Integrated Assessment Models:
Background (I) and Uncertainty (II)
William D. Nordhaus
Yale University
Summer School in Resource and Environmental Economics
Venice International University
June 30 – July 6, 2013
1
Lectures
I. Introduction to Integrated Assessment Models
II. Applications to Uncertainty
2
Integrated Assessment (IA) Models of Climate
Change
• What are IA model?
– These are models that include the full range of cause and
effect in climate change (“end to end” modeling).
– They are necessarily interdisciplinary and involve natural
and social sciences
• Major goals:
– Project the impact of current trends and of policies on
important variables
– Assess the costs and benefits of alternative policies
– Assess uncertainties and priorities for scientific and
project/engineering research
Major Components of Models
Behavioral and
Scientific
Equations
Identities
Value Judgments
(markets, policies,
ethics, etc.)
Person or
nation 1
Pareto Improvement from
Climate Policy
Bargaining
region (Pareto
improving)
Inefficient
initial (nopolicy)
position
Person or nation 25
Elements of IA Models.
To be complete, the model needs to incorporate the
following elements:
- human activities generating emissions
- carbon cycle
- climate system
- biological and physical impacts
- socioeconomic impacts
- policy levers to affect emissions or other parts of cycle.
The Need for IAMs
•
Many areas of the natural and social sciences involve
complicated systems that link together multiple sectors. This is
particularly true for environmental problems, which are intrinsically
ones having strong roots in the natural sciences and require social
and policy sciences to solve in an effective and efficient manner.
• A good example, which will be the subject of this survey, is climate
change science and policy, which involve at a minimum a wide
variety of geophysical sciences such as atmospheric chemistry and
climate sciences, ecology, economics, political science, game theory,
and international law.
•
As understanding progresses in the different areas, it is
increasingly necessary to link together the different areas to develop
effective understanding and efficient policies. In this role, integrated
assessment analysis and models play a key role. Integrated
assessment models (IAMs) can be defined as approaches that
integrate knowledge from two or more domains into a single
framework.
8
There are many kinds of IA models, useful for different
purposes
Policy evaluation models
- Models that emphasize projecting the impacts of different
assumptions and policies on the major systems;
- often extend to non-economic variables
Policy optimization models
- Models that emphasize optimizing a few key control
variables (such as taxes or control rates) with an eye to
balancing costs and benefits or maximizing efficiency;
- often limited to monetized variables
Some important objectives of policy
• When efficiency. Timing of emissions reductions that
minimizes discounted costs.
• Where efficiency. Locate emissions reductions in regions
where reductions are cheapest
• Why efficiency. Policy is set to some ultimate objective
(economic or environmental)
• How efficiency. Point to most effective policy instruments
for reaching target.
• Bargaining efficiency: What are Pareto-improving
allocations of emissions reductions across players?
10
Economic Theory Behind Modeling
1. Basic theorem of “markets as maximization” (Samuelson, Negishi)
Non-linear constrained optimization
Fixed point
Outcome of efficient
competitive market
(however complex
but finite time)
=
Maximization of weighted
utility function:
n
W   i [U i (c ki ,s ,t )]
i 1
for utility functions U; individuals i=1,...,n;
locations k, uncertain states of world s,
i
time periods t; welfare weights  .
2. This allows us (in principle) to calculate the outcome of a market
system by a constrained non-linear maximization.
11
How do we solve IA models?
The structure of the models is the following:
max W 
{  ( t )}
T max
 U[c(t ),L(t )]R(t )
t 1
subject to
c(t )  H[ (t ),s(t ); initial conditions, parameters]
(The H[...] functions are production functions, climate model,
carbon cycle, abatement costs, damages, and so forth.)
We solve using various mathematical optimization techniques.
1. GAMS solver (proprietary). This takes the problem and solves it
using LP-type solvers through successive steps. It is extremely
reliable.
2. Use EXCEL solver. This is available with standard EXCEL and
uses various numerical techniques. It is not 100% reliable for
difficult or complex problems and often doesn’t (next lecture).
3. MATLAB and the like. Useful if you know it.
4. Genetic algorithms. Some like these.
12
Example using Excel-Solver
The most transparent method of solving is using Excel.
For any optimization, can use Solver (free for small
problems) and Risk Solver Platform ($1000 for complicated
problems.
I will show some screenshots of a simple Excel example
using DICE-like model.
13
Example: Minimize cost of emissions
to attain a total emissions constraint
Period
0
10
20
30
Discount rate (per year)
0.10
Output
100.00
148.02
219.11
324.34
Emissions control rate
Constant
0.50
0.50
0.50
0.50
Efficient
0.50
0.50
0.50
0.50
Emissions
Uncontrolled
10.00
14.80
21.91
32.43
Controlled
Constant rate
5.00
7.40
10.96
16.22
Efficient
5.00
7.40
10.96
16.22
Total emissions over time: The target is to achieve 50 percent reductions
Uncontrolled
79.15
Controlled
Constant rate
39.57
Efficient
39.57
Abatement costs (=.1*miu^3*Q)
Constant rate
1.25
1.85
2.74
4.05
Efficient
1.25
1.85
2.74
4.05
Net output
Constant rate
98.75
146.17
216.37
320.29
Efficient
98.75
146.17
216.37
320.29
PV output
Level
205.6241
Efficient
205.6241
THESE ARE CONTROL VARIABLES
THIS WILL BE THE ENVIRONMENTAL
CONSTRAINT
THIS WILL BE THE OBJECTIVE FUNCTION
TO BE MAXIMIZED
14
Setup
Period
0
10
20
30
Discount rate (per year)
0.10
Output
100.00
148.02
219.11
324.34
Emissions control rate
Constant
0.50
0.50
0.50
0.50
Efficient
0.50
0.50
0.50
0.50
Emissions
Uncontrolled
10.00
14.80
21.91
32.43
Controlled
Constant rate
5.00
7.40
10.96
16.22
Efficient
5.00
7.40
10.96
16.22
Total emissions over time: The target is to achieve 50 percent reductions
Uncontrolled
79.15
Controlled
Constant rate
39.57
Efficient
39.57
Abatement costs (=.1*miu^3*Q)
Constant rate
1.25
1.85
2.74
4.05
Efficient
1.25
1.85
2.74
4.05
Net output
Constant rate
98.75
146.17
216.37
320.29
Efficient
98.75
146.17
216.37
320.29
PV output
Level
205.6241
Efficient
205.6241
THESE ARE CONTROL VARIABLES
THIS WILL BE THE ENVIRONMENTAL
CONSTRAINT
Start with an initial feasible
solution, which is equal
reductions in all periods.
THIS WILL BE THE OBJECTIVE FUNCTION
TO BE MAXIMIZED
15
Number crunch….
Period
0
10
20
30
Discount rate (per year)
0.10
Output
100.00
148.02
219.11
324.34
Emissions control rate
Constant
0.50
0.50
0.50
0.50
Efficient
0.50
0.50
0.50
0.50
Emissions
Uncontrolled
10.00
14.80
21.91
32.43
Controlled
Constant rate
5.00
7.40
10.96
16.22
Efficient
5.00
7.40
10.96
16.22
Total emissions over time: The target is to achieve 50 percent reductions
Uncontrolled
79.15
Controlled
Constant rate
39.57
Efficient
39.57
Abatement costs (=.1*miu^3*Q)
Constant rate
1.25
1.85
2.74
4.05
Efficient
1.25
1.85
2.74
4.05
Net output
Constant rate
98.75
146.17
216.37
320.29
Efficient
98.75
146.17
216.37
320.29
PV output
Level
205.6241
Efficient
205.6241
Then THESE
maximize
PV output
ARE CONTROL VARIABLES
Subject to the constraint that:
THIS WILL BE THE ENVIRONMENTAL
CONSTRAINT
the sum of emissions
< target sum of emissions
THIS WILL BE THE OBJECTIVE FUNCTION
TO BE MAXIMIZED
16
This is the solver dialogue box
Objective function
Control variables
Constraints
17
If you have set it up right
and have a good optimization program, then voilà !!!
Period
0
10
20
30
Discount rate (per year)
0.10
Output
100.00
148.02
219.11
324.34
Emissions control rate
Constant
0.50
0.50
0.50
0.50
Efficient
0.17
0.28
0.45
0.73
Emissions
Uncontrolled
10.00
14.80
21.91
32.43
Controlled
Constant rate
5.00
7.40
10.96
16.22
Efficient
8.25
10.63
11.97
8.73
Total emissions over time: The target is to achieve 50 percent reductions
Uncontrolled
79.15
Controlled
Constant rate
39.57
Efficient
39.57
Abatement costs (=.1*miu^3*Q)
Constant rate
1.25
1.85
2.74
4.05
Efficient
0.05
0.33
2.05
12.67
Net output
Constant rate
98.75
146.17
216.37
320.29
Efficient
99.95
147.69
217.06
311.67
PV output
Level
205.6241
Efficient
207.0152
THESE ARE CONTROL VARIABLES
Note that the emissions
controls are generally
“backloaded” because of the
positive discounting
THIS WILL BE of
THE capital)
ENVIRONMENTAL
(productivity
and
CONSTRAINT
because damages are in future.
THIS WILL BE THE OBJECTIVE FUNCTION
TO BE MAXIMIZED
18
Can also calculate the “shadow prices,”
here the efficient carbon taxes
600
Marginal cost of Emissions Reductions ($)
Remember that in a constrained
optimization (Lagrangean), the
multipliers have the
interpretation of
∂[Objective Function]/∂X.
So, in this problem, interpretation
is MC of emissions reduction.
Price = MC in efficiency.
Optimization programs
(particularly those based on
LP) will generate the shadow
prices of carbon emissions in
the optimal path.
For example, in the problem we
just did, we have the following
shadow prices:
500
400
300
200
100
0
0
10
20
30
Period
19
Example of DICE model
20
Basic economic strategy
1. Begin with a Solow-style economic growth model
2. Add the geophysical equations: note these impose an
externality
3. Then add an objective function to be optimized subject
to constraints:
4. So we have:
-
1 + 3 = optimal growth model (Ramsey model)
1 + 2 + 3 = integrated assessment model
5. Then estimate or calibrate the various components.
6. Then do various simulations and policy runs.
21
Macrogeoeconomics
Basic economics behind IA models. Have standard optimal
growth model + geophysics externalities:

(1)
max W   L(t )U[c(t )]e   t dt
{ c( t )}
0
subject to economic and climate constraints:
(2) c(t )  f [K(t ), (t ),T(t ); parameters, exogenous variables ]
(3) T(t )  h[E{Q(t ), (t ); parameters, exogenous variables ]
Then optimize W over emissions and capital stock,
subject to economic and geophysical constraints.
22
Objective Function
(1) W 
T max
 U[c(t ),L(t )]R(t )
t 1
Economics
(2) U [c(t),L(t)] = L(t)[c(t)1- / (1 -  )]
Utility function
(3) Q g (t) = A(t) K(t) L(t)1
Gross output
(4) D(t) =  1TAT (t) + 2TAT (t)
2
2
(5) C(t) = Q (t) 1(t)(t)
(6) Qn (t) =  (t)[1 - (t)]A(t) K(t) L(t)1
g
Damage function
Abatement cost
Net output
(7) EInd (t) =  (t)[1 - (t)]Q (t)
g
Geosciences
(8) M AT (t )  E(t )  11 M AT (t - 1 )  ...
(9) F(t)  {log2 [ MAT (t) / MAT (0)]}  FEX (t)
Industrial emissions
Atmospheric CO2
(10) TAT (t )  TAT (t  1 )  1 { F(t )  ...
Radiative forcings
Global mean temperature
Key variables (in addition to standard from growth theory):
Eind = industrial CO2 emissions
F = radiative forcings
MAT = atmospheric concentrations CO2
Qg = output gross of damages and abatement
Qn = output net of damages and abatement
TAT = global mean surface temperature
C = abatement (mitigation) cost
μ = emissions control rate (fraction of uncontrolled)
σ = emissions/output ratio
D = damages as fraction of output
R = utility discount factor = (1+ρ)-t
23
Slightly Simplified Equations of DICE-type Model
Objective Function
(1) W 
T max
 U[c(t ),L(t )]R(t )
t 1
Economics
(2) U [c(t),L(t)] = L(t)[c(t)1- / (1 -  )]
(3) Q g (t) = A(t) K(t) L(t)1
(4) D(t) =  1TAT (t) + 2TAT (t)2
(5) C(t) = Q g (t) 1(t)(t)2
(6)
Utility function
Gross output
Damage function
Abatement cost
Qn (t) = [( 1  D(T )][1 - C(t)]A(t) K(t) L(t)1 
g
= [( 1  D(T )][1 - C(t)]Q (t)
(7) EInd (t) =  (t)[1 - (t)]Q g (t)
Geosciences
(8) M AT (t)  E(t)  11 M AT (t - 1 )  ...
(9) F(t)  {log2 [ MAT (t) / MAT (0)]}  FEX (t)
(10) TAT (t)  TAT (t  1 )  1 {F(t)  ...
Net output
Industrial emissions
Atmospheric CO2
Radiative forcings
Global mean temperature
Note: For complete listing, see Question of Balance, pp. 205-209.
24
The next few slides are available for students
but will not be reviewed in summer school.
They are necessary components of IAMs and
should be understood by modelers and users.
For today, they go beyond the time-scope of our
studies.
We will pick up the lecture at the slide
“Applications of IA Models”
25
Modeling Strategies (I): Emissions in DICE/RICE
Emissions trajectories:
Start with data base of major countries.
Major issue of whether to use PPP or MER (next slide)
Estimate productivity growth
Estimate CO2 emissions-output ratios
Project these by decade for next two centuries
Then aggregate up by twelve major regions (US, EU, …)
Constrain by global fossil fuel resources
This is probably the largest uncertainty over the long run:
σ(Q) ≈ .01 T, or + factor 2.5 in 100 yrs, +7 in 200 yrs
26
CO2-GDP: Three countries (PPP v. MER)
Sigma: MER (DC)
Sigma: PPP (DC)
US
3.5
3.5
Russia
US
3
Russia
China
3
China
CO2/GDP
2.5
2
1.5
2
1.5
1
1
0.5
0.5
20
02
19
96
19
90
19
84
19
78
19
72
19
66
2000
1995
1990
1985
1980
1975
1970
1965
19
60
0
0
1960
CO2/GDP
2.5
27
Modeling Strategies (II): Climate Models
Climate model
Idea here to use “reduced form” or simplified models.
For example, large models have very fine resolution and
require supercomputers for solution.*
DICE uses two-layers (atmosphere, deep oceans) and 5-year
time steps.
Calibrated to ensemble of models in IPCC FAR science
reports with updates.
*http://www.aip.org/history/exhibits/climate/GCM.htm
28
Actual and predicted global temperature history
.6
.4
.2
.0
-.2
-.4
-.6
1840
1880
1920
1960
2000
YEAR
T_DICE
T_Hadley
T_GISS
29
Projected DICE and IPCC: two scenarios
5
4
3
2
1
0
1920
1960
2000
2040
2080
2120
YE A R
T_A2_DICE
T_A2_IPCC
T_B1_DICE
T_B1_IPCC
30
Modeling Strategies (III): Impacts
• Central difficulty is evaluation of the impact of climate
change on society
• Two major areas:
–
market economy (agriculture, manufacturing, housing, …)
–
non-market sectors
•human (health, recreation, …)
•non-human (ecosystems, fish, trees, …)
31
Summary of Impacts Estimates
Early studies contained a major surprise:
Modest impacts for gradual climate change, market impacts, highincome economies, next 50-100 years:
- Impact about 0 (+ 2) percent of output.
- Further studies confirmed this general result.
BUT, outside of this narrow finding, potential for big problems:
-
many subtle thresholds
abrupt climate change (“inevitable surprises”)
ecological disruptions
stress to small, topical, developing countries
gradual coastal inundation of 1 – 10 meters over 1-5 centuries
OVERALL: “…global mean losses could be 1-5% Gross Domestic Product
(GDP) for 4 ºC of warming.” (IPCC, FAR, April 2007)
32
Estimated Damages from Tol survey, DICE model, and
IPCC Estimate
33
Modeling Strategies (IV): Abatement costs
IA models use different strategies:
–
Some use econometric analysis of costs of reductions
–
Some use engineering/mathematical programming
estimates
–
DICE model generally uses “reduced form” estimates of
marginal costs of reduction as function of emissions
reduction rate
34
Derivation of mitigation cost function
Start with a reduced-form cost function:
C = Qλμ
(1)
where C = mitigation cost, Q = GDP, μ = emissions control rate,
λ,  are parameters.
Take the derivative w.r.t. emissions and substitute σ = E0 /Q
(2)
dC/dE = MC emissions reductions
= Qλβμ-1[dμ/dE] = λβμ-1/σ
Taking logs:
(3)
ln(MC) = constant + time trend + ( β-1) ln(μ)
We can estimate this function from microeconomic/engineering
studies of the cost of abatement.
35
Example from McKinsey Study
36
Reduced form equation: C=.0657*miu^1.66*Q
60
50
40
30
20
10
0
0
5
10
15
20
25
30
35
37
Further discussion
However, there has been a great deal of controversy about
the McKinsey study. The idea of “negative cost” emissions
reduction raises major conceptual and policy issues.
For the DICE model, we have generally relied on more
micro and engineering studies.
The next set of slides shows estimates based on the IPCC
Fourth Assessment Report survey of mitigation costs.
The bottom line is that the exponent is much higher
(between 2.5 and 3).
38
Note that the MC is much more convex than
McKinsey: much more diminishing returns
Source: IPCC, AR4, Mitigation.
39
Alternative abatement cost functions: From IPCC
Parameterized as C/Q = aμ2.8 , with backstop price(2005) = $1100/tC
40
Lecture resumes here:
Applications of IA Models
How can we use IA models to evaluate alternative
approaches to climate-change policy?
I will illustrate analyzing the economic and climatic
implications of several prominent policies.
41
Economic evaluation
We want to examine the economic efficiency of each of the
scenarios.
Some techniques:
- PV of abatement, damages, and total
- PV as percent of PV of total consumption
- Consumption annuity equivalent:


t 0
c(t )e  t 


ˆ  t
ce
t 0
where c(t ) is the actual path and cˆ is the
consumption annuity equivalent.
42
Some results from DICE-2013: Scenarios
•
•
•
•
•
•
Baseline
Optimal
Temperature-limited
Low discounting according to Stern Review.
Low time preference with calibrated interest rates.
Copenhagen Accord.
IMPORTANT NOTE: The ability to do scenario
comparisons is the major advantage of IAMs over other
modeling.
43
Details on Scenarios
• Baseline: No climate-change policies are adopted. The baseline can be
interpreted as the status quo of inaction on climate policies.
• Optimal: Climate-change policies maximize economic welfare, with full
participation by all nations starting in 2015 and without climatic
constraints.
• Temperature-limited: The optimal policies are undertaken subject to a further
constraint that global temperature does not exceed 2 °C above the 1900
average.
• Low discounting according to Stern Review. The Stern Review advocated using
very low discount rates for climate-change policy. This was implemented
using a time discount rate of 0.1 percent per year and a consumption
elasticity of 1.
• Low time preference with calibrated interest rates. Because the Stern Review run
leads to real interest rates that are below the assumed level, we adjust the
parameters of the preference function to match the calibrated real interest
rates.
• Copenhagen Accord. In this scenario, high-income countries are assumed to
implement deep emissions reductions over the next four decades, with
developing countries following gradually.
44
45
46
47
1400
Atmospheric CO2 Concentration
Atmospheric concentrations CO2 (ppm)
1200
1000
800
600
400
200
0
2000
Base
Optimal
Lim2t
Stern
SternCalib
Copen
2020
2040
2060
2080
2100
2120
2140
2160
2180
2200
48
7
Atmospheric
Temperature
Global mean temperature (increase from 1900, °C )
Base
6
Optimal
Lim2t
Stern
5
SternCalib
Copen
4
3
2
1
0
2000
2020
2040
2060
2080
2100
2120
2140
2160
2180
2200
49
Policies
50
120%
Emissions control rate (% of baseline)
Emissions Control Rate
100%
80%
60%
Base
Optimal
40%
Lim2t
Stern
SternCalib
20%
Copen
0%
2000
2010
2020
2030
2040
2050
2060
2070
2080
2090
2100
51
300
Stern
Carbon Price
Price of carbon emissions ($ per ton of CO2)
Lim2t
250
Optimal
SternCalib
Copen
200
Base
150
100
50
0
2000
2010
2020
2030
2040
2050
2060
2070
2080
2090
2100
52
Overall economic impact
53
Three economic results from IAMs
1. Overall cost. An economically oriented climate-change
policy would be relatively inexpensive and have a
substantial impact on long-run climate change.
2. Measures of tightness. The best way to gauge the optimal
“tightness” of climate change policies is the optimal
carbon price or carbon tax.
3. Time path of optimal policies. Most optimal policies ramp
up over time from modest C prices today to very very
high ones in a few decades.
54
Three inconvenient truths from IAMs
4. An inconvenient scientific truth: Global warming is not a
hoax. It will (almost surely) become an increasingly
important and obvious feature of earth systems.
5. An inconvenient economic truth: To be efficient, firms and
consumers must face a market price of carbon emissions that
reflects the social costs.
6. An inconvenient political truth: To be effective, the price
must be universal and harmonized in every sector and
country.
55
Readings for lectures
Lecture 1 on Integrated Assessment
W. Nordhaus,
56
References
General Background: The Stern Review, 2005.
On RICE and DICE models: W.D. Nordhaus, A Question of Balance, Yale Press, 2008; W.D. Nordhaus,
“Copenhagen Accord,” PNAS, 2010; W.D. Nordhaus and P. Sztorc, DICE-2013: A Users Manual.
On IA models: W.D. Nordhaus, “Integrated Assessment Models…”Handbook of CGE Models, available
at cowles.econ.yale.edu/P/cd/d18a/d1839.pdf; P. Kelly and C. Kolstad, “Integrated Assessment
Models For Climate Change Control,” International Yearbook 1999-2000, available as pdf.
On disaggregation: H. Theil, Linear Aggregation of Economic Relations; Grunfeld and Griliches, “Is
Aggregation Necessarily Bad?” ReStat, 1960.
On uncertainty: David Kelly and Charles Kolstad, Bayesian learning, growth, and pollution. Journal
of Economic Dynamics and Control, 23(4):491-518, 1999; Pindyck, R. S. , Fat tails, thin tails, and
climate change policy. Review of Environmental Economics and Policy 5(2): 258-274, 2008; Weitzman,
M. L. On modeling and interpreting the economics of catastrophic climate change. Review of
Economics and Statistics 91(1): 1-19, 2009; Weitzman, M. L. Fat-tailed uncertainty in the economics of
catastrophic climate change. Review of Environmental Economics and Policy 5(2): 275-292, 2011.
57
References (cont)
On uncertainty and discouting: Christian Gollier, Pricing the Planet, Princeton U Press, 2012; M.
Weitzman, “On Modeling and Interpreting Catastrophic Events,” ReStat, Feb 2009; W. Nordhaus,
“Economics of Tail Events,” REEP, 2011.
On learning and decision theory: Basics: A. K. Dixit and R.S. Pindyck, Investments under Uncertainty,
1994, Princeton University Press;
On learning and decision theory: Good applications: A.S. Manne and R.G. Richels, Buying
Greenhouse Insurance: The Economic Costs of Carbon Dioxide Emission Limits. Cambridge, Mass.,
MIT Press, 1992; M. Webster, The Curious Role of ”Learning” in Climate Policy: Should We Wait for
More Data? The Energy Journal, 2002.
Data:
US output data from bea.gov.
CO2 emissions from CDIAC at http://cdiac.ornl.gov/trends/emis/overview_2008.html and IEA.
CO2 concentrations at Mauna Loa from http://www.esrl.noaa.gov/gmd/ccgg/trends/
Mann reconstruction from http://www.ncdc.noaa.gov /paleo/pubs/mann2008/mann2008.html.
Vostok ice core record from
http://www.nature.com/nature/journal/v399/n6735/abs/399429a0.html.
Global output data from Maddison and World Bank use PPP valuation.
Global temperature estimates aveaged from Hadley, GISS, and NCDC.
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