
economics of climate change, not much has been formally done to introduce IAMs to
undergraduate students. Tol (2014) is a notable exception, as a text on climate economics
suitable for a full course in climate economics with a specific focus on the IAM at the masters’
or advanced undergraduate levels. Yet there is little available to introduce undergraduate
students to IAMs for a portion of a climate economics course or for courses in macroeconomic
growth theory, development economics or environmental economics. The existing IAMs are
overly complex for teaching the economics of climate change to undergraduate students. For
example, the Dynamic Integrated Climate Economics (DICE) model is based on the Ramsey
growth model that many economics students do not encounter until graduate school.7 Our
5 Recently, Nordhaus (2015) has proposed the formation of climate clubs to solve the free rider problem. 6 Because of the large amount of uncertainty with respect to climate change and climate damages Pindyck (2013, 2015) concludes that IAMs are not very useful for guiding policy; 7 The closest economic models to the one we have constructed are Nordhaus’ DICE model, Brock and Taylor (2010), and Taylor (2014). None of these three closely related works are aimed at educating undergraduate students
about the economics of climate change.
5
approach adjusts the simple Solow growth model that undergraduate economics students are
familiar with.8 Furthermore, the existing IAMs include a complex representation of the climate
system that takes a significant amount of time to explain to undergraduate students. Our model
replaces the complex climate system with a simple linear relationship between atmospheric
carbon accumulation and expected temperature change demonstrated by Matthews et al (2012).
This paper is aimed at making the simple IAM model available to instructors and undergraduate
students in order to explore the economics of climate change. The model is available in two
possible formats both accompanying this article: an MS Excel workbook or an R code version.
Throughout the paper figures and key points are provided for instructors to highlight to students
and to use as starting points to motivate in-class discussion.
The step-by-step instructions to replicate the simple IAM (included in an accompanying
appendix) and the exercises provided throughout the paper allow students to learn the economic
issues surrounding climate change in a hands-on way. Learning-by-doing, rather than watching
only the instructor’s lectures, is a more effective way to absorb and understand the material
(Findley (2014), Dalton et. al., (2015)). After being exposed to the topic by the instructor,
students learn more when they can use the simple IAM to make and graph the projections and to
explain the results. Visually seeing the pattern students created themselves is a powerful teaching
tool (Watts and Becker, (2008)). Psychological studies show that visuals improve learning
outcomes and learning-by-doing increases knowledge retention and also becomes a more
enjoyable experience to students (Vazquez and Chiang, (2014)). This paper (and accompanying
appendix) guides instructors and students to create visuals of future trajectories of the standard of
living of the world economy with and without climate change under different scenarios.
8 In case students are not exposed to the Solow model, more time can be spent explaining the basics of the Solow
model and the concept of steady state levels.
6
Section 2 describes the basic climate-Solow model for the world economy. Section 3
alters the model to examine damages which are more severe. This section provides direction for
instructors to use the model to illustrate the impact of climate change when damages are more
severe at higher temperatures, and when temperature increases affect the depreciation of capital
and productivity growth. Section 4 uses the model developed in section 2 to illustrate the costs
and benefits of emission reductions by conducting a simple Benefit-Cost Analysis for the 2
degrees target. Finally, concluding remarks and other possible classroom extensions are
mentioned. The appendix provides step-by-step instructions for students to create and run the
base case version of the simple IAM outlined in the paper following an approach similar to
Tebaldi and Elmslie (2010). Students can construct, on their own, the income per capita
trajectories with and without damages, the Environmental Kuznets curve, and the time paths of
other variables over 200 years. Exercises are provided throughout the main text.
2. The Simple Climate-Solow Model
2.1 Economic Growth & Climate Impacts
The economic growth component of the model is a variation on the standard Solow Growth
model. In the standard undergraduate treatment of the Solow model, output is produced by the
combination of capital, Kt labor, Lt and technology, At according to the Cobb-Douglas production
function 𝑌𝑡 = 𝐴𝑡 𝐾𝑡 𝛼 𝐿𝑡
1−𝛼 , which can be rearranged in terms of output per worker as
𝑦𝑡 = 𝐴𝑡 𝑘𝑡 𝛼 .
This is the standard Solow Growth model that students should be already familiar with.
For the purposes of studying climate change, the effect of increased temperatures is added to the
model in a similar way as by Nordhaus (2008) and Fankhauser and Tol (2005). This is a standard
7
assumption in most IAMS. The production function in the model is slightly altered to be the
following
𝑦𝑡 = 𝐷𝑡 𝐴𝑡 𝑘𝑡 𝛼,
where 𝐷𝑡 = 1/(1 + 𝜃1𝑇𝑡 𝜃2 ) ≤ 1 is the damage function and Tt is the temperature anomaly in
year t. The production function looks the same as the standard Cobb-Douglas production
function, except output per worker is now reduced by increased temperatures, i.e., the higher is
Tt, the lower is yt ceteris paribus.
The savings rate, s is constant, leading to investment per worker in period t of 𝑠𝑦𝑡.
Capital depreciates at a constant rate, 𝛿𝐾. To reflect recent UN population projections that
predict global population will plateau around 10.5 billion, total population and the labor force
grow at a decreasing rate over time, 𝑔𝐿,𝑡 = 𝑔𝐿,0/(1 + 𝛿𝐿 ) 𝑡 determined by the parameter 𝛿𝐿 > 0
which reduces the degree of population growth over time. The term gL,0 is the population growth
rate in the base year of 2010. Total factor productivity, At also grows at a decreasing rate over
time: 𝑔𝐴,𝑡 = 𝑔𝐴,0/(1 + 𝛿𝐴) 𝑡.9 This leads to the following difference equation to describe the
transitional dynamics in the model:
𝑘𝑡+1 − 𝑘𝑡 = 𝑠𝑦𝑡 − (𝛿𝐾 + 𝑔𝐿,𝑡 ) 𝑘𝑡.
Given this equation it is easy to show convergence to a balanced growth stable steady
state capital labour ratio 𝑘𝑠𝑠,𝑡 = [ 𝑠𝐴𝑡𝐷𝑡
𝛿𝐾+𝑔𝑛,𝑡 ]
1/(1−𝛼)
for a given time period t.10 Due to population
9 The assumption of a declining growth rates of total factor productivity and population growth as shown above are
also used in Nordhaus (2013). Most undergraduate students will be familiar with the Solow model with constant
rates of population and technology growth; therefore, the diminishing growth rates used here may appear more
complicated at first glance to the students. However, this change has little effect on how an instructor would
traditionally introduce the dynamics of the Solow model. 10 Simulations can also conducted using transitional dynamics but this is a possible extension. The differences between the two paths is not significant and this path will converge to the same unique steady state values when
technology is constant and population growth is constant.
8
growth declining and technology advancing, the balanced growth steady state capital labor ratio
will increase over time (offset by damages). Along the balanced growth path, output per worker,
𝑦𝑠𝑠,𝑡 = 𝐷𝑡 𝐴𝑡 𝑘𝑠𝑠,𝑡 𝛼 grows at a rate dependent on changes in temperature (outlined in subsection
2.3), the growth rate of total factor productivity, gA,t (which grows at a declining rate) and the
growth rate of the capital labour ratio which is weighted by the income share of capital, α. It can
be easily seen that in the absence of climate damages (i.e., 𝐷𝑡 = 1), yt grows at a faster rate.
To identify the impact of Business-As-Usual (BAU) in the model, a simple comparison
of 𝐷𝑡 = 1 for all t (i.e., no climate damages) to 𝐷𝑡 < 1 (i.e., with climate damages) is required.
This comparison is shown in Figure 1 for the parameter values displayed in the appendix. The
figure is very useful to highlight to students the central trade-off involved in the climate change
problem. There are two important aspects of this figure to highlight. First, that the model,
consistent with other IAMs, predicts that future generations are better off despite climate
damages. Second, that the climate change problem is intergenerational in nature; the damages of
climate change, as represented by the wedge between the two lines, are imposed mainly on
future generations.11 Combined, these two aspects highlight that the climate change problem can
be encapsulated by the following trade-off: A relatively poorer current generation is imposing
damages (costs) on relatively richer future generations. This is of course only true in the base
case of the model, and altering either the damage function or where damages enter the model can
lead to future generations being made worse off; which is a useful exercise for instructors to do
for their class using our provided Excel workbook or R code.
11 Damages by 2100 are 5.5 percent (as a % of the income per person without climate change) and increase to 17 percent by 2200. These damages are within the range found in the literature (See Tol (2015)).
9
Figure 1: The Solow Model with and without Climate Impacts
Source: Authors’ calculations.
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Question 1: The base model predicts that future generations will be worse off because of
climate change but that they will still be richer than the current generation. What are the
implications for the climate policy decisions being made by politicians in the current generation?
10
2.2 Carbon Emissions
Carbon emissions, Et are generated in the model by the production process based on a variable,
t that specifies how emissions intensive (i.e., how dirty or clean) the production technology is at
time t. The emissions intensity variable defines how much emissions are released per unit of
output. Carbon emissions in year t are calculated by multiplying the emissions intensity in year t
by the output in year t
𝐸𝑡 = 𝜎𝑡 𝑌𝑡 .
where Et is tonnes of carbon released and Yt is total output. For modelling purposes, emissions
intensity is computed by assuming a level in the base year and then specifying the growth rate of
emissions intensity over time into the future. Figure 2 shows that global emissions intensity has
steadily declined between 1950 and 2010. This decline has occurred for many reasons. Sectors
that have been growing most rapidly, like information technology or health care, are generally
less energy intensive than the sectors that are growing more slowly or stagnating. Also, the
advance of technology improves the efficiency of production, so that it now takes less energy to
produce the same product. There has also been a general shift in the composition of the sources
of energy away from coal and towards natural gas, nuclear, hydroelectricity, and others. Future
declines in emissions intensity take the following relationship
𝑔𝜎,𝑡 = 𝑔𝜎,𝑡−1/(1 + 𝛿𝜎 ),
where 𝑔𝜎,𝑡 < 0 is the growth rate of emissions intensity between periods t and t-1 and 𝛿𝜎 < 0.
The value of emissions intensity in year t can then be calculated as12
𝜎𝑡 = 𝜎𝑡−1(1 + 𝑔𝜎,𝑡 ).
12 Similar assumptions about emissions intensity were made by Nordhaus (2013). For details see http://www.econ.yale.edu/~nordhaus/homepage/documents/DICE_Manual_103113r2.pdf
11
This formula can also be expressed in terms of the base year
𝜎𝑡 = 𝜎0 ∏[1 + 𝑔𝑎,0/(1 + 𝛿𝜎 ) 𝑖 ]
𝑛=𝑡
𝑖=1
.
This information is provided for the benefit of instructors and can be given to especially
interested students; however, the important thing to highlight to students is that emissions
intensity of output is assumed to decline at an increasing rate into the future (consistent with past
history)
Figure 2. Global Emissions Intensity, 1950-2010
Source: CDIAC, 2015; Maddison Project, 2013; authors’ calculations.
The carbon emissions predicted by the model follow an inverse-u shape consistent with the
Environmental Kuznets’ Curve hypothesis and are displayed in Figure 3A and 3B. As income
per capita increases emissions initially increase, peak in the later part of this century when
income per capita reaches approximately twenty eight thousand dollars and then emissions start
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declining. Along a steady state, emissions initially grow because output grows faster than the
rate at which intensity falls but after a certain period the latter becomes stronger than the former
causing emissions to fall. This can be seen as follows (See also Taylor and Brock for a similar
expression, 2010):
𝑔𝐸,𝑡 = 𝑔𝜎,𝑡 + 𝑔𝑌,𝑡 .
This relationship is important as it indicates to students how difficult it is to reduce emissions in
an economy that is growing along a steady state due to population growth, total factor
productivity growth and capital per worker growth.13
This relationship can also be connected to the IPAT equation when expressed in growth
rates. The IPAT equation is used by the IPCC for setting future emission targets. It links
environmental impact (I) to population (𝑃𝑡 ), affluence ( 𝑌𝑡
𝑃𝑡 ) and technology (
𝐸𝑡
𝑌𝑡 ). In our experience,
students find the IPAT equation easy to understand even though it is an identity.14 The IPAT
equation for carbon emissions is usually expressed as follows:
𝐸𝑡 ≡ 𝑃𝑡 𝑌𝑡 𝑃𝑡
𝐸𝑡 𝑌𝑡
.
Carbon dioxide emissions at time t (i.e., 𝐸𝑡) are proportional to population multiplied by
affluence as measured by output per capita at time t and technology as measured by carbon
emissions per dollar of output (recall that in the model 𝐸𝑡 /𝑌𝑡 = 𝜎𝑡). In growth rates, after
cancelling out the growth of population, this identity becomes:
𝑔𝐸,𝑡 ≡ 𝑔𝜎,𝑡 + 𝑔𝑌,𝑡
which is identical to the growth rate of emissions from the model. The difference now is that the
Solow model provides a theory that explains why output grows. Emissions grow because
13 This is offset partially by the growth rate of the damage that occurs with increasing temperature. 14 Students can download yearly data from Gapminder.org to explore this relationship for individual countries.
13
affluence grows along a steady state that in the Solow model is due to population growth, growth
in total factor productivity and growth of capital per worker offset by the impact on growth from
damages growing over time. The emissions growth rate is also affected by the emissions
intensity falling over time (i.e., 𝑔𝜎,𝑡 < 0). Hence the IPAT equation in growth rates arises from
the long run properties of the Solow model and can explain why the model produces an inverse
u-shaped emissions path over time (as displayed in Figure 3A). At first, −𝑔𝜎,𝑡 < 𝑔𝑌,𝑡 but over
time the growth rate of output slows down (due to the assumed diminishing TFP and population
growth) and eventually −𝑔𝜎,𝑡 > 𝑔𝑌,𝑡 producing negative emissions growth (i.e., 𝑔𝐸,𝑡 < 0).
Figure 3A. Predicted Global Carbon Emissions, 2010-2200
Source: Authors’ calculations.
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2000 2050 2100 2150 2200
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rb o
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m is
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(b il
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Figure 3B. Environmental Kuznet’s Curve
𝐸𝑡 ≡ 𝑃𝑡 𝑌𝑡 𝑃𝑡
𝐸𝑡 𝑌𝑡
to find what it takes in terms of technology to reduce emissions in 2050 by 50% below 2010 levels with
an assumed population growth of 1.5 percent and growth of affluence as measured by income per person
by 2.5% per year.
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rb o
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Income per person with damages (000s of constant $s)
Question 2: Use the IPAT equation
15
2.3 Carbon Accumulation & Temperature Change
One of the aspects that make this model so useful for teaching is the simplicity of how the
climate system is modelled.15 The simple proportional stable linear relationship between carbon
accumulation and global temperature change found by Matthews et al. (2012) is used in the
model. They found that temperature increases by approximately 1.8 Celsius per 1000 billion
tonnes of carbon (i.e., 1000 PgC) emitted with a 95 percent confidence band between 1 and 2.5
degrees Celsius. This relationship is found to be independent of both time and the level of
stabilization of atmospheric carbon concentration (i.e., the emissions scenario). Using this
scientifically based relationship avoids modelling much of the complexity of the climate system
done by other IAM models.16 The following relationship shows the cumulative emissions from
pre-industrial levels to 2010. The cumulative emissions from the pre-industrial levels to 2010
(the base year for the simulations) are labelled as C0, i.e., these are the sum of past emission
releases. The global temperature change relationship to carbon accumulation into the future is:
𝑇𝑡 = 𝛽 [𝐶0 + ∑ 𝐸𝑖
𝑡
𝑖=1
],
where t ≥ 1. The first term, 𝛽 𝐶0, is the impact on global temperature change relative to pre-
industrial levels due to the accumulated carbon emissions that were released prior to 2010 (i.e.,
there are 530 billion tonnes already accumulated). The second term, 𝛽 ∑ 𝐸𝑖 𝑡 𝑖=1 , is the impact on
global temperature at any time t in the future due to the additional emissions accumulated since
15 It is important to give students a basic understanding of the science of climate change before exposing them to the
modelling of temperature anomaly. Basics understanding of climate change can be found at the U.S. EPA
http://www.epa.gov/climatechange/basics/ or showing students the IPCC AR5 short video on the physical science
basis at https://www.youtube.com/watch?v=6yiTZm0y1YA. For students that want to go beyond the basics on the
science of climate change, Professor Archer’s video lectures are recommended:
http://forecast.uchicago.edu/lectures.html. 16 This complexity arises because there is uncertainty associated with the path of carbon emissions towards affecting
the atmospheric concentration level, through carbon sensitivity. Also there is uncertainty as to the impact of the
concentration level of carbon to temperature anomaly change via the climate sensitivity parameter.
16
2010. Because of a growing economy, as shown in the previous section, emissions will continue
to accumulate resulting in a higher temperature change.
Note that the above relationship is independent of the emissions pathway selected. What
matters in terms of temperature change anomaly is the cumulative carbon emissions and the
targeted budget. For example, to keep global temperature anomaly below 2 degrees Celsius
relative to pre-industrial levels then cumulative emissions should not increase more than
approximately 1110 billion tonnes (i.e., the budget). If they increase by 470 billion tonnes over
the next 50 years which is within the current BAU pathway (See Figure 3A and 3B) they will
reach 1000 billion tonnes. This will result in a temperature increase of 1.8 degrees Celsius
relative to pre-industrial level given that Matthew et al. found 𝛽 to be 0.0018 per 1 billion tonnes
of cumulative carbon emitted. Figure 4 shows the path of cumulative carbon emissions starting
from 530 billion tonnes. Figure 4 also shows the corresponding temperature increases as well as
the 2degC target. With business as usual, 2 degrees Celsius will be reached just before 2050 and
surpass 2000 billion tonnes by 2100 leading to a temperature increase of 4 degrees Celsius which
is considered dangerous climate change.17
17 There is an estimated 6000 PgC that can be accumulated given the fossil fuels available. Recently, the relationship
has been found to be stable within 5000 PgC (Tokarska et al. 2016).
17
Figure 4. Predicted Cumulative Carbon Emissions and temperature anomaly
Source: authors’ calculations.
to keep the accumulation of carbon below 1000 billion tones by 2100. Note that each box in Figure 3A is
250 billion tonnes of carbon and that 530 billion tonnes since 2010 have already been accumulated. Can
emissions increase in the short run? Does stabilizing emissions reduce the concentration? What are the
implications of the alternative paths?
0
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2010 2060 2110 2160
Te m
p e
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Year
Cummulative Carbon Emisiosns two degrees Temperature Anomaly
Question 3: Find different paths, using Figure 3A, in order
18
3 Additions to the base model for discussion
Damages in the base model enter multiplicatively in the production function as in Nordhaus
(2013). It is assumed that climate change causes losses to production in the same period only via
the damage function. Temperature increases are assumed not to affect the depreciation of
physical capital nor any other form of capital such as environmental, social and organizational
capital. In addition, climate change is assumed not to impact the factors of production
individually nor the growth rate of total factor productivity. Also, the damage function used in
the base model has been calibrated for losses when temperature increases to 2.5-3 degrees
Celsius but it does not apply for higher temperature changes which are a real possibility under
BAU (Stern, 2013). Furthermore, catastrophic damages are not incorporated into the base model
(See Pyndick (2013), Weitzman (2013)). Below some of these additions are incorporated into the
base model. This enriches the simple model in terms of illustrating impacts to students. 18
First consider the depreciation rate of physical capital. It is easy to conceive that
increased temperatures and more severe weather will lead to capital having a shorter life span. It
was mentioned as a possibility by Fankhauser and Tol (2005) and by Stern (2013). Recently, it
has been incorporated into the DICE model by Dietz and Stern (2014) as well as Moore and Diaz
(2015). Climate change can affect the durability and the longevity of stock of capital, for
example, increased temperatures cause increased frequency of storms, more extreme weather,
rising sea levels, and many other impacts. Such events can cause permanent damage to capital
infrastructure. 19 Capital will require more maintenance to keep it from further wear and tear due
18 Stern (2013) suggests four alterations to the basic model to make it more relevant. First, damages to social,
organizational and environmental capital. Secondly damages to the stock of capital and land. Third, damages to
overall factor productivity. Finally, damages to learning and endogenous growth. 19 Stern states: “Climate events such as storms or inundation can do permanent or long term damages to capital and
land. If it is necessary to abandon certain areas, capital, infrastructure and land have zero use value and are
19
to temperature rising. Capital could even be stranded if people move far away from the ocean
shores due to sea level rising. Extremely powerful storms could destroy capital which then needs
replacement. With temperature increasing, a larger fraction of investment spending will be
allocated towards depreciation (and to adaptation measures) than to new investment which is the
engine of economic growth. This increased spending on necessary investment, to keep the capital
labour ratio constant, reduces the steady state capital per person and hence the steady state
income per capita. A simple way to introduce the impact of temperature on the depreciation rate
is as follows:20
𝛿𝐾 = 𝛿0 + 𝛿1 𝑇𝑡 .
For the simulations we suggest that the depreciation rate increase by 1 percent per 1degC
temperature increase (i.e., 𝛿1=0.01). Currently, on average, capital is replaced after 10 years
assuming the depreciation rate is at the base rate of 0.1. If temperature increases to 2degC
(5degC) then capital will need to be replaced on average every 8.3 years (6.7 years) as it wears
out faster.
Second, temperature increases could affect the growth rate of total factor productivity
(Moyer et al, (2013), Dietz and Stern (2014), Moore and Diaz, (2015)). The growth rate of total
factor productivity could be impacted negatively because resources will be diverted away from
R&D and instead used for climate adaptation and for the reconstruction of capital due to climate
damages. Furthermore, output per hour of input (labour or capital) could decline if inputs need
more hours to produce the same output level due to a different climate environment. Also, a
warmer climate will increase the likelihood of human conflict (Hsiang et al. (2013)). This in turn
essentially lost. This could be incorporated via a permanent damage or a reduction in capital occurring in period t as
a result of temperature and events in that period.” pg. 849. 20 Stern (2013) suggested an equation along these lines (See page 850).
20
negatively impacts the institutions that protect property rights causing a possible reduction in the
growth rate of total factor productivity. There is also evidence that economic growth is lower
with higher temperatures (Dell, et. al. (2012)). Total factor productivity is modelled along the
following lines:21
𝑔𝐴,𝑡 = 𝑔𝐴,0
(1 + 𝛿𝐴) 𝑡
− 𝛾𝑇𝑡 .
For the simulations we set 𝛾 = 0.001. It reduces the growth rate of total factor
productivity by 0.001 for every 1 degree Celsius increase. Although this might seem like a small
effect, it accumulates over time into a significant impact as it directly affects current production
and the future growth rate of output per person. Note that total factor productivity is assumed to
decline over time even without climate change but remains positive.22 But with temperature
rising beyond 3 degrees Celsius, the impact from the warmer temperature can offset the
exogenous growth rate of total factor productivity.
Figure 5 shows the path of steady state output under base case damages (slight gray path),
base case and depreciation impact (darker gray path), base case and impact on the growth rate of
total factor productivity (darkest gray path), and the black path under all damages. It is clear that
adding only the depreciation effect does not cause a large reduction in steady state income per
person. Income per person is rising throughout but is always at a lower level relative to the base
case. Steady state income per person in the year 2200 increases to $42,000 rather than to
$50,000. Adding the impact of climate change on the growth rate of total factor productivity
amplifies the damages relative to the base case. In this case, steady state income increases
initially, reaches a maximum after 2100, and then starts falling. Given that the growth rate of
21 This formulation is similar to Dell et al. (2013). 22 Dietz and Stern argued that this is due to depreciation of productivity (i.e., displacement of skills and know how)
being stronger than institutional innovations that promote growth in productivity.
21
total factor productivity is assumed to decline even without climate change eventually there will
be a temperature level after which the growth rate of total factor productivity will be negative
and this will cause the reduction in steady state output per person. As seen from the path, the
losses to output per person are significant if temperature increases affect the growth rate of total
factor productivity. Taken all together the damages approach 60 percent of income by 2200
relative to no climate change and about 27 percent lower by 2100.23 Still income per person is
higher, at approximately $21,670, than the current income level of approximately $10,000. Note
that the slower growth in output per person and the decline in income per person from 2100
helps in slowing down emissions and hence temperature increases. The temperature increases to
5.5 degrees Celsius by 2200.
23 Burke et al. (2015) estimate a 23 percent decrease in average global incomes by 2100 relative to world income without climate change.
22
Figure 5: The Solow Model with climate impacts on depreciation and TFP growth
Source: authors’ calculations.
Another extension that can be incorporated into the simple model is associated with the
type of damage function. The standard damage function in the DICE model represented in the
base model has parameter values that have been calibrated for temperature increases not
exceeding 3 degrees Celsius relative to pre-industrial levels.24 Higher temperature levels than 3
Celsius are highly likely with BAU as seen from Figure 4. Figure 1 shows very low damages at
temperatures which reach 6-7 degrees Celsius by 2200 with BAU. But such changes in the
24 More recently Nordhaus (2008, 2013) has altered the damage function to be 𝐷𝑡 = 1/(1 + 𝜋1𝑇𝑡 + 𝜋2𝑇𝑡
2) but calibrated the parameters to yield similar results to the damage function used in the base model which was the
Nordhaus, (1994) damage function.
0
10
20
30
40
50
60
70
1960 2010 2060 2110 2160 2210
S te
a d
y S
ta te
I n
co m
e p
e r
p e
rs o
n (
0 0
0 s
o f
2 0
0 5
U .S
. $
s)
Year
base case damages no damages
base case and depreciation impact base case and tfp impact
with TFP and depreciation damages
23
climate have not been observed for millions of years and could have profound impacts on the
planet. Weitzman (2012) states that “six degrees of extra warming is about the upper limit of
what the human mind can envision for how the planet might change” (pg. 226). He speculates on
further temperature increases
A temperature change of 12C therefore represents an extreme threat to human
civilization and global ecology as we know it, even if, conceivably, it might not
necessarily mean the end of Homo sapiens as a species. (pg. 232)
He envisions an increase of 18 degrees Celsius as the “death temperature.” Weitzman calibrates
𝜃1 to be 0.00238 so that it conforms to the Nordhaus (2008) DICE model. Nordhaus’ model
results in an eight percent loss from a temperature rise to 6 Celsius relative to pre-industrial
levels and only a 26 percent loss with temperature increasing to 12 degrees. According to
Weitzman this type of temperature increase (12 Celsius) was observed during the Eocene epoch,
55-34 million years ago with an ice free planet and alligators living near the North Pole. While
for low temperature increases the damages are not high, and the Nordhaus parameters are fine,
he argues that temperature increases of 6 Celsius can result in at least a 50 percent loss of output
and that a 12 Celsius increase will result in a 99 percent loss of output. He recommends to
capture this with the damage function along the following lines:
𝐷𝑡 = 1/(1 + 𝜃1 𝑇𝑡 𝜃2 + 𝜃3𝑇𝑡
𝜃4 )
where the parameters assigned to 𝜃1and 𝜃2 are as the Nordhaus damage function but 𝜃3=0.507E-
05 and 𝜃4= 6.754 as given in Table 1 of the Appendix. A simple way to show students the
fundamental difference between the Nordhaus and Weitzman damage functions is to use a
traditional, static marginal damage graph with marginal damages on the vertical axis and
emissions on the horizontal axis. Environmental economics students will be readily familiar with
24
this type of graph. In this space, the Nordhaus marginal damage curve is linear and upwards
sloping, whereas the Weitzman marginal damage curve is convex.
Figure 6 illustrates the different cases in the model. The path without climate change is
the green trajectory, followed by the slight gray color path with Nordhaus type of damages,
followed by Weitzman’s path (darker grey path) which closely follows the Nordhaus path until
temperature increases reach 3 Celsius and then the last term in the damage function takes on a
more important role and the two income paths start diverging. This can be considered a tipping
point. When temperature reaches 4degC steady state income per person starts declining reaching
$21,000 by 2200. Note that the current generation is poorer than the generation living in 2200
but the 2200 generation is poorer than a generation living around the year 2100.25 Adding in
possible damage impacts on the depreciation rate and the growth rate of TFP, 𝛿1=0.01 and 𝛾 =
0.001, the steady state income per person follows the black path. Steady state income per capita
increases initially reaches a maximum of approximately $25,000 (i.e., a 30 percent damage
relative to no climate change) and then drops. This drop is due to the new damage function and
the impact climate change has on the growth rate of total factor productivity. Steady state income
per person declines and reaches $15,000 by 2200 (i.e., a 74 percent damage).
25 With different parameter values it could be possible that the current generation is richer than one living in 2200.
25
Figure 6: The Solow Model with expanded climate impacts and new damage function
Source: authors’ calculations.
Notes: The lines referred to as no damages and base case damage function are as displayed in
Figure 5.
0
10
20
30
40
50
60
70
2000 2020 2040 2060 2080 2100 2120 2140 2160 2180 2200
S te
a d
y S
ta te
I n
co m
e p
e r
P e
rs o
n (
0 0
0 s
o f
2 0
0 5
U .S
. $
s)
Year
no damages base case damages with Weitzman's damages with all damages
26
4 Emissions Mitigation, Benefit-Cost Analysis & the 2 Degree Target
At COP16 in Cancun in 2010, attending nations agreed to a long-term goal of limiting the
increase in average global temperatures to below 2 degrees Celsius (UNFCCC, 2011). In a
recent meeting, the leaders of the G7 countries reiterated their commitment to this target (Carrel
and Martin, 2015). To achieve this goal in the model, global cumulative carbon emissions should
be limited to 1.111 trillion tons. Given that 530 billion tons have already been emitted by 2010,
this leaves a carbon budget from 2010 on of 581 billion tons. If this budget is exceeded,
temperature will have more than a 50% chance of exceeding 2 degrees.
Question 4: What would the consequences be if damages negatively affected the
growth rate of population?
Question 5: Using the instructions from the Appendix, reproduce the results in table 2
and conduct a sensitivity analysis by changing the parameters as well as Figure 1.
Question 6: What would the economic consequences be if there was a one-time impact arising
from climate change such as a sudden destruction of the capital stock and/or population due to
a tipping point occurring after 4 degrees Celsius? Would the steady state path be relevant to
follow or would the transitional dynamic path be more appropriate in this case?
Question 7: Using the transitional dynamic path in lieu of the balanced growth steady state
path reproduce the income per person path when all damages are present. Can the
future generation be worse off than the current generation?
27
The simple model can be used to teach students about the costs and the benefits of the
proposed target relative to business-as-usual as well as the importance of the choice of the
discount rate for evaluating climate policy. The model is very well suited to incorporating the 2
degree target and its carbon budget because temperature change is based directly on cumulative
emissions. Capping cumulative emissions is therefore relatively straightforward.
We choose an emissions reduction path for which government emissions regulation (the
control rate, Mt) increases at a constant growth rate, m. Assume that in 2010, the emissions
control rate is 9% and grows annually at a 4.267% growth rate (i.e., 𝑀𝑡 = 𝑀𝑡−1(1 + 𝑚) ). The
choice of emission control path here is relatively arbitrary and certainly not universally
optimal.26 The following equation describes how emissions control enters the model:
𝐸𝑡 = (1 − 𝑀𝑡 )𝜎𝑡 𝑌𝑡
Given the assumed parameter values (𝑀0 = 0.09 and 𝑚 = 0.04267), the annual emissions path
is displayed in Figure 7. Emissions continue to increase but peak around 2035 and decline to
zero by 2068. The emissions peak to reach the two degree target is almost half of the peak in the
BAU scenario displayed in Figure 3A.
\
26 The path is optimal (maximizes net present value) among all control paths that follow exponential growth assuming a 5% discount rate. Changing the discount rate affects which path is optimal.
28
Figure 7. Predicted emissions path to achieve 2 Degree Limit
Source: authors’ calculations.
Reducing emissions to zero and limiting temperature increase to 2 degrees reduces
temperature damages on future generations; however, reducing emissions is not costless. To
reflect the immediate cost of reducing emissions, use the convex abatement cost function of
Nordhaus and Sztorc (2013),
𝐴𝐶𝑡 = Ω𝑡 𝑀𝑡 2,
where Ω0 = 0.06 is an abatement cost coefficient that declines over time at the rate at which
TFP grows (i.e., t declines at the rate –gA,t). Total income per capita net of abatement cost in
year t is
𝑦𝑡 = (1 − 𝐴𝐶𝑡 )𝐷𝑡 𝐴𝑡 𝑘𝑡 .
The reduction in per capita income of reducing emissions is difficult to show by just
plotting the income path of BAU and the income path when the 2 degree limit is imposed.
0
2
4
6
8
10
12
14
2010 2050 2090
C a
rb o
n e
m is
si o
n s
(b il
li o
n s
o f
to n
n e
s)
Year
29
Instead Figure 8A shows the annual net benefits in present value of the 2 degree limit versus
BAU over time (2010-2200) assuming a 5% discount rate. The present value of total net benefits
with a 5% discount rate is -$449 per capita. However, the net present value depends critically on
the choice of discount rate. The annual net benefits of the 2 degree mitigation policy are
displayed in Figure 8B, but in present values using the 1.4% discount rate selected by the Stern
Review. In this case, the future annual net benefits do not limit to zero by 2200 and the present
value of total net benefits is positive ($40,665 per capita). The Internal Rate of Return for
limiting to 2 degrees is 4.08%.
Figure 8A. Annual Net Benefits in Present Value using 5% Discount Rate
Source: authors’ calculations.
What is interesting to highlight to students between the two figures, is that the discount
rate determines if the 2 degree mitigation target is a potential Pareto improvement, but not a pure
-35
-30
-25
-20
-15
-10
-5
0
5
10
2010 2050 2090 2130 2170
a n
n u
a l n
e t
b e
n e
fi ts
( 2
0 0
0 5
$
) in
p re
se n
t va
lu e
(r
= 5
% )
Years
30
Pareto improvement. For both discount rate values, the annual net benefits still show the inter-
generational trade-off in which the costs of emissions reductions are imposed on early
generations and the benefits of lower temperature increases accrue to later generations. Unlike
with intra-generational policy evaluation, in this case there is no potential mechanism for a future
generation to compensate earlier generations to make them at least as well off. Mitigation policy
is not asking the current generation to incur costs to benefit their children; it is asking them to
incur costs to benefit their great- great-grandchildren.
Figure 8B. Annual Net Benefits in Present Value using 1.4% Discount Rate
Source: authors’ calculations.
-300
-200
-100
0
100
200
300
400
500
600
2010 2050 2090 2130 2170
a n
n u
a l n
e t
b e
n e
fi ts
( 2
0 0
5
$ )
in p
re se
n t
va lu
e (
r= 1
.4 %
)
Years
31
5 Extensions and Concluding Remarks
This paper presents an extension of the Solow model that includes a simple climate model. To
the best of our knowledge, we have developed the simplest Integrated Assessment Model. The
simplicity of the model serves well for introducing the economics of climate change, which
relies heavily on IAMs, to undergraduate students. Even though the model is very basic its
predictions match well with Nordhaus’ more complex DICE model.
In addition to introducing the model to students and highlighting the central intra-
generational trade-off inherent in the climate change problem, two other classroom applications
are outlined. The model can be used to teach the academic controversy over how damages from
increased temperatures should enter the model and the implications of changing the standard
assumptions on damages. The model is also very useful for teaching students about how
economists approach evaluating the 2 degree Celsius target and the importance of the discount
rate.
Although only two specific applications of the model have been highlighted, many other
learning applications are possible both in and out of the classroom. For example, students can be
Question 8: What factors determine the social discount rate? Does it make sense to have
the social discount rate decline over time?
Question 9: Read the article by Arrow (2007) and using an Excel spreadsheet verify
Arrow’s cost benefit analysis on page 4-5. Discuss the implications of this finding.
Question 10: Increase carbon emissions per person by 1 tonne in 2020 only and compare the
net present value of income per person with the additional tonne and without. Repeat the
exercise by adding an additional tonne of carbon in 2050 only. Discuss your findings.
32
presented with the basic model and then asked to do their own simulations and to write up an
essay on the economics of climate change. Students will observe that climate change will affect
the standard of living of the world economy in the future and that climate change caused by
humans is a global externality and as a result requires coordinated, corrective action by
governments. They will also observe the intra-generational trade-offs involved, i.e., economic
activity today inflicts a cost on future generation and that future generations will be richer than
today’s generation due to increases in productivity and the stabilization of global population.
However, they will also observe that the richer future generations will not be as rich as they
would have been without climate change. Since cost of action is absorbed by the current
generation and the benefits of action accrue to future generations students can conduct a cost-
benefit analysis and explore the importance of the discount rate. But future generations could be
poorer than the current generation if temperature anomaly exceeds 3 Celsius and under a
different damage function; different model parameters; when climate change affects the
depreciation of capital and especially when temperature affects total factor productivity. In this
case, the benefits of action relative to the costs in present value terms increase and stronger
action is needed.
There are many extensions students can explore with the model beyond what is
mentioned above (i.e., different damage impacts and a cost-benefit analysis). One example
would be for students to conduct a sensitivity analysis by changing the parameters of the model
and observing the impact on the standard of living. Secondly, in this paper the balanced growth
path is used; however, simulations can be conducted according to the transitional growth path
given by 𝑘𝑡+1 = 𝑘𝑡 + 𝑠𝑦𝑡 − (𝛿𝐾 + 𝑔𝑛,𝑡 ) 𝑘𝑡). Namely, the capital stock (per person) next period
is equal to the capital stock per capita this period plus the difference between actual investment
33
per person and the necessary investment that is needed to maintain the capital labor ratio
constant. The new stock of capital is then channeled into the actual path of output per person
through the production function 𝑦𝑡 = 𝐷𝑡 𝐴𝑡 𝑘𝑡 𝛼 . Using this approach students and instructors can
explore the impact of sudden destructions of the capital stock or population due to tipping points.
In the balanced growth model, the impact of sudden shocks are only for one period as the system
returns to the steady state in the next period, but with the introduction of transitional dynamics
the impact could last for over a decade until the system reaches a new steady state. This can be
either done in-class led by the instructor or be given as an assignment so that students can
compare and contrast the different paths of the economy.
Another option is to use the model to compute the social cost of carbon. We have found
that students easily grasp the general idea of the social cost of carbon, but have difficulty
understanding how the various estimates are calculated. Although the true social cost of carbon
may include non-market values not best reflected in a damage function that is multiplicative on
production; the model is still useful to give students a hands-on demonstration of how the social
cost of carbon is actually calculated using integrated assessment models. This is done easily by
comparing the change in the net present value of steady state income per person under the base
case carbon emissions path relative to a path when 1 additional tonne of carbon per person is
added in a particular year. Students will see that the social cost of carbon increases if the tonne of
carbon is released further into the future. Students will also immediately see why the choice of
discount rate matters so crucially for the resulting social cost of carbon value.
A further application is to ask students to find parameters that are relevant to a particular
nation and to examine the impact climate change will have on that particular nation. They can
also examine the impact on economies that are emerging and growing faster relative to the
34
industrial nations. For more advanced courses they could also change the economic model to an
endogenous growth model and examine the impact of climate change.
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38
APPENDIX:
INSTRUCTIONS FOR STUDENTS: EXCEL SPREADSHEET
Below are set of instructions for you (the student) to setup the Excel spreadsheet of the simple
Climate-Solow IAM. Setting up the model will help you examine different future trajectories of
the economic system under the base model (as outlined in section 2). Once set up, you can then
use the model to explore other more complicated scenarios as assigned by your instructor.
To set up the model, you will use Excel with two worksheets initially. One sheet will contain the
parameter values of the model. The other sheet will be used to compute the steady state values,
such as income per person, over time. We assume you have a basic understanding of Excel.
STEP 1: The Parameters Worksheet
The first step is to setup the first worksheet called “parameters” similar to table 1 below. You
should enter the information as reported in table 1. The worksheet will have:
1. Column A contains descriptions of the parameters
2. Column B contains the symbols of the parameters. (Note: This is not necessary and can be skipped.)
3. Column C contains the assigned values.
This is very easy to setup but very important as all other sheets used will reference this
worksheet named “parameters”.
39
Table 1: Variables, symbols and values used for base case. “Parameters” sheet
Column A B C
R o
w
1 Description Symbol Value
2 Capital’s share of income α 0.3
3 Savings rate s 0.25
4 Depreciation rate 𝛿0 0.1
5 Impact of temperature on depreciation rate 𝛿1 0.01
6
7 Initial 2010 population (in billions) 𝐿0 6.838
8 Initial 2010 population growth rate 𝑔𝐿,0 0.023
9 Parameter affecting population growth 𝛿𝐿 .052
10
11 Initial 2010 total factor productivity 𝐴0 3.955
12 Initial productivity growth rate 𝑔𝐴,0 0.015
13 Parameter affecting productivity growth 𝛿𝐴 0.011
14 Temperature impact on productivity growth 0.001
15
16 Initial world GDP (trillions of 2005 US $) 𝑌0 63.69
17 Initial world capital (trillion of 2005 US $s) 𝐾0 135
18
19 Initial emission intensity 𝜎0 0.549
20 Initial 2010 growth of emissions intensity 𝑔𝜎,0 -0.01
21 Parameter affecting emissions intensity growth 𝛿𝜎 -0.0002
23 Damage parameter 𝜃1 0.002384
24 Damage parameter 𝜃2 2
25 Damage parameter 𝜃3 0.00000507
26 Damage parameter 𝜃4 6.754
27
28 CCR per trillion tonnes 𝛽 0.0018
29 Initial Carbon (billions of tonnes) 530
STEP 2: The Model Worksheet
This step requires the creation of a new worksheet named “model 1”. In this sheet you will
compute the values of the various variables of the model over 200 years into the future (See table
2). The first row will contain the variable symbols or names. In order to create this sheet you will
need to create the data for the next two rows. The second row will contain the initial values for
each variable and the third row contains the formulas that calculate the value of the variable for
the next year. Once these two rows are completed, students can copy and paste the third row
until the year 2200 (row 192).
40
1. Column A: This column lists the years from 2010 to 2200. These cells correspond to the subscript t in the model’s mathematical equations. Enter 2010 in cell A2 and then in cell
A3 enter the formula =A2+1.
2. Column B: To compute the growth rate of population over time, gL,t, the first value in cell B2 is the initial population growth rate of 2.3 percent in the sheet titled “parameters”
under cell $C$8. The formula to enter in cell B2 is: =parameters!$C$8. The value in cell
B3 is calculated by using the population growth formula, 𝑔𝐿,𝑡 = 𝑔𝐿,𝑡−1/(1 + 𝛿𝐿 ). The formula to enter in cell B3 is: =B2/(1+parameters!$C$9).
3. Column C: To compute the world population level over the years, starting from the initial value of 6.838 billion obtained from parameters!$C$7 and entered in cell C2 with
the formula =parameters!$C$7. In cell C3 the new population level in 2011 is computed
by multiplying the value in cell C2 by the growth rate of that same year in column B3.
The formula to enter is: =C2*(1+B3).
4. Column D: To compute the growth rate of carbon intensity, the first value of the growth rate of carbon intensity is taken from table 1 as -0.01 in cell parameters!$C$20. The
formula to enter in cell D2 is =parameters!$C$20. In cell D3 the value of this variable is
generated using the formula 𝑔𝜎,𝑡 = 𝑔𝜎,𝑡−1
1+𝛿𝜎 where 𝛿𝜎 is from table 1 cell parameters!$C$21,
while 𝑔𝜎,𝑡−1 is the previous value of the growth rate of carbon intensity. For cell D3 the formula is =D2/(1+parameters!$C$21).
5. Column E: To compute the carbon intensity, the formula to enter in E2 is =
parameters!$C$19. The value in E3 is found using: 𝜎𝑡 = 𝜎𝑡−1(1 + 𝑔𝜎,𝑡 ) where 𝑔𝜎,𝑡 is from column D and 𝜎𝑡−1 is the previous carbon intensity value. In cell E3 the formula is =E2*(1+D3).
6. Column F: This column shows world output/income per capita lagged one year (𝑦𝑡−1) and is used to calculate emissions per capita. The formula to enter in cell F2 is:
=parameters!$C$16/parameters!$C$7. This value should be aligned as closely as possible
with the steady state value in 2010. Cell F3 is computed as =P2. There will be no value in
F3 as column P has not been constructed yet but will appear once there is a value in P2.
7. Column G: Carbon dioxide emissions per person is computed by multiplying carbon intensity in column E with output per person in column F. The formula to enter in cell G2
is =E2*F2. For cell G3 enter =E3*F2. Note that column F does not have values yet, but
will be computed soon.
8. Column H: To compute total annual carbon emissions, multiply carbon dioxide emissions per person (column G) by the total population (column C) and divide by 3.67
(the conversion factor between carbon and carbon dioxide). The formula to enter in cell
H2 is =G2*C2/3.67. For cell H3 the formula is =G3*C3/3.67.
41
9. Column I: Cumulative carbon emissions start in 2010 as 530 billion tonnes of carbon found in $C$29 of the parameters sheet. This is entered in cell I2 as =parameters!$C$29.
Cell I3 requires the formula =I2+H2.
10. Column J: To compute the temperature anomaly (𝑇𝑡 = 𝛽[𝐶0 + ∑ 𝐸𝑖 ] 𝑡 𝑖=1 ), multiply the
cumulative carbon emissions in column I by the cell parameters!$C$28 (this is the
selected value for ). The formula to enter in cell J2 is =I2*parameters!$C$28. For cell J3
the formula is =I3*parameters!$C$28.
11. Column K: This column computes the growth rate of total factor productivity, 𝑔𝐴,𝑡 = 𝑔𝐴,0
(1+𝛿𝐴) 𝑡. The first value in cell K2 is computed using the formula =parameters!$C$12 (this
is the initial value gA,t). The other value in cell K3 is computed according to the formula: =$K$2/((1+parameters!$C$13)^(A3-$A$2)).
12. Column L: To calculate total factor productivity, At, the initial value in cell L2 is taken from $C$11 in the parameters sheet. The next value grows at the rate calculated in
column K, the formula in cell L3 is =L2*(1+K3).
13. Column M: This column computes the depreciation rate, K. For all cells in this column the formula is the same: =parameters!$C$4.
Column N: This column computes the damage function for the base model for each year
(𝐷𝑡 = 1/(1 + 𝜃1𝑇𝑡 𝜃2 )). It is obtained using the values for the damage coefficients in the
parameters sheet and the corresponding temperature anomaly (column J). In cell N2 the
following formula is entered: =1/(1+parameters!$C$23*(J3^parameters!$C$24)). Copy
and paste this formula in cell N3.
14. Column O: The steady state level of capital per person for a particular year is then
computed as 𝑘𝑠𝑠,𝑡 = [ 𝑠𝐴𝑡𝐷𝑡
𝛿𝐾+𝑔𝑛,𝑡 ]
1/(1−𝛼)
. The formula is entered in cell O2 as follows:
= ((parameters!$C$3*L2*N2)/(M2+B2))^(1/(1- parameters!$C$2)). Copy and paste this
formula in cell O3.
Column P: The steady state output/income per person for a particular year is computed
using the formula 𝑦𝑠𝑠,𝑡 = 𝐷𝑡 𝐴𝑡 𝑘𝑠𝑠,𝑡 𝛼 . The formula is entered in cell P2 as
=N2*L2*O2^parameters!$C$2. Copy and paste this formula in cell P3.
15. FINAL STEP: Copy and paste all row 3 cells to row 4 until row 192. Check that all values in your worksheet correspond with table 2. Congrats, you have created an
Integrated Assessment Model!
42
Exercise: Graph the path of income per person with damages and without
In order to graph the path of output per person with and without climate damages (i.e., reproduce
Figure 1), and be available for other simulations, create a new worksheet labelled “graph data”. It
will utilize three columns. The first row in the worksheet will indicate the name of each variable:
Years, Income per person with damages, and Income per person without damages.
1. Column A: This will be the column indicating Years, it is the same as column A of the “model 1” sheet. In cell A2 enter =model 1!A2. Copy and paste the formula from cells
A3 to A192.
2. Column B: This column is labeled Income per person with damages. You have already calculated the values for this column in column P of the model 1 worksheet. In cell B2
enter the following: =model 1!P2. Copy and paste this formula in cells B3 to B192.
3. For income per person without damages you will need to enter two new columns in the model 1 worksheet:
Column R: Compute capital per person without damages as 𝑘𝑠𝑠,𝑡 = [ 𝑠𝐴𝑡
𝛿𝐾+𝑔𝑛,𝑡 ]
1/(1−𝛼)
.
This formula entered into cell R2 is as follows:
=((parameters!$C$3*L2)/(M2+B2))^(1/(1- parameters!$C$2)).
Copy and paste this formula in cells R3 to R192.
Column S: Compute income per person without damages using the formula 𝑦𝑠𝑠,𝑡 = 𝐴𝑡 𝑘𝑠𝑠,𝑡
𝛼 . This formula entered in cell S2 is =K2*P2^parameters!$C$2. Copy and paste
this formula in cells S3 to S192.
Now return to the “graph data” worksheet to fill in column C.
Column C: This column displays income per person without climate damages. In cell C2
enter the formula =model 1!S2. Copy and paste the formula in cells C3 to C192.
4. Finally use Excel to graph the two variables over time.
Note: Other figures can be similarly created. Careful when including damages for depreciation
and total factor productivity as these will affect the no climate scenario.
43
Table 2: The path of the variables from 2010 to 2200
R
o w
Column
A B C D E F G H I J K L N O P Q
1 Year gL,t Lt gs,t σt yt-1 C02/Lt Et
Tt gA,t At δk Dt kt yt
2 2010 0.023 6.838 -0.010 0.549 9.314 5.113 9.527 530.000 0.954 0.015 3.955 0.100 0.998 19.578 9.632 3 2011 0.022 6.988 -0.010 0.544 9.632 5.235 9.968 539.527 0.971 0.015 4.014 0.100 0.998 20.259 9.875 4 2012 0.021 7.133 -0.010 0.538 9.875 5.314 10.327 549.495 0.989 0.015 4.073 0.100 0.998 20.948 10.120 5 2013 0.020 7.274 -0.010 0.533 10.120 5.391 10.685 559.822 1.008 0.015 4.132 0.100 0.998 21.643 10.368 6 2014 0.019 7.410 -0.010 0.527 10.368 5.467 11.039 570.507 1.027 0.014 4.191 0.100 0.997 22.345 10.617 7 2015 0.018 7.542 -0.010 0.522 10.617 5.543 11.391 581.546 1.047 0.014 4.251 0.100 0.997 23.054 10.868 8 2016 0.017 7.670 -0.010 0.517 10.868 5.617 11.740 592.938 1.067 0.014 4.310 0.100 0.997 23.768 11.120 9 2017 0.016 7.794 -0.010 0.512 11.120 5.690 12.084 604.677 1.088 0.014 4.370 0.100 0.997 24.488 11.375
10 2018 0.015 7.914 -0.010 0.507 11.375 5.762 12.425 616.762 1.110 0.014 4.430 0.100 0.997 25.213 11.632 11 2019 0.015 8.029 -0.010 0.501 11.632 5.833 12.761 629.186 1.133 0.014 4.490 0.100 0.997 25.943 11.890 12 2020 0.014 8.140 -0.010 0.496 11.890 5.903 13.092 641.947 1.156 0.013 4.551 0.100 0.997 26.677 12.149
… … … … … … … … … … … … … … … …
2197 0.000 10.615 -0.010 0.081 48.931 3.957 11.444 3835.483 6.904 0.002 12.902 0.100 0.898 122.558 49.024 2198 0.000 10.616 -0.010 0.080 49.024 3.923 11.347 3846.928 6.924 0.002 12.926 0.100 0.897 122.787 49.115 2199 0.000 10.616 -0.010 0.079 49.115 3.889 11.250 3858.275 6.945 0.002 12.951 0.100 0.897 123.013 49.206
192 2200 0.000 10.616 -0.010 0.078 49.206 3.856 11.154 3869.525 6.965 0.002 12.975 0.100 0.896 123.237 49.296
𝐶0 + ∑ 𝐸𝑖
𝑡
𝑖=1
