Integrated Assessment Models of Economics of Climate Change

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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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1

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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1400

1600

1800

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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0.30

0.40

0.50

0.60

Carbon prices for major scenarios

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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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100

200

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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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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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