1 economics 331b spring 2009 integrated assessment models of economics of climate change
TRANSCRIPT
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Economics 331bSpring 2009
Integrated Assessment Modelsof Economics of Climate Change
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Slightly Simplified Equations of DICE-2007 Model: Revised
Objective Function
(1) 1
T max
t
W U[c(t),L(t)]R(t)
Economics (2) 1-U [c(t),L(t)] =L(t)[c(t) / (1- )] Utility function
(3) 1gQ (t) = A(t) K(t) L(t) Gross output
(4) 21 AT 2 ATD(t)= T (t)+ T (t) Damage function
(5) 2g1C(t) = Q (t) (t) (t) Abatement cost
(6) 11
1
n
g
Q (t) =[( D(T )][1-C(t)]A(t) K(t) L(t)
=[( D(T )][1-C(t)]Q (t)
Net output
(7) gIndE (t) = (t)[1- (t)]Q (t) Industrial emissions
Geosciences (8) 11 1AT ATM (t) E(t) M (t - ) ... Atmospheric CO2
(9) 2 AT AT EXF(t) {log [M (t) / M (0)]} F (t) Radiative forcings (10) 11 AT ATT (t) T (t ) {F(t) ... Global mean temperature
Note: For complete listing, see Question of Balance, pp. 205-209.
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Basic economics behind IA models. Have standard optimal
growth model + geophysics externalities:
(1)
subject to economic and climate co
t
{c(t)}max W L(t)U[c(t)]e dt
Macrogeoeconomics
nstraints:
(2) parameters, exogenous variables
(3) parameters, exogenous variables
Then optimize W over emissions and capital stock,
subject to economic
c(t) f [K(t), (t),T(t); ]
T(t) h[E{Q(t), (t); ]
and geophysical constraints.
How do we solve IA models numerically?
We take discrete version of model, simplified as follows.
We solve using various mathematical optimization techniques.
1. GAMS solver (proprietary). This takes the problem and solves it using linear programming (LP) 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.
3. MATHLAB. Useful if you know it.4. Genetic algorithms. Some like these.
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subject to
initial conditions, parameters]
(The functions are production functions, climate model,
carbon cycle, abatement costs, damages, and
T max
{ (t)}t
max W U[c(t),L(t)]R(t)
c(t) H[ (t),s(t);
H[...]
so forth.)
Example: Minimize cost of emissions to limit the sum of emissions over time
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Period 0 10 20 30Discount rate (per year) 0.10Output 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 THESE ARE CONTROL VARIABLESEmissions 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 16.22Total emissions over time: The target is to achieve 50 percent reductions Uncontrolled 79.15 Controlled Constant rate 39.57 THIS WILL BE THE ENVIRONMENTAL Efficient 39.57 CONSTRAINTAbatement costs (=.1*miu^3*Q) Constant rate 1.25 1.85 2.74 4.05 Efficient 1.25 1.85 2.74 4.05Net output Constant rate 98.75 146.17 216.37 320.29 Efficient 98.75 146.17 216.37 320.29PV output THIS WILL BE THE OBJECTIVE FUNCTION Level 205.6241 TO BE MAXIMIZED Efficient 205.6241
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Period 0 10 20 30Discount rate (per year) 0.10Output 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 THESE ARE CONTROL VARIABLESEmissions 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 16.22Total emissions over time: The target is to achieve 50 percent reductions Uncontrolled 79.15 Controlled Constant rate 39.57 THIS WILL BE THE ENVIRONMENTAL Efficient 39.57 CONSTRAINTAbatement costs (=.1*miu^3*Q) Constant rate 1.25 1.85 2.74 4.05 Efficient 1.25 1.85 2.74 4.05Net output Constant rate 98.75 146.17 216.37 320.29 Efficient 98.75 146.17 216.37 320.29PV output THIS WILL BE THE OBJECTIVE FUNCTION Level 205.6241 TO BE MAXIMIZED Efficient 205.6241
Start with an initial feasible solution, which is
equal reductions in all periods.
Setup
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Period 0 10 20 30Discount rate (per year) 0.10Output 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 THESE ARE CONTROL VARIABLESEmissions 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 16.22Total emissions over time: The target is to achieve 50 percent reductions Uncontrolled 79.15 Controlled Constant rate 39.57 THIS WILL BE THE ENVIRONMENTAL Efficient 39.57 CONSTRAINTAbatement costs (=.1*miu^3*Q) Constant rate 1.25 1.85 2.74 4.05 Efficient 1.25 1.85 2.74 4.05Net output Constant rate 98.75 146.17 216.37 320.29 Efficient 98.75 146.17 216.37 320.29PV output THIS WILL BE THE OBJECTIVE FUNCTION Level 205.6241 TO BE MAXIMIZED Efficient 205.6241
Then maximize PV output
Subject to the constraint that:
the sum of emissions < target sum of emissions
Number crunch….
This is the solver dialogue box
Objective function
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Control variables
Constraints
If you have set it up right and have a good optimization program, then
voilà !!!
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Period 0 10 20 30Discount rate (per year) 0.10Output 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 THESE ARE CONTROL VARIABLESEmissions 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 8.73Total emissions over time: The target is to achieve 50 percent reductions Uncontrolled 79.15 Controlled Constant rate 39.57 THIS WILL BE THE ENVIRONMENTAL Efficient 39.57 CONSTRAINTAbatement costs (=.1*miu^3*Q) Constant rate 1.25 1.85 2.74 4.05 Efficient 0.05 0.33 2.05 12.67Net output Constant rate 98.75 146.17 216.37 320.29 Efficient 99.95 147.69 217.06 311.67PV output THIS WILL BE THE OBJECTIVE FUNCTION Level 205.6241 TO BE MAXIMIZED Efficient 207.0152
Note that the emissions controls are generally “backloaded” because of the positive discounting (productivity of capital) and because damages are in future.
Can also calculate the “shadow prices,” here the efficient carbon taxes
Remember that in a constrained optimization (Lagrangean), the multipliers have the interpretation of d[Objective Function]/dX.
So, in this problem, interpretation is MC of emissions reduction.
Optimization programs (particularly 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:
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1000
1200
1400
1600
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0 10 20 30
Mar
gina
l co
st o
f Em
issi
ons
Redu
ction
s ($
)
Period
With a little work, you can show that the rate of growth of prices = interest rate for this case.
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Applications of IA Models
Major applications of IA Models:1. Project the impact of current trends and of policies
on important variables.2. Assess the costs and benefits of alternative
policies3. Determine efficient levels of policy variables
(carbon taxes, emissions control rates, emissions, …)
For these, I will illustrate using the DICE-2007 model:– Full analysis Question of Balance (see reading list).– There is a “beta” version using an Excel
spreadsheet at http://www.econ.yale.edu/~nordhaus/homepage/DICE2007.htm (both an *.xls and *.xlsx version)
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1. No controls ("baseline"). No emissions controls.2. Optimal policy. Emissions and carbon prices set for
economic optimum.3. Climatic constraints with CO2 concentration constraints.
Concentrations limited to 550 ppm4. Climatic constraints with temperature constraints.
Temperature limited to 2½ °C 5. Kyoto Protocol. Kyoto Protocol without the U.S. 6. Strengthened Kyoto Protocol. Roughly, the Obama/EU policy
proposals.7. Geoengineering. Implements a geoengineering option that
offsets radiative forcing at low cost.
Illustrative Policies for DICE-2007
Snapshot of DICE-Excel model
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WORLD0 1 2 3 4 5 6 7 8 9 10
2005 2015 2025 2035 2045 2055 2065 2075 2085 2095PARAMETERS AND EXOGENOUS VARIABLES
OUTPUT AND CAPITAL ACCUMULATIONcapital share 0.300damage coefficient on temperature 0.000damage coefficient on temperature squared 0.0028388Exponent on damages 2.0rate of depreciation (percent per year) 0.100Initial capital stock ($ trillion) 137.000Abatement cost function coefficient 0.056068 0.051082 0.046660 0.042728 0.039225 0.036095 0.033295 0.030782 0.028524 0.026489 pback 1.17000 backrat 2.00000 gback 0.05000 limmiu 1.00000Backstop price 1.170000 1.141469 1.114330 1.088514 1.063957 1.040598 1.018379 0.997243 0.977137 0.958012 Initial cost function coefficient 0.045 Initial rate of decline in cost of abatement function (percent per decade) 26.000 Rate of decline in decline rate of cost of abatement function (percent per year) 2.000
Rate of decline in cost of abatement function (percent per decade) 0.260 0.213 0.174 0.143 0.117 0.096 0.078 0.064 0.052 0.043Exponent of control cost function 2.800
EMISSIONSSigma (industrial CO2 emissions/output -- MTC/$1000) 0.1342 0.1253 0.1172 0.1099 0.1032 0.0971 0.0915 0.0864 0.0817 0.0774 Initial sigma 0.1342 Initial growth rate of sigma (percent per decade) -7.300
Rate of decrease in the growth rate of sigma (percent per year) 0.300 Accleration parameter of growth rate of sigma 0.000 Growth rate of sigma (percent per decade) -7.300 -0.071 -0.069 -0.067 -0.065 -0.063 -0.061 -0.059 -0.057 -0.056Carbon emissions from land use change (GTC per year) 1.100 0.990 0.891 0.802 0.72171 0.650 0.585 0.526 0.474 0.426 Initial carbon emissions from land use change (GTC per year) 1.100
CARBON LIMITSMaximum carbon use 6000.000
CONCENTRATIONSInitial atmospheric concentration of CO2 (GTC) 808.900Initial concentration of CO2 in biosphere/shallow oceans (GTC) 1255.000Initial concentration of CO2 in deep oceans (GTC) 18365.000
http://www.econ.yale.edu/~nordhaus/homepage/DICE2007.htm
Per capita GDP: history and projections
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1
10
100
1960 1980 2000 2020 2040 2060 2080 2100
Per c
apita
GD
P (2
000$
PPP
)
US WE OHI
Russia EE/FSU Japan
China India World
CO2-GDP ratios: history
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.0
.1
.2
.3
.4
.5
.6
.7
80 82 84 86 88 90 92 94 96 98 00 02 04
ChinaRussiaUS
WorldWestern/Central Europe
CO
2-G
DP
rat
io (
tons
per
con
stan
t P
PP
$)
IPCC AR4 Model Results: History and Projections
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DICE-2007model
2-sigma rangeDICE model
DICE-2007 model results
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Concentrations profiles: DICE 2007
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300
400
500
600
700
800
900
1000
1100
1200
1300C
arbo
n c
once
ntr
atio
ns
(ppm
)Optimal
Baseline
< 2 deg C
Strong Kyoto
Temperature profiles
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0
1
2
3
4
5
6
Tem
per
ature
cha
nge
(C)
Optimal Baseline
2x CO2 Strong Kyoto
2o C limit
Policy outcomes variables
Overall evaluationTwo major policy variables are
- emissions control rate- carbon tax
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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:
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0 0
ˆ( )
ˆwhere ( ) is the actual path and is the
consumption annuity equivalent.
t t
t t
c t e ce
c t c
Evaluation: PV trillions of 2000 $
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Run
Difference from base: Objective function
Present value
environ-mental
damages
Present value abate-ment costs
Net present value abate-ment costs
plus damages
Trillions of 2005 US $
No controls 0.0 22.5 0.0 22.5Optimal 3.4 17.3 2.2 19.5Concentration limits
Limit to 1.5X CO2 -14.9 9.9 27.2 37.1Limit to 2X CO2 2.9 16.0 3.9 19.9Limit to 2.5X CO2 3.4 17.3 2.2 19.5
Temperature limitsLimit to 1.5 degree C -14.7 10.0 27.0 37.0Limit to 2 degree C -1.6 13.1 11.3 24.3Limit to 2.5 degree C 2.3 15.3 5.2 20.6Limit to 3 degree C 3.2 16.7 2.9 19.5
Kyoto ProtocolKyoto with US 0.7 21.4 0.5 21.9Kyoto w/ o US 0.1 22.4 0.0 22.5Strengthened 1.0 16.0 5.8 21.8
Low discounting -17.0 9.0 27.7 36.7Low-cost backstop 17.2 4.9 0.4 5.4
Evaluation: as percent of PV consumption
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Run
Difference from base: Objective function
Present value
environ-mental
damages
Present value abate-ment costs
Net present value abate-ment costs
plus damages
As percent of discounted world consumptionNo controls - 1.13 - 1.13 Optimal 0.17 0.87 0.11 0.98 Concentration limits
Limit to 1.5X CO2 (0.75) 0.50 1.37 1.87 Limit to 2X CO2 0.14 0.80 0.20 1.00 Limit to 2.5X CO2 0.17 0.87 0.11 0.98
Temperature limitsLimit to 1.5 degree C (0.74) 0.50 1.36 1.86 Limit to 2 degree C (0.08) 0.66 0.57 1.22 Limit to 2.5 degree C 0.11 0.77 0.26 1.03 Limit to 3 degree C 0.16 0.84 0.14 0.98
Kyoto ProtocolKyoto with US 0.04 1.07 0.03 1.10 Kyoto w/ o US 0.01 1.13 0.00 1.13 Strengthened 0.05 0.80 0.29 1.10
Low discounting (0.85) 0.45 1.39 1.84 Low-cost backstop 0.86 0.25 0.02 0.27
Evaluation: Consumption annuity per capita
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Run
Difference from base: Objective function
Environ-mental
damages
Abatement costs
Abatement costs plus climate damages
Consumption annuity equivalent (billions of 2000$ per year)No controls - 121 - 121 Optimal 18 93 12 105 Concentration limits - - - -
Limit to 1.5X CO2 (80) 54 146 200 Limit to 2X CO2 15 86 21 107 Limit to 2.5X CO2 18 93 12 105
Temperature limits - - - - Limit to 1.5 degree C (79) 54 145 199 Limit to 2 degree C (9) 70 61 131 Limit to 2.5 degree C 12 82 28 111 Limit to 3 degree C 17 90 15 105
Kyoto Protocol - - - - Kyoto with US 4 115 3 118 Kyoto w/ o US 1 121 0 121 Strengthened 5 86 31 118
Low discounting (91) 49 149 198 Low-cost backstop 93 26 2 29
Emissions control rate (industrial CO2), 2015
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0.00
0.10
0.20
0.30
0.40
0.50
0.60
Carbon prices for major scenarios
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0
100
200
300
400
500
600
700
800
900
1000
2005 2015 2025 2035 2045 2055 2065 2075 2085 2095 2105
Car
bon
pric
e (2
005
US$
per
ton
C)
Optimal
Baseline
< 2 degrees C
Strong Kyoto
Carbon prices for major scenarios
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0
100
200
300
400
500
600
700
800
900
1000
2005 2015 2025 2035 2045 2055 2065 2075 2085 2095
Car
bon
pri
ce (2
005
US$
per
ton
C)
Optimal Baseline < 2 degrees C
< 2x CO2 Low discounting
Carbon prices 2010 for major scenarios ($/tC)
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0
10
20
30
40
50
60
70
80
90
100190 140 305
What do carbon prices mean in practice?
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Carbon tax, 2010 Increase, price of energy, US
[$/tC] GasolineAll energy
expenditures
Kyoto: global average $2 0.2% 0.3%
"Optimal" 35 3.3% 5.4%
Climate constrained 50 4.8% 7.7%
"Ambitious" 200 19.0% 30.7%
Impact on PCE expenditures of $50 C tax
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.028
.032
.036
.040
.044
.048
.052
.056
.060
1975 1980 1985 1990 1995 2000 2005 2010 2015
Carbon tax + energy expendituresEnergy expenditures
Policy question
The impact of efficient/climate target carbon taxes is relatively modest:– abatement/output circa 0.1 – 0.6 % of output– net impact -0.1 to +0.2 % of output
Why is the debate so strident? Why are some people so opposed?
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