Transcript Slide 1

Integrated Assessment Models
of Economics of Climate Change
Economics 331b
Spring 2009
1
Slightly Simplified Equations of DICE-2007 Model: Revised
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 
= [( 1  D(T )][1 - C(t)]Q (t)
(7) EInd (t) =  (t)[1 - (t)]Q g (t)
g
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.
2
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.
3
How do we solve IA models numerically?
We take discrete version of model, simplified as follows.
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 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.
4
Example: Minimize cost of emissions
to limit the sum of emissions over time
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
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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
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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
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This is the solver dialogue box
Objective function
Control variables
Constraints
8
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
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Can also calculate the “shadow prices,”
here the efficient carbon taxes
1800
Marginal cost of Emissions Reductions ($)
Remember that in a constrained
optimization (Lagrangean), the
multipliers have the
interpretation of
d[Objective Function]/dX.
So, in this problem, interpretation
is MC of emissions reduction.
Optimization programs
(particularly LP) will generate
the shadow prices of carbon
emissions in the optimal path.
For example, in the problem we
just did, we have the following
shadow prices:
1600
1400
1200
1000
800
With a little
work, you can
show that the
rate of growth of
prices = interest
rate for this case.
600
400
200
0
0
10
20
30
Period
10
Applications of IA Models
Major applications of IA Models:
1. Project the impact of current trends and of policies on
important variables.
2. Assess the costs and benefits of alternative policies
3. Determine efficient levels of policy variables (carbon
taxes, emissions control rates, emissions, …)
For these, I will illustrate using the DICE-2007 model:
– Full analysis Question of Balance (see reading list).
– There is a “beta” version using an Excel spreadsheet at
http://www.econ.yale.edu/~nordhaus/homepage/DICE
2007.htm (both an *.xls and *.xlsx version)
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1. No controls ("baseline"). No emissions controls.
2. Optimal policy. Emissions and carbon prices set for economic
optimum.
3. Climatic constraints with CO2 concentration constraints.
Concentrations limited to 550 ppm
4. Climatic constraints with temperature constraints. Temperature
limited to 2½ °C
5. Kyoto Protocol. Kyoto Protocol without the U.S.
6. Strengthened Kyoto Protocol. Roughly, the Obama/EU policy
proposals.
7. Geoengineering. Implements a geoengineering option that offsets
radiative forcing at low cost.
Illustrative Policies for DICE-2007
12
Snapshot of DICE-Excel model
WORLD
0
PARAMETERS AND EXOGENOUS VARIABLES
OUTPUT AND CAPITAL ACCUMULATION
capital share
damage coefficient on temperature
damage coefficient on temperature squared
Exponent on damages
rate of depreciation (percent per year)
Initial capital stock ($ trillion)
Abatement cost function coefficient
pback
backrat
gback
limmiu
Backstop price
Initial cost function coefficient
Initial rate of decline in cost of abatement function (percent per
decade)
Rate of decline in decline rate of cost of abatement function
(percent per year)
Rate of decline in cost of abatement function (percent per decade)
Exponent of control cost function
EMISSIONS
Sigma (industrial CO2 emissions/output -- MTC/$1000)
Initial sigma
Initial growth rate of sigma (percent per decade)
Rate of decrease in the growth rate of sigma (percent per year)
Accleration parameter of growth rate of sigma
Growth rate of sigma (percent per decade)
Carbon emissions from land use change (GTC per year)
Initial carbon emissions from land use change (GTC per year)
CARBON LIMITS
Maximum carbon use
CONCENTRATIONS
Initial atmospheric concentration of CO2 (GTC)
Initial concentration of CO2 in biosphere/shallow oceans (GTC)
Initial concentration of CO2 in deep oceans (GTC)
1
2005
2
2015
3
2025
4
2035
5
2045
6
2055
7
2065
8
2075
9
2085
1
209
0.300
0.000
0.0028388
2.0
0.100
137.000
0.056068
1.17000
2.00000
0.05000
1.00000
1.170000
0.045
0.051082
0.046660
0.042728
0.039225
0.036095
0.033295
0.030782
0.028524
0.02648
1.141469
1.114330
1.088514
1.063957
1.040598
1.018379
0.997243
0.977137
0.95801
0.260
2.800
0.213
0.174
0.143
0.117
0.096
0.078
0.064
0.052
0.04
0.1342
0.1342
-7.300
0.1253
0.1172
0.1099
0.1032
0.0971
0.0915
0.0864
0.0817
0.077
-0.071
0.990
-0.069
0.891
-0.067
0.802
-0.065
0.72171
-0.063
0.650
-0.061
0.585
-0.059
0.526
-0.057
0.474
-0.05
0.42
26.000
2.000
0.300
0.000
-7.300
1.100
1.100
6000.000
808.900
1255.000
18365.000
http://www.econ.yale.edu/~nordhaus/homepage/DICE2007.htm
13
Per capita GDP: history and projections
Per capita GDP (2000$ PPP)
100
10
US
WE
OHI
Russia
EE/FSU
Japan
China
India
World
1
1960
1980
2000
2020
2040
2060
2080
2100
14
CO2-GDP ratios: history
CO2-GDP ratio (tons per constant PPP $)
.7
.6
China
Russia
US
World
Western/Central Europe
.5
.4
.3
.2
.1
.0
80 82 84 86 88 90 92 94 96 98 00 02 04
15
IPCC AR4 Model Results: History and Projections
DICE-2007
model
2-sigma
range
DICE
model
16
DICE-2007 model results
17
Concentrations profiles: DICE 2007
1300
Carbon concentrations (ppm)
1200
1100
1000
Optimal
Baseline
< 2 deg C
Strong Kyoto
900
800
700
600
500
400
300
18
Temperature profiles
6
Temperature change (C)
5
Optimal
Baseline
2x CO2
Strong Kyoto
2o C limit
4
3
2
1
0
19
Policy outcomes variables
Overall evaluation
Two major policy variables are
- emissions control rate
- carbon tax
20
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.
21
Evaluation: PV trillions of 2000 $
Run
Difference
from base:
Objective
function
Present
value
environmental
damages
Net present
Present
value abatevalue abate- ment costs
ment costs
plus
damages
Trillions of 2005 US $
No controls
Optimal
Concentration limits
Limit to 1.5X CO2
Limit to 2X CO2
Limit to 2.5X CO2
Temperature limits
Limit to 1.5 degree C
Limit to 2 degree C
Limit to 2.5 degree C
Limit to 3 degree C
Kyoto Protocol
Kyoto with US
Kyoto w/o US
Strengthened
Low discounting
Low-cost backstop
0.0
3.4
22.5
17.3
0.0
2.2
22.5
19.5
-14.9
2.9
3.4
9.9
16.0
17.3
27.2
3.9
2.2
37.1
19.9
19.5
-14.7
-1.6
2.3
3.2
10.0
13.1
15.3
16.7
27.0
11.3
5.2
2.9
37.0
24.3
20.6
19.5
0.7
0.1
1.0
-17.0
17.2
21.4
22.4
16.0
9.0
4.9
0.5
0.0
5.8
27.7
0.4
21.9
22.5
21.8
36.7
5.4
22
Evaluation: as percent of PV consumption
Run
No controls
Optimal
Concentration limits
Limit to 1.5X CO2
Limit to 2X CO2
Limit to 2.5X CO2
Temperature limits
Limit to 1.5 degree C
Limit to 2 degree C
Limit to 2.5 degree C
Limit to 3 degree C
Kyoto Protocol
Kyoto with US
Kyoto w/o US
Strengthened
Low discounting
Low-cost backstop
Difference
from base:
Objective
function
Present
value
environmental
damages
Net present
Present value abatevalue abate- ment costs
ment costs
plus
damages
As percent of discounted world consumption
1.13
1.13
0.17
0.87
0.11
0.98
(0.75)
0.14
0.17
0.50
0.80
0.87
1.37
0.20
0.11
1.87
1.00
0.98
(0.74)
(0.08)
0.11
0.16
0.50
0.66
0.77
0.84
1.36
0.57
0.26
0.14
1.86
1.22
1.03
0.98
0.04
0.01
0.05
(0.85)
0.86
1.07
1.13
0.80
0.45
0.25
0.03
0.00
0.29
1.39
0.02
1.10
1.13
1.10
1.84
0.27
23
Evaluation: Consumption annuity per capita
Run
No controls
Optimal
Concentration limits
Limit to 1.5X CO2
Limit to 2X CO2
Limit to 2.5X CO2
Temperature limits
Limit to 1.5 degree C
Limit to 2 degree C
Limit to 2.5 degree C
Limit to 3 degree C
Kyoto Protocol
Kyoto with US
Kyoto w/o US
Strengthened
Low discounting
Low-cost backstop
Difference
from base:
Objective
function
Environmental
damages
Abatement
costs
Abatement
costs plus
climate
damages
Consumption annuity equivalent (billions of 2000$ per year)
121
121
18
93
12
105
(80)
54
146
200
15
86
21
107
18
93
12
105
(79)
54
145
199
(9)
70
61
131
12
82
28
111
17
90
15
105
4
115
3
118
1
121
0
121
5
86
31
118
(91)
49
149
198
93
26
2
29
24
Emissions control rate (industrial CO2), 2015
0.60
0.50
0.40
0.30
0.20
0.10
0.00
25
Carbon prices for major scenarios
1000
Carbon price (2005 US$ per ton C)
900
800
700
Optimal
Baseline
< 2 degrees C
Strong Kyoto
600
500
400
300
200
100
0
2005 2015 2025 2035 2045 2055 2065 2075 2085 2095 2105
26
Carbon prices for major scenarios
1000
Carbon price (2005 US$ per ton C)
900
800
700
600
500
400
300
200
100
0
2005 2015 2025 2035 2045 2055 2065 2075 2085 2095
Optimal
Baseline
< 2x CO2
Low discounting
< 2 degrees C
27
Carbon prices 2010 for major scenarios ($/tC)
100
190
140
305
90
80
70
60
50
40
30
20
10
0
28
What do carbon prices mean in practice?
Carbon tax,
2010
Kyoto: global average
"Optimal"
Climate constrained
"Ambitious"
Increase, price of energy, US
[$/tC]
Gasoline
All energy
expenditures
$2
35
50
200
0.2%
3.3%
4.8%
19.0%
0.3%
5.4%
7.7%
30.7%
29
Impact on PCE expenditures of $50 C tax
.060
Carbon tax + energy expenditures
Energy expenditures
.056
.052
.048
.044
.040
.036
.032
.028
1975
1980
1985
1990
1995
2000
2005
2010
2015
30
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?
31